# Formalism of general boundary conditions for continuum models

Maxim Kharitonov1,2 1Institute for Theoretical Physics and Astrophysics,
University of Würzburg, 97074 Würzburg, Germany
2Donostia International Physics Center (DIPC), Manuel de Lardizabal 4, E-20018
San Sebastian, Spain

###### Abstract

Continuum models are particularly appealing for theoretical studies of bound
states, due to simplicity of their bulk Hamiltonians. The main challenge on
this path is a systematic description of the boundary, which comes down to
determining proper boundary conditions (BCs). BCs are a consequence of the
fundamental principle of quantum mechanics: norm conservation of the wave
function, which leads to the conservation of the probability current at the
boundary. The notion of general BCs arises, as a family of all possible BCs
that satisfy the current-conservation principle. Ahari, Ortiz, and Seradjeh
formulated a systematic derivation procedure of the general BCs from the
current-conservation principle for the 1D Hamiltonian of the most general
form. The procedure is based on the diagonalization of the current and leads
to the universal “standardized” form of the general BCs, parameterized in a
nonredundant one-to-one way by unitary matrices. In this work, we
substantiate, elucidate, and expand this formalism of general boundary
conditions for continuum models, addressing in detail a number of important
physical and mathematical points. We provide a detailed derivation of the
general BCs from the current-conservation principle and establish the
conditions for when they are admissible in the sense that they describe a
well-defined boundary, which is directly related to a subtle but crucial
distinction between self-adjoint (hermitian) and only symmetric operators. We
provide a natural physical interpretation of the structure of the general BCs
as a scattering process and an essential mathematical justification that the
formalism is well-defined for Hamiltonians of momentum order higher than
linear. We discuss the physical meaning of the general BCs and outline the
application schemes of the formalism, in particular, for the study of bound
states in topological systems.

##  I Introduction

Bound states at boundaries, interfaces, and defects of bulk crystalline
electron systems have by now been firmly established as a widespread feature
in a multitude of real materials and theoretical models [1, 2, 3, 4, 5, 6] (we
will use the term “bound states” for bound, edge, or surface states in a
generalized sense for any dimension, when system dimension is not explicitly
specified or does not matter). When the bulk has nontrivial topology, certain
bound states are guaranteed, as per the concept of bulk-boundary
correspondence [5]. Moreover, even systems that are topologically trivial in a
rigorous sense have been theoretically shown to exhibit robust bound states
[7, 8, 9].

Continuum models (CMs), whose bulk Hamiltonians are polynomials in momentum,
are particularly appealing for theoretical studies of bound states. Most
commonly, a CM is intended as a simpler version of a more complicated (“more”)
microscopic system that arises in the low-energy limit of the latter and still
captures its essential properties. For a system with discrete translation
symmetry, such as a crystal, the Hamiltonian of the CM typically describes the
low-energy expansion of the underlying microscopic Hamiltonian about certain
special points of interest in the Brillouin zone. The multicomponent wave
function of the CM consists of the envelopes of the quantum states at those
points. For nodal semimetals or superconductors, these special points of
interest can be the nodes; for topological insulators or nodeless
superconductors with a generally gapped spectrum, these can be points of gap
closing at the topological phase transition. CMs are thus particularly well-
suited for the study of the vicinity of a topological phase transition, which
is often also the most interesting regime. For high enough spatial symmetry of
the system, these special points of interest often happen to be the high-
symmetry points in the Brillouin zone.

The main appeal of CMs is the relative analytical simplicity of their bulk
Hamiltonians. For a CM, one may take into account just enough “degrees of
freedom”, i.e., wave-function components and momentum powers in the
Hamiltonian (see Sec. III.1 for more details), that capture the desired
physical properties. Often the lowest-order, linear-in-momentum Hamiltonians
are sufficient to capture the behavior of interest in the bulk; in particular,
such Hamiltonians were studied earlier [30, 5] as representative models for
various topological symmetry classes.

Figure 1:  (a) The considered 1D half-infinite system occupying the region
$x\geq 0$ and (b) the physical interpretation of the general boundary
conditions (BCs) (52) for the Hamiltonian $\hat{H}(\hat{p})$ [Eq. (1)] of the
most general form, polynomial in momentum up to some order $N$, for the
multicomponent wave function $\hat{\psi}(x)$. The wave-function components and
their derivatives $\hat{p}^{n}\psi_{m}(x)$ with $m=1,\ldots,M$ and
$n=0,\ldots,N_{m}$ are arranged into the vectors
$\hat{\bar{\Psi}}_{+}[\hat{\psi}(x)]$ and
$\hat{\bar{\Psi}}_{-}[\hat{\psi}(x)]$ [Eq. (32)], in terms of the
$\mathcal{N}_{\pm}$ components of which the probability current
$j[\hat{\psi}(x)]$ is diagonalized and normalized [Eq. (31)] with the
eigenvalues $\pm 1$, respectively. These components can be interpreted as
chiral right- and left-moving “waves” (Sec. VI), which in this geometry are
reflected from and incident upon the boundary, respectively. The general BCs
(52), well-defined only in the case of equal numbers
$\mathcal{N}_{+}=\mathcal{N}_{-}=\mathcal{N}/2$, is the matrix relation
between the vectors $\hat{\bar{\Psi}}_{+}[\hat{\psi}(0)]$ and
$\hat{\bar{\Psi}}_{-}[\hat{\psi}(0)]$ at the boundary via a unitary matrix
$\hat{U}\in{\text{U}}(\mathcal{N}/2)$. This form has a natural physical
interpretation as a scattering process between the chiral right- and left-
moving waves. The family of general BCs is parameterized in a nonredundant
one-to-one way by all possible unitary matrices $\hat{U}$.

Since CMs can provide a tremendous simplification of the underlying
microscopic system, it would be desirable to exploit this advantage not only
for the analysis of the bulk properties, but also for the study of bound
states. However, despite this appeal, application of CMs to the study of bound
states has generally been rather underdeveloped and non-systematic, the main
reason being the challenge of describing the sample boundary within CMs. In
fact, it is quite common, especially in the studies of topological systems,
that the analysis of the bulk properties is performed within a CM, but the
bound states are studied via some “regularization” procedure, e.g., using
instead a lattice model, whose Hamiltonian reduces to that of the CM in the
low-energy limit. The reason is that for lattice models the termination of the
sample can easily be described as the absence of sites. However, such an
approach has the following drawbacks. (i) The analysis of bound states is
usually done numerically for a large finite-size system (where finite-size
effects impose additional limitations). (ii) Typically, some specific choice
of the lattice termination is made. If one resorts to just one or a few
possibilities, some possible bound-state structures could be overlooked. On
the other hand, considering different lattice terminations can be
computationally costly. (iii) Sometimes, the lattice model introduced in the
regularization procedure is different (in structure and sometimes even in
symmetry) from the true original lattice of the system of interest (because
the former is still simpler than the latter). This way, in the end, the bound
states are actually studied not within a CM, but within a different model.
Such approach to calculating bound states therefore partially defeats the
point of considering the CM in the first place, since the main advantage of
the latter as being a technically much simpler model of the system is lost.

Meanwhile, a description of the boundary fully within CMs is entirely
possible: the problem reduces to deriving proper boundary conditions (BCs). It
has been understood for quite some time [10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 7, 8, 23, 24, 25, 29, 26, 27, 28, 9] that BCs arise as a
consequence of the fundamental principle of quantum mechanics: conservation of
the wave-function norm upon the time evolution described by the Schrödinger
equation. For a system with a boundary, this imposes constraints not only on
the form of the Hamiltonian operator, but also on the Hilbert space of allowed
wave functions. The latter constraint has the form of the nullification of the
probability current at the boundary. This constraint can be resolved in the
form of linear homogeneous relations between the wave-function components and
their derivatives at the boundary, and these relations are commonly referred
to as the boundary conditions (BCs). This understanding naturally leads to the
notion of general BCs, as a family of all possible BCs (for a given
Hamiltonian) that resolve the current-nullification constraint.

Previously, general BCs have been derived from the current-conservation
principle and used to study the bound-, edge-, or surface-state structures for
specific continuum models: for the 1D quadratic-in-momentum one-component
model [14] (a textbook nonrelativistic Schrödinger particle); for the 3D
quadratic-in-momentum multi-component model [15], in the context of
semiconductors; for the linear-in-momentum two-component model, in the context
of 2D semimetals [12, 7, 9], 2D quantum anomalous Hall (QAH) systems [8],
graphene [24], 1D insulators [22, 9], and 3D Weyl semimetals [20, 21, 9]; for
the linear-in-momentum four-component model of graphene [16, 17, 18, 19] and
3D Dirac materials [25]; for the 3D quadratic-in-momentum two-component model
of Weyl semimetals [23]; for the 1D linear-in-momentum four-component model of
superconductors with one Fermi surface [29]. Also, general BCs with some
symmetry constraints have been derived for quadratic- and linear-in-momentum
four-component models of 2D and 3D topological insulators and Dirac materials
[26, 27] and two-node 3D Weyl semimetals [28].

Though the general understanding of the origin of BCs for CMs has existed for
some time, the approaches of the above works (the procedure of finding BCs,
the form and parametrization of BCs) have been tailored to specific, often
simplest Hamiltonians. On the other hand, in Ref. 22, Ahari, Ortiz, and
Seradjeh formulated a systematic derivation procedure of the general BCs from
the current-conservation principle for the 1D continuum model with the
translation-symmetric Hamiltonian of the most general form, with any number of
wave-function components and any order of momentum. The central technical
advancement was to present the probability current in the universal diagonal
and normalized form. This leads to the “standardized” universal form of the
family of the general BCs, which are parameterized in a nonredundant one-to-
one way by unitary matrices.

##  II Structure of the paper and summary of the results

In this work, we substantiate, elucidate, and expand this formalism of general
boundary conditions (BCs) for continuum models (CMs), initiated in Ref. 22,
addressing in detail and clarifying a number of important physical and
mathematical points. The paper is organized as follows.

In Sec. III, we present the Hamiltonian for the 1D translation-symmetric CM of
the most general form and the half-infinite system, for which the general BCs
will be derived.

In Sec. IV, we demonstrate how the current-conservation principle arises from
the fundamental principle of quantum mechanics, the norm conservation of the
wave function.

Sec. V is devoted to deriving the general BCs and providing the necessary
justification. In Sec. V.1, the main properties of the probability current are
presented. In Sec. V.2, we reproduce the key initial step of the derivation
that brings the current to the universal diagonal and normalized form. In Sec.
V.3, we derive the families of general BCs that nullify the current at the
boundary. In Secs. V.4.1 and V.4.2, we demonstrate that BCs nullifying the
current are still not always admissible in the sense that they represent a
well-defined boundary and establish the corresponding conditions. In Sec.
V.4.3, we demonstrate that this distinction between which BCs that nullify the
current are admissible or not coincides with the distinction between the
Hilbert spaces over which the Hamiltonian is self-adjoint (hermitian) or only
symmetric.

In Sec. VI, we provide a natural physical interpretation of the structure of
the general BCs as a scattering process at the boundary between the incident
(left-moving) and reflected (right-moving) modes of the wave function,
illustrated in Fig. 1(b).

In Sec. VII, we study in detail the structure of the current matrix and
provide an essential mathematical justification that the formalism is well-
defined for Hamiltonians of momentum order higher than linear; this inevitably
requires introducing a fictitious length scale.

In Sec. VIII, we present the general, analytical or semi-analytical, method of
calculating the bound states that becomes available within the framework of
CMs with BCs.

In Sec. IX, we explain the physical meaning of the general BCs in terms of
their relation to the underlying microscopic models and outline the related
systematic low-energy-expansion procedure of deriving low-energy continuum
models with BCs from microscopic models.

In Sec. X, we explain that the general BCs lead to general bound-state
structures and discuss other related advantages that the formalism provides
for the study of bound states.

In Sec. XI, we present two important application schemes of the formalism to
families of bulk systems. In the first scheme, the Hamiltonian is of the most
general form satisfying some set of symmetries and the BCs are of the most
general form, satisfying only the current-conservation principle. In the
second scheme (already formulated recently in Ref. [9]), both the Hamiltonian
and BCs are of the most general form satisfying the same set of symmetries.
The second scheme is perfectly suited for the study of bound states in
topological systems.

In Secs. XII and XIII, we illustrate the application of the formalism to the
two models with the minimal number of degrees of freedom necessary to have a
well-defined boundary: one-component quadratic-in-momentum Hamiltonian and
two-component linear-in-momentum Hamiltonian. In both models, a bound state is
present in a half of the ${\text{U}}(1)$ parameter space of the general BC.
For the quadratic model, we present two examples of possible realization of
the general BC by potentials. Using these two examples, in Sec. XIV, we
demonstrate the relation in the structure of the general BCs between the
number of wave-function components and the order of momentum in the
Hamiltonian.

In Sec. XV, we discuss possible generalizations and extensions of the
formalism of general BCs to higher dimensions and hybrid systems with
junctions and interfaces.

Concluding remarks are presented in Sec. XVI.

##  III General 1D continuum model

###  III.1 Hamiltonian

This work is devoted to the formalism of general BCs specifically in one
dimension (1D); generalization to higher dimensions should be possible, as
briefly discussed in Sec. XV.

In 1D, the most general form of the single-particle Hamiltonian of a CM with a
translation-symmetric bulk is a polynomial matrix function

$\hat{H}(\hat{p})=\sum_{n=0}^{N}\hat{h}_{n}\hat{p}^{n}$ (1)

of the momentum operator

$\hat{p}=-{\text{i}}\partial_{x}$ (2)

(we use the units in which the Planck constant $\hbar=1$) for a multicomponent
wave function

$\hat{\psi}(x)=\left(\begin{array}[]{c}\psi_{1}(x)\\\ \psi_{2}(x)\\\ \ldots\\\
\psi_{M-1}(x)\\\ \psi_{M}(x)\end{array}\right).$ (3)

Throughout, $M$ will denote an arbitrary number of wave-function components
$\psi_{m}(x)$, $m=1,\dots,M$; hence, $\hat{h}_{n}$ are $M\times M$ matrices.
The top order of momentum $\hat{p}$ in $\hat{H}(\hat{p})$ for the component
$\psi_{m}(x)$ will be denoted as $N_{m}\geq 1$; in general, $N_{m}$ could
differ for different $m$. The top order of momentum in $\hat{H}(\hat{p})$
among all components is the maximum $N=\max_{m}N_{m}$.

Although the (real) momentum eigenvalues $p$ are formally unrestricted in a
CM, it should be kept in mind that, if such CM represents a crystal with only
a discrete translation symmetry, the CM is valid only for small enough momenta
and respective small enough energies. In particular, for a gapped system, the
gaps between the bands must be much smaller than the full widths of the bands.
Within this validity range, the CM is a rigorous asymptotic limit of the
underlying microscopic model. The momentum operator $\hat{p}$ corresponds to
the deviation of quasimomentum from the expansion points of interest of the
latter.

###  III.2 Half-infinite system, inaccessible region

We consider a half-infinite system occupying the region $x\geq 0$, Fig. 1. The
wave function $\hat{\psi}(x)$ [Eq. (3)] is defined only in this region and is
undefined in $x<0$. Physically, this means the following. As already mentioned
above, typically, a continuum model arises as a low-energy limit of some
(“more”) microscopic model. The wave function of this underlying microscopic
model (which is at least implied, even when not specified) is defined for all
$x$. In the region $x\geq 0$, this model has low-energy excitations, which are
described by the continuum model of interest, with the wave function
$\hat{\psi}(x)$ [Eq. (3)] and the Hamiltonian $\hat{H}(\hat{p})$ [Eq. (1)].
Whereas in the region $x<0$, the microscopic model has a gap [31] in the
spectrum that is much larger than the relevant low-energy scale. As a result,
the microscopic wave function decays rapidly into the region $x<0$. The half-
space $x<0$ is therefore the region “inaccessible” for the low-energy
excitations that the wave function $\hat{\psi}(x)$ describes.

##  IV Conservation of wave-function norm as the origin of boundary conditions

In this section, we demonstrate how BCs arise as a consequence of the
fundamental principle of quantum mechanics: conservation of the wave-function
norm.

As the starting point of our analysis, we assume no constraints on the form of
the Hamiltonian operator $\hat{H}(\hat{p})$ [Eq. (1)], i.e., that
$\hat{h}_{n}$ therein are initially arbitrary complex matrices. Also, no
initial boundary constraints on the wave functions $\hat{\psi}(x)$ are
assumed: all wave functions initially belong to the larger “embedding” Hilbert
space $\Upsilon_{0}$ of normalizable wave functions with appropriate
smoothness properties for a given $\hat{H}(\hat{p})$. All necessary
restrictions on both the Hamiltonian and Hilbert space should and will follow
from the norm-conservation principle.

A fundamental principle of the theory of quantum mechanics is that the norm

$\langle\hat{\psi}(t),\hat{\psi}(t)\rangle=\int_{0}^{+\infty}{\text{d}}x\,\hat{\psi}^{\dagger}(x,t)\hat{\psi}(x,t)$
(4)

of the time-dependent wave function $\hat{\psi}(t)=\hat{\psi}(x,t)$ is
conserved upon time evolution described by the Schrödinger equation

${\text{i}}\partial_{t}\hat{\psi}(x,t)=\hat{H}(\hat{p})\hat{\psi}(x,t).$ (5)

In other words, the norm (4) is a time-independent constant, which is
equivalent to its time derivative being zero:

$\partial_{t}\langle\hat{\psi}(t),\hat{\psi}(t)\rangle=0.$ (6)

Here,

$\langle\hat{\psi}_{a},\hat{\psi}_{b}\rangle=\int_{0}^{+\infty}{\text{d}}x\,\hat{\psi}_{a}^{\dagger}(x)\hat{\psi}_{b}(x)$

is the scalar product in the embedding Hilbert space $\Upsilon_{0}$, given by
the integral over the region $x\geq 0$ of the half-infinite system, described
in Sec. III.1. The norm and the scalar product are well-defined only for
normalizable wave functions, that decay $\hat{\psi}(x)\rightarrow\hat{0}$ at
$x\rightarrow+\infty$ and in such a way that the integral converges (in
reality, only the “wave packets” consisting of normalizable wave functions are
ever realized). Throughout, $\dagger$ denotes hermitian conjugation (complex
conjugation ∗ and transposition) of a matrix of any size, including the column
vector (3) of the wave function; so,
$\hat{\psi}^{\dagger}(x)=(\psi_{1}^{*}(x),\ldots,\psi_{M}^{*}(x))$ is a row
vector and $\hat{p}^{\dagger}=\hat{p}^{*}=-\hat{p}$ [Eq. (2)] below.

Using the Schrödinger equation, one obtains that this is equivalent to the
following condition being satisfied:

${\text{i}}\partial_{t}\langle\hat{\psi}(t),\hat{\psi}(t)\rangle=\langle\hat{\psi}(t),\hat{H}\hat{\psi}(t)\rangle-\langle\hat{H}\hat{\psi}(t),\hat{\psi}(t)\rangle=0.$
(7)

For this to be satisfied for any time-dependent solution $\hat{\psi}(x,t)$,
the condition

$\langle\hat{\psi},\hat{H}\hat{\psi}\rangle=\langle\hat{H}\hat{\psi},\hat{\psi}\rangle$
(8)

has to be satisfied for any wave function $\hat{\psi}(x)\in\Upsilon$ in the
sought Hilbert space $\Upsilon\subset\Upsilon_{0}$. The latter is the
definition of a symmetric operator $\hat{H}(\hat{p})$ over the Hilbert space
$\Upsilon$; we forewarn that this is not the definition of a hermitian
operator, the term we use synonymously to self-adjoint; this important
distinction will be discussed in Sec. V.4.3. We note that, importantly, Eq.
(8) is a constraint on both the operator $\hat{H}(\hat{p})$ and the Hilbert
space $\Upsilon$.

Consider the time evolution of the more general quantity than the norm (4)
(this is necessary for dealing with the hermitian Hamiltonian operator in Sec.
V.4.3): the local-in-coordinate sesquilinear form

$\rho[\hat{\psi}_{a}(x,t),\hat{\psi}_{b}(x,t)]=\hat{\psi}_{a}^{\dagger}(x,t)\hat{\psi}_{b}(x,t)$

of two different wave functions $\hat{\psi}_{a,b}(x,t)$, both satisfying the
Schrödinger equation (5). Using the Schrödinger equation, we obtain
(temporarily dropping the arguments for brevity)

${\text{i}}\partial_{t}(\hat{\psi}_{a}^{\dagger}\hat{\psi}_{b})=\sum_{n=0}^{N}\left[\hat{\psi}_{a}^{\dagger}\hat{h}_{n}\hat{p}^{n}\hat{\psi}_{b}-(\hat{p}^{n}\hat{\psi}_{a})^{\dagger}\hat{h}_{n}^{\dagger}\hat{\psi}_{b}\right]=\sum_{n=0}^{N}\left\\{\hat{\psi}_{a}^{\dagger}\hat{h}_{n}^{\prime}\hat{p}^{n}\hat{\psi}_{b}-(\hat{p}^{n}\hat{\psi}_{a})^{\dagger}\hat{h}_{n}^{\prime}\hat{\psi}_{b}-{\text{i}}[\hat{\psi}_{a}^{\dagger}\hat{h}_{n}^{\prime\prime}\hat{p}^{n}\hat{\psi}_{b}+(\hat{p}^{n}\hat{\psi}_{a})^{\dagger}\hat{h}_{n}^{\prime\prime}\hat{\psi}_{b}]\right\\},$
(9)

where the initially arbitrary complex matrices

$\hat{h}_{n}=\hat{h}_{n}^{\prime}-{\text{i}}\hat{h}_{n}^{\prime\prime}$ (10)

are presented in terms of hermitian matrices:
$\hat{h}_{n}^{\prime\dagger}=\hat{h}_{n}^{\prime}$,
$\hat{h}_{n}^{\prime\prime\dagger}=\hat{h}_{n}^{\prime\prime}$. We recognize
that the first part on the right-hand side of Eq. (9) is presentable as the
full coordinate derivative of the quantity

$j[\hat{\psi}_{a}(x,t),\hat{\psi}_{b}(x,t)]=\sum_{n=1}^{N}j_{n}[\hat{\psi}_{a}(x,t),\hat{\psi}_{b}(x,t)],\mbox{
}j_{n}[\hat{\psi}_{a}(x,t),\hat{\psi}_{b}(x,t)]=\sum_{n^{\prime}=0}^{n-1}(\hat{p}^{n^{\prime}}\hat{\psi}_{a}(x,t))^{\dagger}\hat{h}_{n}^{\prime}(\hat{p}^{n-1-n^{\prime}}\hat{\psi}_{b}(x,t)),$
(11)

so that Eq. (9) can be presented as

$\partial_{t}\rho[\hat{\psi}_{a}(x,t),\hat{\psi}_{b}(x,t)]=-\partial_{x}j[\hat{\psi}_{a}(x,t),\hat{\psi}_{b}(x,t)]+q[\hat{\psi}_{a}(x,t),\hat{\psi}_{b}(x,t)]$
(12)

with

$q[\hat{\psi}_{a}(x,t),\hat{\psi}_{b}(x,t)]=-\sum_{n=0}^{N}\left[\hat{\psi}_{a}^{\dagger}(x,t)\hat{h}_{n}^{\prime\prime}\hat{p}^{n}\hat{\psi}_{b}(x,t)+(\hat{p}^{n}\hat{\psi}_{a}(x,t))^{\dagger}\hat{h}_{n}^{\prime\prime}\hat{\psi}_{b}(x,t)\right].$

In particular, for the same wave function
$\hat{\psi}_{a}(x,t)=\hat{\psi}_{b}(x,t)=\hat{\psi}(x,t)$, this form

$\rho[\hat{\psi}(x,t)]=\hat{\psi}^{\dagger}(x,t)\hat{\psi}(x,t)$

is the probability density and Eq. (12) takes the form of the continuity
equation

$\partial_{t}\rho[\hat{\psi}(x,t)]=-\partial_{x}j[\hat{\psi}(x,t)]+q[\hat{\psi}(x,t)]$
(13)

with the real probability current

$j[\hat{\psi}(x,t)]=\sum_{n=1}^{N}j_{n}[\hat{\psi}(x,t)],\mbox{
}j_{n}[\hat{\psi}(x,t)]=\sum_{n^{\prime}=0}^{n-1}(\hat{p}^{n^{\prime}}\hat{\psi}(x,t))^{\dagger}\hat{h}_{n}^{\prime}(\hat{p}^{n-1-n^{\prime}}\hat{\psi}(x,t))$
(14)

and source

$q[\hat{\psi}(x,t)]=-\sum_{n=0}^{N}[\hat{\psi}^{\dagger}(x,t)\hat{h}_{n}^{\prime\prime}\hat{p}^{n}\hat{\psi}(x,t)+(\hat{p}^{n}\hat{\psi}(x,t))^{\dagger}\hat{h}_{n}^{\prime\prime}\hat{\psi}(x,t)]$
(15)

terms.

Using Eqs. (7) and (13), for the deviation from the equality (8), we have

$\langle\hat{\psi},\hat{H}\hat{\psi}\rangle-\langle\hat{H}\hat{\psi},\hat{\psi}\rangle={\text{i}}\int_{0}^{+\infty}{\text{d}}x\,q[\hat{\psi}(x)]+{\text{i}}j[\hat{\psi}(x=0)],$
(16)

where the current $j[\hat{\psi}(x=+\infty)]=0$ at infinity vanishes for the
normalizable wave function.

There are two contributions on the right-hand side of Eq. (16), which must
vanish individually. The first one is a bulk contribution: an integral of the
source term (15). For the source term $q[\hat{\psi}(x)]\equiv 0$ to vanish
identically for any wave function, the matrices

$\hat{h}_{n}^{\prime\prime}=\hat{0}$

have to vanish, which means that the matrices
$\hat{h}_{n}=\hat{h}_{n}^{\prime}$ at each power $\hat{p}^{n}$ of momentum in
the Hamiltonian must be hermitian [Eq. (10)]:

$\hat{h}_{n}^{\dagger}=\hat{h}_{n},\mbox{ }n=0,\ldots,N.$ (17)

As expected, this is equivalent to the Hamiltonian $\hat{H}(p)$ being a
hermitian matrix when momentum $p$ is a real number:

$\hat{H}^{\dagger}(p)=\hat{H}(p).$ (18)

Nullification of the bulk contribution to the time derivative (7) of the norm
therefore provides a constraint on the form of the Hamiltonian as a local
differential operator. This standard requirement is assumed satisfied in the
rest of the paper.

For a system with a boundary, this is, however, only a necessary, but not a
sufficient condition for the wave-function norm to be conserved. The second
contribution to Eq. (16) is a boundary contribution given by the flow of the
probability current. For the time derivative of the norm of any solution to
the Schrödinger equation to vanish [Eq. (6)], the current through the boundary
also has to vanish for every wave function $\hat{\psi}(x)$ in the sought
Hilbert space(s) $\Upsilon$:

$j[\hat{\psi}(x=0)]=0.$ (19)

We see that while the vanishing of the bulk contribution to the time
derivative (7) [via Eq. (16)] is a restriction on the form of the Hamiltonian
as a local differential operator, the vanishing of the boundary contribution
is ultimately a restriction on the Hilbert space
$\Upsilon\subset\Upsilon_{0}$.

Since the current $j[\hat{\psi}(x=0)]$ involves only the wave-function
components and their derivatives at the boundary [Eq. (14), see also Sec.
V.1], such Hilbert spaces $\Upsilon$ can be specified in the form of linear
homogeneous relations between these quantities that resolve the current
nullification constraint (19). And these relations are commonly referred to as
the boundary conditions (BCs). Therefore, BCs are essentially a way of
specifying Hilbert spaces over which the current at the boundary is nullified.
According to Eqs. (8) and (16), these are the Hilbert spaces over which the
Hamiltonian is symmetric.

The first subsections of Sec. V are devoted to finding all general BCs, i.e.,
all families of all possible BCs that resolve the current-nullification
constraint (19). Next, we observe that the situation turns out to be more
subtle, as not any BCs nullifying the current are “admissible” and deliver an
“admissible” Hilbert space, in the sense that they represent a system with a
well-defined boundary. For some systems, there are no admissible BCs at all,
which means that for such 1D systems a boundary cannot be introduced. Which
BCs and the Hilbert spaces they specify should be deemed admissible has itself
to be argued. We demonstrate that these nuances are directly related to the
subtle but crucial distinction between self-adjoint (hermitian) and only
symmetric Hamiltonian operators. Establishing these conditions and formulating
appropriate arguments is part of the development of the formalism, presented
in this work.

##  V Formalism of general boundary conditions

###  V.1 Probability current

Here, we summarize the main properties of the probability current. Under the
assumed standard constraints (17) and (18) on the form of the Hamiltonian
operator, the quadratic form of the probability current for one wave function
$\hat{\psi}(x)$ for the Hamiltonian $\hat{H}(\hat{p})$ [Eq. (1)] reads

$j[\hat{\psi}(x)]=\sum_{n=1}^{N}j_{n}[\hat{\psi}(x)],$ (20)
$j_{n}[\hat{\psi}(x)]=\sum_{n^{\prime}=0}^{n-1}(\hat{p}^{n^{\prime}}\hat{\psi}(x))^{\dagger}\hat{h}_{n}(\hat{p}^{n-1-n^{\prime}}\hat{\psi}(x)).$
(21)

Here, $j_{n}[\hat{\psi}(x)]$ is the respective contribution from the term
$\hat{h}_{n}\hat{p}^{n}$ in the Hamiltonian; in particular,

$j_{1}[\hat{\psi}]=\hat{\psi}^{\dagger}\hat{h}_{1}\hat{\psi},\mbox{
}j_{2}[\hat{\psi}]=\hat{\psi}^{\dagger}\hat{h}_{2}(\hat{p}\hat{\psi})+(\hat{p}\hat{\psi})^{\dagger}\hat{h}_{2}\hat{\psi},\mbox{
}j_{3}[\hat{\psi}]=\hat{\psi}^{\dagger}\hat{h}_{3}(\hat{p}^{2}\hat{\psi})+(\hat{p}\hat{\psi})^{\dagger}\hat{h}_{3}(\hat{p}\hat{\psi})+(\hat{p}^{2}\hat{\psi})^{\dagger}\hat{h}_{3}\hat{\psi}.$
(22)

The structure (21) of $j_{n}[\hat{\psi}(x)]$ can be readily understood: for a
plane-wave wave function $\hat{\psi}(x)\propto{\text{e}}^{{\text{i}}px}$, when
the momentum operator $\hat{p}\rightarrow p$ becomes a real number, the
current is the derivative
$\partial_{p}\hat{H}(p)=\sum_{n=1}^{N}\hat{h}_{n}np^{n-1}$ of the Hamiltonian
and Eq. (21) is the properly symmetrized operator version of each contribution
$\hat{h}_{n}np^{n-1}$. Note that the zero-order term $\hat{h}_{0}$ in the
Hamiltonian does not contribute to the current. The current (20) is a local-
in-coordinate real quadratic form of wave-function components ($n=0$) and
their derivatives ($n>0$)

$\hat{p}^{n}\psi_{m}(x)\mbox{ with $m=1,\ldots,M$ and $n=0,\ldots,N_{m}-1$},$
(23)

which should be treated as its independent variables. Note that it will be
technically more convenient not to spell out the momentum operator (2) in
terms of the derivative $\partial_{x}$; for brevity, we will still refer to
these quantities as derivatives. The total number of the wave-function
components and their derivatives (23) entering the current (20) is

$\mathcal{N}=\sum_{m=1}^{M}N_{m}.$ (24)

These quantities (23) can be viewed as and will be referred to as the degrees
of freedom of the CM. Their total number $\mathcal{N}$ is equal to the number
of linearly independent particular solutions to the stationary Schrödinger
equation $\hat{H}(\hat{p})\hat{\psi}(x)=\epsilon\hat{\psi}(x)$ at some energy
$\epsilon$.

###  V.2 Diagonalization of the probability current

The first technical goal of the formalism is finding all BCs that resolve the
current-nullification constraint (19). The central technical advancement made
in Ref. [22] was to present the probability current [Eqs. (20) and (21)] in
the diagonal normalized form. We reproduce this step here. The quadratic form
of the probability current is first presented in the matrix form

$j[\hat{\psi}(x)]=\hat{\Psi}^{\dagger}[\hat{\psi}(x)]\hat{J}\hat{\Psi}[\hat{\psi}(x)],$
(25)

where the wave-function components and their derivatives (23) are arranged
into a vector $\hat{\Psi}[\hat{\psi}(x)]$ of size $\mathcal{N}$ [Eq. (24)] and
$\hat{J}$ is a hermitian $\mathcal{N}\times\mathcal{N}$ matrix,
$\hat{J}^{\dagger}=\hat{J}$. These quantities will be specified explicitly and
explored in detail in Sec. VII. Importantly, as also discussed in Sec. VII,
for a well-defined bulk spectrum, $\hat{J}$ has to be nondegenerate. The
current matrix $\hat{J}$ can therefore be diagonalized in the form

$\hat{\mathcal{J}}=\hat{S}^{\dagger}\hat{J}\hat{S}=\text{diag}(\mathcal{J}_{+,1},\ldots,\mathcal{J}_{+,\mathcal{N}_{+}},\mathcal{J}_{-,1},\ldots,\mathcal{J}_{-,\mathcal{N}_{-}})$

with $\mathcal{N}_{+}$ positive $\mathcal{J}_{+,\nu_{+}}>0$
($\nu_{+}=1,\ldots,\mathcal{N}_{+}$) and $\mathcal{N}_{-}$ negative
$\mathcal{J}_{-,\nu_{-}}<0$ ($\nu_{-}=1,\ldots,\mathcal{N}_{-}$) eigenvalues,
such that

$\mathcal{N}_{+}+\mathcal{N}_{-}=\mathcal{N}.$ (26)

Here,

$\hat{S}=\left(\hat{s}_{+,1},\ldots,\hat{s}_{+,\mathcal{N}_{+}},\hat{s}_{-,1},\ldots,\hat{s}_{-,\mathcal{N}_{-}}\right)$
(27)

is a unitary matrix whose columns $\hat{s}_{\pm,\nu_{\pm}}$ are the normalized
eigenvectors of $\hat{J}$ with positive and negative eigenvalues
$\mathcal{J}_{\pm,\nu_{\pm}}$, respectively. The current is then presented in
the diagonal form

$j[\hat{\psi}(x)]=\sum_{\nu_{+}=1}^{\mathcal{N}_{+}}\mathcal{J}_{+,\nu_{+}}\Psi_{+,\nu_{+}}^{*}[\hat{\psi}(x)]\Psi_{+,\nu_{+}}[\hat{\psi}(x)]+\sum_{\nu_{-}=1}^{\mathcal{N}_{-}}\mathcal{J}_{-,\nu_{-}}\Psi_{-,\nu_{-}}^{*}[\hat{\psi}(x)]\Psi_{-,\nu_{-}}[\hat{\psi}(x)],$
(28)

where

$\Psi_{\pm,\nu_{\pm}}[\hat{\psi}(x)]=\hat{s}_{\pm,\nu_{\pm}}^{\dagger}\hat{\Psi}[\hat{\psi}(x)]$
(29)

are the projections of the vector $\hat{\Psi}[\hat{\psi}(x)]$ onto the
eigenvectors $\hat{s}_{\pm,\nu_{\pm}}$.

One can further perform the nonunitary “stretch” transformation (note that it
also changes the physical dimension)

$\bar{\Psi}_{\pm,\nu_{\pm}}[\hat{\psi}(x)]=\sqrt{|\mathcal{J}_{\pm,\nu_{\pm}}|}\Psi_{\pm,\nu_{\pm}}[\hat{\psi}(x)],$
(30)

to bring the current to the quadratic “normalized” form

$j[\hat{\psi}(x)]=\hat{\bar{\Psi}}_{+}^{\dagger}[\hat{\psi}(x)]\hat{\bar{\Psi}}_{+}[\hat{\psi}(x)]-\hat{\bar{\Psi}}_{-}^{\dagger}[\hat{\psi}(x)]\hat{\bar{\Psi}}_{-}[\hat{\psi}(x)]$
(31)

with $\pm 1$ eigenvalues in terms of the rescaled projections
$\bar{\Psi}_{\pm,\nu_{\pm}}[\hat{\psi}(x)]$, which we join into the vectors

$\hat{\bar{\Psi}}_{\pm}[\hat{\psi}(x)]=\left(\begin{array}[]{c}\hat{\bar{\Psi}}_{\pm,1}[\hat{\psi}(x)]\\\
\ldots\\\
\hat{\bar{\Psi}}_{\pm,\mathcal{N}_{\pm}}[\hat{\psi}(x)]\end{array}\right)$
(32)

of sizes $\mathcal{N}_{\pm}$, respectively.

Note that according to the signs of their contributions
$\pm\bar{\Psi}_{\pm,\nu_{\pm}}^{*}[\hat{\psi}(x)]\bar{\Psi}_{\pm,\nu_{\pm}}[\hat{\psi}(x)]\gtrless
0$ to the current (31), $\bar{\Psi}_{\pm,\nu_{\pm}}[\hat{\psi}(x)]$ can
already now be interpreted as chiral “waves” (or “modes”) propagating in the
positive and negative $x$ direction, respectively, i.e., as right- and left-
moving waves (or, for brevity, simply, right- and left-movers), in the
geometry of Fig. 1. We adopt this terminology from now on and substantiate
this interpretation more in Sec. VI.

###  V.3 Derivation of the general boundary conditions

As the next step, we now find the families of all possible BCs that resolve
the current nullification constraint (19). Presenting the current in the
universal diagonal normalized form (31), as performed in Ref. 22, completely
“standardizes” and unifies the problem: instead of the initial degrees of
freedom, the wave-function components and their derivatives
$\hat{p}^{n}\psi_{m}(0)$ [Eq. (23)], their linear combinations
$\bar{\Psi}_{\pm,\nu_{\pm}}[\hat{\psi}(0)]$ [Eq. (30)] now become the
independent variables, in terms of which the current is presented in the
universal form (31), which is fully specified for any Hamiltonian just by
their numbers $\mathcal{N}_{\pm}$. Consequently, finding the BCs in terms of
$\bar{\Psi}_{\pm,\nu_{\pm}}[\hat{\psi}(0)]$ variables then solves the problem
for any Hamiltonian. The BCs can then be simply expressed in terms of the
original degrees of freedom (23) via diagonalization formulas (29) and (30).

We first look for all largest subspaces (of the highest dimension) of the
vector space of

$\left(\begin{array}[]{c}\hat{\bar{\Psi}}_{+}[\hat{\psi}(0)]\\\
\hat{\bar{\Psi}}_{-}[\hat{\psi}(0)]\end{array}\right),$

over which the current form (31) is nullified identically. Each such subspace
can be specified by $\Lambda$ linearly independent relations for the
components of $\hat{\bar{\Psi}}_{\pm}[\hat{\psi}(0)]$ (the to-be-derived
correct form of which will become the BCs). The “right” number $\Lambda$ of
such relations (and what that means in the first place), which determines the
dimension $\mathcal{N}_{+}+\mathcal{N}_{-}-\Lambda$ of these subspaces, is
itself to be determined and is a nontrivial question; arguments will follow.
The system of $\Lambda$ relations of the most general form can be presented in
the matrix form

$\hat{C}_{+}\hat{\bar{\Psi}}_{+}[\hat{\psi}(0)]=\hat{C}_{-}\hat{\bar{\Psi}}_{-}[\hat{\psi}(0)],$
(33)

where $\hat{C}_{\pm}$ are arbitrary matrices of dimensions
$\Lambda\times\mathcal{N}_{\pm}$, respectively.

We first notice that the matrices $\hat{C}_{\pm}$ cannot have the ranks lower
than $\mathcal{N}_{\pm}$, respectively. Indeed, suppose $\hat{C}_{+}$ has a
rank lower than $\mathcal{N}_{+}$. Then, taking
$\hat{\bar{\Psi}}_{-}[\hat{\psi}(0)]=\hat{0}$ as a null vector (of size
$\mathcal{N}_{-}$), the system (33) reduces to
$\hat{C}_{+}\hat{\bar{\Psi}}_{+}[\hat{\psi}(0)]=\hat{0}$ (where $\hat{0}$ is a
null vector of size $\Lambda$), which has a nonzero solution, for which the
current
$j[\hat{\psi}(0)]=\hat{\bar{\Psi}}_{+}[\hat{\psi}(0)]\hat{\bar{\Psi}}_{+}[\hat{\psi}(0)]$
[Eq. (31)] is nonzero. The proof for the minimal rank $\mathcal{N}_{-}$ of
$\hat{C}_{-}$ is the same. Hence, the number $\Lambda$ of linearly independent
relations must be equal or greater than the dimension of the larger vector:

$\Lambda\geq\max(\mathcal{N}_{+},\mathcal{N}_{-}),$

while $\Lambda<\max(\mathcal{N}_{+},\mathcal{N}_{-})$ is the situation of not
enough BCs, when the current cannot be nullified identically.

Without loss of generality, we can and will assume
$\mathcal{N}_{+}\geq\mathcal{N}_{-}$ from now on, since
$\hat{\bar{\Psi}}_{\pm}[\hat{\psi}(0)]$ enter the current form (31) in a
completely equivalent way, as far as its nullification is concerned. Consider
the case $\Lambda=\mathcal{N}_{+}$, when the number of BCs is equal to the
larger of $\mathcal{N}_{\pm}$. In this case, $\hat{C}_{+}$ has to be a square
nondegenerate matrix. Then, the system of $\mathcal{N}_{+}$ relations of the
most general form can be presented as

$\hat{\bar{\Psi}}_{+}[\hat{\psi}(0)]=\hat{V}\hat{\bar{\Psi}}_{-}[\hat{\psi}(0)],$
(34)

with an arbitrary $\mathcal{N}_{+}\times\mathcal{N}_{-}$ matrix $\hat{V}$
(related to the previous matrices as $\hat{V}=\hat{C}_{+}^{-1}\hat{C}_{-}$,
which are, however, no longer needed). This convention also removes the
unwanted redundancy of the BCs parametrization due to equivalence
transformations present in Eq. (33), since these do not change the subspace
that the relations define.

Substituting the form (34) into the expression (31) for the current, we obtain
the form

$j[\hat{\psi}(0)]=\hat{\bar{\Psi}}_{-}^{\dagger}[\hat{\psi}(0)](\hat{V}^{\dagger}\hat{V}-\hat{1})\hat{\bar{\Psi}}_{-}[\hat{\psi}(0)]$

with only $\hat{\Psi}_{-}[\hat{\psi}(0)]$ left, which have now become the only
remaining independent variables. Here, $\hat{1}$ is an
$\mathcal{N}_{-}\times\mathcal{N}_{-}$ unit matrix. This form has to vanish
[Eq. (19)] for every $\hat{\bar{\Psi}}_{-}[\hat{\psi}(0)]$, i.e., it must be a
null form, for which the matrix $\hat{V}$ has to satisfy the condition

$\hat{V}^{\dagger}\hat{V}=\hat{1}.$ (35)

If the numbers $\mathcal{N}_{+}=\mathcal{N}_{-}=\mathcal{N}/2$ of right- and
left-moving waves are equal, in which case the total number $\mathcal{N}$ has
to necessarily be even, then the square matrix

$\hat{V}=\hat{U}\in{\text{U}}(\mathcal{N}/2)$ (36)

is unitary. The number of BCs (34) then also equals $\Lambda=\mathcal{N}/2$.
This case of equal numbers $\mathcal{N}_{\pm}$ of right- and left-movers turns
out to be the only case of $\mathcal{N}_{\pm}$ when a boundary can be
introduced. We continue with this case in Sec. V.5, while in the remainder of
this section we explore the remaining cases and argue that in the case of
unequal numbers $\mathcal{N}_{\pm}$ the boundary cannot be introduced at al.

For unequal numbers of right- and left-moving waves,
$\mathcal{N}_{+}>\mathcal{N}_{-}$, one subset of the solutions to Eq. (35) is
the matrix of the form

$\hat{V}=\left(\begin{array}[]{c}\hat{U}\\\ \hat{0}\end{array}\right),$ (37)

where $\hat{U}\in{\text{U}}(\mathcal{N}_{-})$ is an arbitrary unitary
$\mathcal{N}_{-}\times\mathcal{N}_{-}$ matrix and $\hat{0}$ is an
$(\mathcal{N}_{+}-\mathcal{N}_{-})\times\mathcal{N}_{-}$ null matrix. In the
BCs (34), this corresponds to the vector

$\hat{\bar{\Psi}}_{+}[\hat{\psi}(0)]=\left(\begin{array}[]{c}\hat{\bar{\Psi}}_{+}^{\prime}[\hat{\psi}(0)]\\\
\hat{\bar{\Psi}}_{+}^{\prime\prime}[\hat{\psi}(0)]\end{array}\right)$ (38)

being split into two parts, with the first $\mathcal{N}_{-}$ components
related as

$\hat{\bar{\Psi}}^{\prime}_{+}[\hat{\psi}(0)]=\hat{U}\hat{\Psi}_{-}[\hat{\psi}(0)],$
(39)

and the last $\mathcal{N}_{+}-\mathcal{N}_{-}$ components nullified:

$\hat{\bar{\Psi}}^{\prime\prime}_{+}[\hat{\psi}(0)]=\hat{0},$ (40)

where $\hat{0}$ is the null vector of size $\mathcal{N}_{+}-\mathcal{N}_{-}$.

This splitting of the space of $\hat{\bar{\Psi}}_{+}[\hat{\psi}(0)]$ into
subspaces of $\mathcal{N}_{-}$ and $\mathcal{N}_{+}-\mathcal{N}_{-}$
dimensions can be arbitrary and, by its very definition, can be parameterized
by a Grassmannian
$\text{Gr}(\mathcal{N}_{-},\mathcal{N}_{+}+\mathcal{N}_{-})$. All possible
solutions to Eq. (35) are covered by the following family

$\hat{V}=\hat{T}^{\dagger}\left(\begin{array}[]{c}\hat{U}\\\
\hat{0}\end{array}\right)\hat{T}$ (41)

with an arbitrary unitary matrix $\hat{T}$ in which, however, in accord with
the notion of Grassmannian, rotations within the chosen subspaces are
redundant, since they all are already spanned by all possible $\hat{U}$ in the
subspace of $\hat{\bar{\Psi}}_{+}^{\prime}[\hat{\psi}(0)]$ and the vector
$\hat{\bar{\Psi}}_{+}^{\prime\prime}[\hat{\psi}(0)]$ of the other subspace is
nullified.

###  V.4 Admissible boundary conditions and Hilbert spaces

For any numbers $\mathcal{N}_{\pm}$ of right- and left-movers, we have
determined above the families of all BCs with the minimal sufficient number
$\Lambda=\max(\mathcal{N}_{+},\mathcal{N}_{-})$ that specify the Hilbert
spaces over which the current at the boundary is nullified identically. It
could therefore seem that for any $\mathcal{N}_{\pm}$ these BCs describe a
quantum-mechanical system with a well-defined boundary. This is, however, not
true.

As we demonstrate in Sec. V.4.3, the distinction between the cases of
$\mathcal{N}_{\pm}$ for which the boundary is well-defined or not corresponds
precisely to the distinction between self-adjoint (hermitian) and only
symmetric (but not self-adjoint) Hamiltonian operators. In term of the latter,
this point has previously been understood. In Ref. 11, it is formulated as “it
is only self-adjoint operators that may be exponentiated”, i.e., the
exponential operator ${\text{e}}^{-{\text{i}}\hat{H}t}$, describing the time
evolution of the Schrödinger equation (5), is well-defined. The proof of this
statement is rather mathematical. Although this argument is definitely
sufficient (upon establishing the correspondence in Sec. V.4.3), here we also
present alternative (but apparently related) arguments and explanations that
are perhaps less mathematically rigorous but more practical and physical.

These arguments are based on the properties of the spectrum: if the
Hamiltonian operator is only symmetric but not self-adjoint (hermitian), its
set of eigenvectors is incomplete. A system of functions is called complete
within a Hilbert space if any function from it can be expanded as a linear
combination of these functions (with appropriate mathematical requirements of
convergence of the series), i.e., they form a proper basis that spans the
whole Hilbert space.

####  V.4.1 Consideration of two independent boundaries

First, we consider a system occupying a finite-size segment $x\in[0,X]$ with
two boundaries. Repeating the same procedure (7) for such a system, the
condition for the wave-function norm to be conserved is that the total net
current through the two boundaries vanishes:

$j[\hat{\psi}(0)]-j[\hat{\psi}(X)]=0.$

There are two subcases here: (i) when the currents through the two boundaries
vanish individually,

$j[\hat{\psi}(0)]=0,\mbox{ }j[\hat{\psi}(X)]=0,$ (42)

and (ii) when the individual currents are nonzero and therefore have to be
equal,

$j[\hat{\psi}(0)]=j[\hat{\psi}(X)]\neq 0.$ (43)

The case (ii) is relevant to physical systems in which the coordinates $x=0$
and $x=X$ actually correspond to the same or very close points in the physical
real space, like a closed loop with a junction. Another often-used special
case of (ii) are artificial periodic BCs, where the wave function (for a
quadratic-in-momentum Hamiltonian) at $x=0$ and $x=X$ is taken equal (possibly
up to a phase factor); these facilitate calculations of extensive quantities
(proportional to the volume of the system) in statistical physics, for which
boundary effects are negligible.

When the points $x=0$ and $x=X$ are truly well-separated in the physical real
space (e.g., when the coordinate $x$ describes the actual quasi-1D geometry),
there is no physical reason for the case (ii) and the currents should vanish
individually [Eq. (42)]. We refer to this case (i) as the case of independent
boundaries.

For such a natural finite-size system with independent boundaries, described
by the found BCs [Eqs. (37) and (41)] at each boundary with generally
different matrices $\hat{V}_{0}$ and $\hat{V}_{X}$, the problem with these BCs
(even though they do nullify the individual currents) in the case of unequal
numbers $\mathcal{N}_{+}>\mathcal{N}_{-}$ of right- and left-movers becomes
apparent by simple counting of the degrees of freedom. Consider a stationary
Schrödinger equation

$\hat{H}(\hat{p})\hat{\psi}(x)=\epsilon\hat{\psi}(x)$ (44)

for such a system. The general solution to it is a linear combination of
$\mathcal{N}=\mathcal{N}_{+}+\mathcal{N}_{-}$ linearly independent particular
solutions.

For $\mathcal{N}_{+}=\mathcal{N}_{-}=\mathcal{N}/2$, the
$\Lambda=\mathcal{N}/2$ BCs at each of the two boundaries $x=0$ and $x=X$ will
produce a system of $\mathcal{N}$ linear homogeneous relation for
$\mathcal{N}$ coefficients of the general solution. For an arbitrary energy
$\epsilon$, this system is nondegenerate and there no nontrivial solutions. At
some energies $\epsilon$, the determinant of the system turns to zero, and
there are nontrivial solutions, which are the eigenstates of the discrete
spectrum of the finite-size system. This is a standard situation, in which one
can expect this eigenvector set to be complete, i.e. to span the whole Hilbert
space specified by the BCs.

On the other hand, for $\mathcal{N}_{+}>\mathcal{N}_{-}$, the
$\mathcal{N}_{+}$ BCs (37) at each boundary will produce $2\mathcal{N}_{+}$
linear homogeneous relations for $\mathcal{N}=\mathcal{N}_{+}+\mathcal{N}_{-}$
coefficients of the general solution. The excess of
$2\mathcal{N}_{+}-(\mathcal{N}_{+}+\mathcal{N}_{-})=\mathcal{N}_{+}-\mathcal{N}_{-}>0$
of relations will lead to the absence of the eigenstate solutions: of course,
there cannot be more that $\mathcal{N}$ linearly independent relations between
$\mathcal{N}$ coefficients, but this excess will ensure that there will be no
accidental degeneracies of the system as a function of $\epsilon$ since the
number of linearly independent relations will never drop below $\mathcal{N}$.
As a result, for the system with two independent boundaries and
$\mathcal{N}\neq\mathcal{N}_{-}$, there is no spectrum at all, the eigenvector
set is empty.

We arrive at the general conclusion that, for a finite-size system with two
independent boundaries, admissible BCs nullifying the probability currents at
each boundary exist only in the case of equal numbers
$\mathcal{N}_{+}=\mathcal{N}_{-}=\mathcal{N}/2$ of right- and left-moving
waves (in which case $\mathcal{N}$ is necessarily even).

Moreover, we realize that there must be “just the right” number of BCs. For
equal numbers $\mathcal{N}_{+}=\mathcal{N}_{-}$, this number is the minimal
sufficient number $\Lambda=\mathcal{N}_{+}=\mathcal{N}_{-}$ to nullify the
current: less BCs do not nullify the current; more BCs
$\Lambda>\mathcal{N}_{+}=\mathcal{N}_{-}$ will keep the current nullified, but
will also nullify the wave-function solution identically for a system with two
independent boundaries, via the same argument as above.

For unequal numbers $\mathcal{N}_{+}\neq\mathcal{N}_{-}$, there is no “right”
number of BCs: the minimal sufficient number
$\Lambda=\max(\mathcal{N}_{+},\mathcal{N}_{-})$ of BCs that nullify the
current is already too many, as it leads to the vanishing of the whole wave-
function solution for a system with two independent boundaries. Considering
even more (linearly independent) BCs
$\Lambda>\max(\mathcal{N}_{+},\mathcal{N}_{-})$ (restricting the Hilbert space
further) will only keep the wave-function solution nullified. This
impossibility to introduce a boundary for unequal $\mathcal{N}_{\pm}$ is also
consistent with the physical interpretation of the BCs we provide in Sec. VI.

####  V.4.2 Half-infinite system with a linear-in-momentum decoupled
Hamiltonian

Next, for a half-infinite system $x\geq 0$, we explicitly prove that the
eigenvector set is incomplete for $\mathcal{N}_{+}>\mathcal{N}_{-}$ for the
following linear-in-momentum Hamiltonian

$\hat{H}(\hat{p})=\hat{h}_{1}\hat{p}=\left(\begin{array}[]{cc}\hat{h}_{1+}&\hat{0}\\\
\hat{0}&\hat{h}_{1-}\end{array}\right)\hat{p}.$ (45)

The wave function

$\hat{\psi}(x)=\left(\begin{array}[]{c}\hat{\psi}_{+}(x)\\\
\hat{\psi}_{-}(x)\end{array}\right)=\left(\begin{array}[]{c}\hat{\psi}_{+}^{\prime}(x)\\\
\hat{\psi}_{+}^{\prime\prime}(x)\\\ \hat{\psi}_{-}(x)\end{array}\right)$

consists of the parts $\hat{\psi}_{\pm}(x)$ of sizes $\mathcal{N}_{\pm}$,
respectively, and $\hat{\psi}_{+}(x)$ is further split into the subparts
$\hat{\psi}_{+}^{\prime}(x)$ and $\hat{\psi}_{+}^{\prime\prime}(x)$ of sizes
$\mathcal{N}_{-}$ and $\mathcal{N}_{+}-\mathcal{N}_{-}$, respectively. The
energy matrix $\hat{h}_{0}$ is assumed absent. The velocity matrices
$\hat{h}_{1+}$ and $\hat{h}_{1-}$ are assumed diagonal (such basis always
exists, see Sec. VI) with positive $h_{1+,\nu_{+}}>0$ and negative
$h_{1-,\nu_{-}}<0$ eigenvalues, respectively. In this case, the chiral right-
and left-moving modes
$\bar{\Psi}_{\pm,\nu_{\pm}}[\hat{\psi}(x)]\equiv\hat{\bar{\psi}}_{\pm,\nu_{\pm}}(x)=\sqrt{|h_{1\pm,\nu_{\pm}}|}\psi_{\pm,\nu_{\pm}}(x)$
are the wave-function components themselves and the BCs (39) and (40) read

$\hat{\bar{\psi}}_{+}^{\prime}(0)=\hat{U}\hat{\bar{\psi}}_{-}(0),$ (46)
$\hat{\bar{\psi}}_{+}^{\prime\prime}(0)=\hat{0}.$ (47)

Looking at the eigenvalue problem (44), since all basis states are decoupled
in the Hamiltonian (45), the “remainder” part, satisfying the BCs (47), has to
be nullified identically in the whole system for any eigenstate:
$\hat{\psi}_{+}^{\prime\prime}(x)\equiv\hat{0}$, $x\geq 0$. Therefore, the
same has to hold for the general wave function belonging to the space spanned
by the eigenvectors,

$\hat{\psi}(x)=\left(\begin{array}[]{c}\hat{\psi}_{+}^{\prime}(x)\\\
\hat{0}\\\ \hat{\psi}_{-}(x)\end{array}\right),\mbox{ }x\geq 0,$

which means that the set of eigenvectors satisfying the BCs that do nullify
the current is incomplete.

This result is intuitively clear: uncompensated chiral modes can be nullified
at the boundary only if they are nullified in the whole system. This removes
this part of the wave function from all eigenstates, which makes their set
incomplete. We expect a similar effect for the Hamiltonian of the most general
form. This is the part that we leave without an explicit proof here, since all
other presented arguments suggest that this is indeed the case.

We arrive at the same conclusion for the half-infinite system as for the
system with two independent boundaries: the eigenvector set within the Hilbert
space specified by the BCs that nullify the current at the boundary is
complete only for equal numbers $\mathcal{N}_{+}=\mathcal{N}_{-}$ of right-
and left-moving mode and incomplete for unequal numbers
$\mathcal{N}_{+}\neq\mathcal{N}_{-}$. Clearly, if the eigenvector set of a
Hamiltonian is not complete in a given Hilbert space, such model cannot
describe a well-defined quantum-mechanical system [Note that the problem here
is with the boundary (the Hilbert space), not the bulk (Hamiltonian): an
infinite system without boundaries described by the Hamiltonian with any
$\mathcal{N}_{\pm}$ is well-defined.] In particular, in a well-defined
quantum-mechanical system, it is implied that any wave function
$\hat{\psi}_{0}(x)$ from the Hilbert space in question can be used as an
initial wave function in the time-evolution (Cauchy) problem:
$\hat{\psi}(x,t_{0})=\hat{\psi}_{0}(x)$. Since the eigenvector set is
incomplete, neither the initial wave function $\hat{\psi}_{0}(x)$, nor the
time-evolved wave function $\hat{\psi}(x,t)$ can be expanded in terms of the
eigenvector set, which spans only a smaller subspace of the Hilbert space in
question. Such a model is clearly ill-defined.

####  V.4.3 Symmetric versus hermitian Hamiltonian operator

It turns out that the above-demonstrated distinction between which BCs that do
nullify the current are admissible and which are not based on the completeness
of the eigenvector set coincides with the distinction between a self-adjoint
(hermitian) and a only symmetric (but not hermitian) Hamiltonian operator.

The constraint (7) defines a symmetric Hamiltonian operator $\hat{H}(\hat{p})$
over some Hilbert space $\Upsilon$: i.e., if the Hamiltonian
$\hat{H}(\hat{p})$ and the Hilbert space $\Upsilon$ are such that Eq. (7) is
satisfied identically for any $\hat{\psi}(x)\in\Upsilon$, then
$\hat{H}(\hat{p})$ is said to be symmetric over that Hilbert space $\Upsilon$.
We stress again that this is a constraint on both the form of the Hamiltonian
$\hat{H}(\hat{p})$ and the Hilbert space $\Upsilon$. Therefore, by definition,
the procedure of finding the BCs that nullify the current carried out above is
simultaneously the procedure of finding the Hilbert spaces $\Upsilon$ over
which the Hamiltonian $\hat{H}(\hat{p})$ is symmetric.

A self-adjoint operator is a special case of a symmetric operator satisfying
additional, more subtle requirements. In physical literature, the distinction
between a self-adjoint and symmetric Hamiltonian operator is often overlooked
and the operator is called hermitian regardless, see the last paragraph of
this section. In some mathematical literature [11], a the term hermitian is
made synonymous to symmetric, but not to self-adjoint. Here, we will adopt a
terminology where hermitian will be synonymous to self-adjoint. This seems
more meaningful, since self-adjoint operators are the ones that describe a
well-defined quantum-mechanical system, while those that are only symmetric
(but not self-adjoint) do not.

We provide a more practical definition of a hermitian operator, which may be
less rigorous than in mathematical literature, but which can immediately be
used to check if the operator is hermitian. To define a hermitian operator,
two different wave functions $\hat{\psi}_{a,b}(x)$ have to necessarily be
considered. Suppose $\Upsilon_{b}$ is some Hilbert space for
$\hat{\psi}_{b}(x)$ (specified by some set of BCs), over which the current
$j[\hat{\psi}_{b}(0)]$ is nullified, and hence the Hamiltonian is symmetric
over $\Upsilon_{b}$. One looks for the largest Hilbert space $\Upsilon_{a}$
for $\hat{\psi}_{a}(x)$, (i.e., specified by the minimal number of BCs) over
which the condition

$\langle\hat{\psi}_{a},\hat{H}\hat{\psi}_{b}\rangle=\langle\hat{H}\hat{\psi}_{a},\hat{\psi}_{b}\rangle$
(48)

is satisfied identically, i.e., for all $\hat{\psi}_{a}(x)\in\Upsilon_{a}$ and
$\hat{\psi}_{b}(x)\in\Upsilon_{b}$. Since the Hamiltonian is symmetric over
$\Upsilon_{b}$, the space $\Upsilon_{a}$ at least includes $\Upsilon_{b}$. If
these spaces actually happen to be same, $\Upsilon_{a}=\Upsilon_{b}$, then the
Hamiltonian is said to be hermitian over this one space. Due to Eqs. (9)-(12),
as for a single wave function [Eq. (16)], the relation

$\langle\hat{\psi}_{a},\hat{H}\hat{\psi}_{b}\rangle-\langle\hat{H}\hat{\psi}_{a},\hat{\psi}_{b}\rangle={\text{i}}j[\hat{\psi}_{a}(0),\hat{\psi}_{b}(0)]$
(49)

holds for the Hamiltonian satisfying the standard bulk requirement (18);
therefore, the condition (48) being satisfied is equivalent to the current
$j[\hat{\psi}_{a}(0),\hat{\psi}_{b}(0)]=0$ for two wave functions being
nullified at the boundary.

Intuitively, hermiticity means a “balance” between the Hilbert spaces
$\Upsilon_{a}$ and $\Upsilon_{b}$. The smaller the chosen space $\Upsilon_{b}$
is, the larger $\Upsilon_{a}$ will be, and vice versa. By taking larger spaces
$\Upsilon_{b}$, one can work towards the hermitian Hamiltonian. However, if
the chosen space $\Upsilon_{b}$ is already the largest possible over which the
current $j[\psi_{b}(0)]$ is nullified (i.e., over which the Hamiltonian is
symmetric) and the other space $\Upsilon_{a}$ still happens to be larger, then
there is no Hilbert space at all over which the Hamiltonian could be
hermitian; the largest possible space $\Upsilon_{b}$ is still too constrained.

We now use this definition to check the Hilbert spaces specified by the
derived BCs (34) with Eqs. (39) and (40) for $\mathcal{N}_{+}>\mathcal{N}_{-}$
and Eq. (36) for $\mathcal{N}_{+}=\mathcal{N}_{-}$ for the hermiticity of the
Hamiltonian; in the definition above, $\Upsilon_{b}$ are now such Hilbert
spaces. The formulas of Sec. V.2 presenting the current in the universal
normalized diagonal form also hold for the sequilinear form of the current for
two different wave functions:

$j[\hat{\psi}_{a}(0),\hat{\psi}_{b}(0)]=\hat{\bar{\Psi}}_{+}^{\dagger}[\hat{\psi}_{a}(0)]\hat{\bar{\Psi}}_{+}[\hat{\psi}_{b}(0)]-\hat{\bar{\Psi}}_{-}^{\dagger}[\hat{\psi}_{a}(0)]\hat{\bar{\Psi}}_{-}[\hat{\psi}_{b}(0)].$
(50)

For unequal numbers $\mathcal{N}_{+}>\mathcal{N}_{-}$ of right- and left-
movers, using the splitting (38), we have

$j[\hat{\psi}_{a}(0),\hat{\psi}_{b}(0)]=\hat{\bar{\Psi}}_{+}^{\prime\dagger}[\hat{\psi}_{a}(0)]\hat{\bar{\Psi}}_{+}^{\prime}[\hat{\psi}_{b}(0)]+\hat{\bar{\Psi}}_{+}^{\prime\prime\dagger}[\hat{\psi}_{a}(0)]\hat{\bar{\Psi}}_{+}^{\prime\prime}[\hat{\psi}_{b}(0)]-\hat{\bar{\Psi}}_{-}^{\dagger}[\hat{\psi}_{a}(0)]\hat{\bar{\Psi}}_{-}[\hat{\psi}_{b}(0)].$
(51)

For $\hat{\psi}_{b}(x)\in\Upsilon_{b}$ satisfying the BCs (39) and (40), of
which there are $\mathcal{N}_{+}$ relations, no restriction on
$\hat{\bar{\Psi}}^{\prime\prime}_{+}[\hat{\psi}_{a}(0)]$ is required to
nullify the whole current. The largest space $\Upsilon_{a}$ over which the
current (51) is nullified is therefore specified by the minimal number
$\mathcal{N}_{-}$ of BCs of the form

$\hat{\bar{\Psi}}^{\prime}_{+}[\hat{\psi}_{a}(0)]=\hat{U}\hat{\Psi}_{-}[\hat{\psi}_{a}(0)],$

while $\hat{\bar{\Psi}}^{\prime\prime}_{+}[\hat{\psi}_{a}(0)]$ can be
arbitrary. Hence, this Hilbert space $\Upsilon_{a}$ of $\hat{\psi}_{a}(x)$ is
larger than $\Upsilon_{b}$ of $\hat{\psi}_{b}(x)$ ($\mathcal{N}_{-}$ versus
$\mathcal{N}_{+}$ BCs, respectively) and the Hamiltonian is not hermitian over
$\Upsilon_{b}$, only symmetric. This, of course, also applies to the general
form (41). Since $\Upsilon_{b}$ is the largest subspace (specified by the
minimal sufficient number $\Lambda=\mathcal{N}_{+}$ of BCs) over which the
current $j[\hat{\psi}_{b}(0)]$ is nullified, there is no Hilbert space at all
over which the Hamiltonian would be hermitian for
$\mathcal{N}_{+}\neq\mathcal{N}_{-}$.

For equal numbers $\mathcal{N}_{+}=\mathcal{N}_{-}$ of right- and left-movers,
the remainder part $\hat{\bar{\Psi}}^{\prime\prime}[\hat{\psi}_{a}(0)]$ is
absent, and the largest space $\Upsilon_{a}$ of $\hat{\psi}_{a}(x)$ coincides
with the space $\Upsilon_{b}$ of $\hat{\psi}_{b}(x)$, specified by the BCs
(52). Hence, the Hamiltonian over such Hilbert space is not only symmetric but
also hermitian.

Therefore, we recognize that the cases of BCs nullifying the current that are
admissible (only $\Lambda=\mathcal{N}_{+}=\mathcal{N}_{-}$) or not (any
$\Lambda>\mathcal{N}_{+}=\mathcal{N}_{-}$ for
$\mathcal{N}_{+}=\mathcal{N}_{-}$ and any
$\Lambda\geq\max(\mathcal{N}_{+},\mathcal{N}_{-})$ for
$\mathcal{N}_{+}\neq\mathcal{N}_{-}$) based on the completeness of the
eigenvector set (Secs. V.4.1 and V.4.2) precisely coincide with the cases of
Hilbert spaces over which the symmetric Hamiltonian is also hermitian or not.

We conclude that the question of which BCs (and the Hilbert spaces they
specify) are admissible in the sense that they represent a system with a well-
defined boundary turns out to be more subtle than the mere nullification of
the current at the boundary. In Ref. 22, this part of analysis has not been
presented.

In physical literature, this subtle but crucial distinction between symmetric
and hermitian (self-adjoint) operators is oftentimes overlooked and the
definition (8) of symmetric operator is used as the (inaccurate) definition of
a hermitian operator. This inaccuracy does not lead to erroneous conclusions
for systems with $\mathcal{N}_{+}=\mathcal{N}_{-}$, which is typical, and when
the right number $\Lambda=\mathcal{N}_{+}=\mathcal{N}_{-}$ of linearly
independent BCs is correctly “guessed”, since this is precisely the only case
when symmetric and hermitian operators do become equivalent.

####  V.4.4 Comment on the relation of the “self-adjoint extension” to the
problem of general boundary conditions

In some studies [11, 22, 23] of the general BCs for CMs, the notion of “self-
adjoint extension” is brought up. The unabbreviated name for the “self-adjoint
extension” procedure reads “extension of a symmetric operator to a self-
adjoint operator”. In accord with the definition of a self-adjoint (hermitian)
operator given above, this procedure is realized when one starts with a
Hilbert space $\Upsilon_{b}$ over which the Hamiltonian operator is symmetric,
but it is not the largest such space (there is room “to extend”) and one tries
to find such space; hence the term “extension”. Note that the form of the
operator is not changed at all during this procedure, only the Hilbert space
is; what is “extended” is the Hilbert space. In terms of BCs, in the self-
adjoint-extension procedure, one therefore starts with possibly “too many” BCs
that do nullify the current and works towards the number of BCs that is less
than the initial one.

On the other hand, in the natural formulation of the BCs problem for a
quantum-mechanical system, as presented in Sec. IV, one starts with the
embedding Hilbert space $\Upsilon_{0}$ of wave functions with no BCs at all,
and recognizes that such restrictions are necessary. One then finds the BCs
with the minimal sufficient number to nullify the current. Therefore, since
$\Upsilon_{b}$ is already by construction the largest space over which the
current is nullified and the Hamiltonian is symmetric, one can only check if
the Hamiltonian is also hermitian over this Hilbert space, as done in Sec.
V.4.3; it is already known that there is “nowhere to extend”.

Therefore, while the end goal of the two procedures is the same, these are
opposite starting points and the starting point of the “self-adjoined
extension” procedure is rather artificial for the problem at hand: some BCs
that do nullify the current, but there are too many of them. Therefore, we
believe that specifically the use of the term “extension” is not relevant to
the natural formulation of the problem of general BCs and is somewhat
misleading; “finding admissible Hilbert spaces over which the Hamiltonian
becomes hermitian” seem like an accurate and more appropriate description of
the procedure.

###  V.5 Standardized universal form of the general boundary conditions

Summarizing the results of the previous subsections, for a system described by
a continuum model with the translation-symmetric Hamiltonian
$\hat{H}(\hat{p})$ of the most general form (1), a boundary can be introduced
only for equal numbers

$\mathcal{N}_{+}=\mathcal{N}_{-}=\frac{\mathcal{N}}{2}$

of positive and negative eigenvalues of the current matrix (right- and left-
moving waves, respectively, in the geometry of Fig. 1). Only in this case
admissible Hilbert spaces, over which the Hamiltonian is not only symmetric
[which is equivalent to the current at the boundary being nullified, Eqs. (8)
and (16)] but also hermitian [Eq. (48)], exist. The family of general BCs
specifying all such admissible Hilbert spaces reads

$\hat{\bar{\Psi}}_{+}[\hat{\psi}(x=0)]=\hat{U}\hat{\bar{\Psi}}_{-}[\hat{\psi}(x=0)].$
(52)

This is a matrix form combining $\Lambda=\mathcal{N}/2$ linearly independent
homogeneous scalar relations between the wave-function components and their
derivatives. All possible BCs are parameterized by all possible unitary
matrices

$\hat{U}\in{\text{U}}(\mathcal{N}/2)$ (53)

of order $\mathcal{N}/2$, with $(\mathcal{N}/2)^{2}$ real parameters. This
reproduces the central result of Ref. 22 (although no substantiation for it
was provided therein, most importantly, why this is the only well-defined
case).

The two most valuable properties of this form of general BCs are:

(i) The universal standardized form of the BCs (52). The current [Eqs. (28)
and (31)] and, as a consequence, the BCs (52) are presented in the universal
forms in terms of the linear combinations
$\bar{\Psi}_{\pm,\nu_{\pm}}[\hat{\psi}(0)]$ [Eq. (30)] of the wave-function
components and their derivatives $\hat{p}^{n}\psi_{m}(0)$ [Eq. (23)] that
diagonalize and normalize the current. These forms are fully characterized for
any Hamiltonian just by the numbers $\mathcal{N}_{\pm}$ of positive and
negative eigenvalues of the current matrix (25), interpreted as the numbers or
right- and left-moving waves. The BCs can then be simply expressed in terms of
the original degrees of freedom (23) via diagonalization formulas (29) and
(30).

(ii) Nonredundant one-to-one (bijective) parametrization of the general BCs:
one instance of $\hat{U}$ provides one distinct set of possible BCs,
specifying one admissible Hilbert space; possible redundancies of BCs due
equivalence transformations or additional linearly dependent relations (see
Appendix A), which do not change the Hilbert space, are eliminated in this
formalism by the imposed structure.

The physical meaning behind the general BCs will be discussed in Sec. IX.

##  VI Physical interpretation of the general boundary conditions as a
scattering process

Figure 2:  The bulk spectrum of the part $\hat{h}_{1}\hat{p}$ of the linear-
in-momentum ($N=1$) Hamiltonian
$\hat{H}_{1}(\hat{p})=\hat{h}_{1}\hat{p}+\hat{h}_{0}$ for a multi-component
wave function, illustrated with the case of $\mathcal{N}_{\pm}=2$ right- and
left-moving waves (modes). The “energy” matrix $\hat{h}_{0}$ is not involved
in the current [Eq. (55)] and hence, in the general BCs (52). The momentum-
independent eigenstates of the linear part $\hat{h}_{1}p$ can be directly
identified as right- and left-moving waves and the interpretation of the
general BCs (52) as a scattering process between them [Sec. VI, Fig. 1(b)] for
the linear-in-momentum Hamiltonian becomes particularly transparent.

The form (52) of the general BCs immediately suggests a natural physical
interpretation, which we now provide. Namely, as already mentioned in Sec.
V.2, the projections $\Psi_{\pm,\nu_{\pm}}[\hat{\psi}(x)]$ [Eq. (29)] of the
vector $\hat{\Psi}[\hat{\psi}(x)]$ onto the eigenvectors
$\hat{s}_{\pm,\nu_{\pm}}$ of the current matrix $\hat{J}$ [Eq. (25)] with
positive and negative eigenvalues $\mathcal{J}_{\pm,\nu_{\pm}}\gtrless 0$ can
be interpreted as chiral “waves” (or “modes”) propagating in the right and
left directions, respectively, in the geometry of Fig. 1, according to the
signs of their contributions to the diagonal form of the current [Eqs. (28)
and (31)]. For the $x\geq 0$ system, these are the waves reflected from and
incident upon the boundary $x=0$, respectively. The general BCs (52) can
therefore be seen as a scattering process between these waves at the boundary,
with $\hat{U}$ being the scattering matrix, as illustrated in Fig. 1.

The interpretation is most transparent in the case of the linear-in-momentum
($N=1$) Hamiltonian

$\hat{H}(\hat{p})=\hat{h}_{0}+\hat{h}_{1}\hat{p},$ (54)

with the “energy” $\hat{h}_{0}$ and “velocity” $\hat{h}_{1}$ matrices, Fig. 2.
In this case [see Eqs. (58) and (59) in the next section], the vector

$\hat{\Psi}[\hat{\psi}(x)]=\hat{\psi}(x)$

in Eq. (25) is equal to the wave-function vector (3) itself and the current
matrix

$\hat{J}=\hat{h}_{1}$ (55)

is given by the velocity matrix. The number $\mathcal{N}=NM=M$ of the degrees
of freedom equals to the number of the wave-function components. The
eigenvector $\hat{s}_{\pm,\nu_{\pm}}$ and eigenvalues
$\mathcal{J}_{\pm,\nu_{\pm}}=h_{1\pm,\nu_{\pm}}$ of the current matrix (55)
are therefore those of the velocity matrix $\hat{h}_{1}$. The boundary is
well-defined only for equal $\mathcal{N}_{+}=\mathcal{N}_{-}=M/2$ and the
general BCs have the form (52) with the projections

$\Psi_{\pm,\nu_{\pm}}[\hat{\psi}(x)]=\hat{s}^{\dagger}_{\pm,\nu_{\pm}}\hat{\psi}(x),$
$\bar{\Psi}_{\pm,\nu_{\pm}}[\hat{\psi}(x)]=\sqrt{|h_{1\pm,\nu_{\pm}}|}\Psi_{\pm,\nu_{\pm}}[\hat{\psi}(x)].$

As already mentioned in Sec. V.1, since the energy matrix $\hat{h}_{0}$ does
not enter the current (55), the BCs (52) are the same for any $\hat{h}_{0}$,
including the absent one. For $\hat{h}_{0}=\hat{0}$, the system is gapless;
the bulk spectrum consists of the linearly dispersing bands
$\varepsilon_{\pm,\nu_{\pm}}(p)=h_{1\pm,\nu_{\pm}}p$, all crossing the point
$(p,\epsilon)=(0,0)$, and the eigenvectors $\hat{s}_{\pm,\nu_{\pm}}$ of the
current matrix (55) coincide with the respective momentum-independent
eigenvectors of the Hamiltonian $\hat{H}(p)=\hat{h}_{1}p$.

For the linear-in-momentum Hamiltonian (54), the right- and left-moving waves
can therefore be identified as the bulk eigenstates of its linear part
$\hat{h}_{1}\hat{p}$. The interpretation of the general BCs (52) as a
scattering process is very natural in this case. For momentum order $N>1$
higher than linear, this interpretation is a little less intuitive, but
nonetheless still meaningful according to the structure of the current [Eqs.
(28) and (31)] and the resulting BCs (52).

From this scattering-process interpretation, one can also better understand
the conditions for when the boundary can be introduced and admissible BCs
exist: from the causality argument, the set of reflected waves should be
completely determined by the set of incident waves (and the reflected and
incident waves can also be interchanged by time reversal). The derived
requirements that the numbers $\mathcal{N}_{\pm}$ of the reflected and
incident waves must be equal, $\mathcal{N}_{-}=\mathcal{N}_{+}=\mathcal{N}/2$,
and the number $\Lambda=\mathcal{N}/2$ of BCs (relations between them) must
also be equal to that number are therefore fully consistent with the causality
argument. If the numbers $\mathcal{N}_{\pm}$ of right- and left-moving waves
are unequal, there are “uncompensated” waves; physically, this means that such
a system cannot be terminated in 1D. The prime example would be an edge of a
quantum Hall system, having, in the simplest case, one edge-state mode,
described by the Hamiltonian $\hat{H}(\hat{p})=h_{1}\hat{p}$ with the
linearized spectrum at low energies, with $\mathcal{N}_{+}=1$ and
$\mathcal{N}_{-}=0$. Physically, as is well-known from the quantum Hall
physics, attempting to terminate the 1D edge of a 2D sample by “cutting” the
sample will force the edge state to propagate along the newly created edge.
This is also the simplest case illustrating the general considerations of Sec.
V.4.3. The largest and only Hilbert space $\Upsilon_{b}$, over which the
Hamiltonian is symmetric, i.e., the current
$j[\psi_{b}(0)]=h_{1}\psi_{b}^{*}(0)\psi_{b}(0)$ is nullified identically, is
that of the wave functions satisfying the BC $\psi_{b}(0)=0$. The current
$j[\psi_{a}(0),\psi_{b}(0)]=h_{1}\psi_{a}^{*}(0)\psi_{b}(0)\equiv 0$ is then
identically zero without any restrictions on $\psi_{a}(0)$. Therefore, there
is no Hilbert space over which the Hamiltonian could be hermitian.

##  VII Structure of the current matrix and justification of the formalism for
top momentum order $N>1$

In the derivation of the general BCs presented in Ref. 22 and reproduced with
more details in Sec. V.2, the diagonalization of the current matrix $\hat{J}$
[Eq. (25)] was assumed to be a well-defined procedure. However, as we now
show, upon closer inspection, there is an important subtlety in this procedure
for the top momentum order $N>1$ higher than linear that needs to be
addressed. Simultaneously, we explore the structure of the current matrix.

We notice that for any Hamiltonian with $N>1$ presenting the current
$j[\hat{\psi}(x)]$ in the matrix form (25) and subsequent diagonalization of
the current matrix $\hat{J}$ inevitably involves introducing an auxiliary
fictitious length scale $l$. Indeed, these operations are well-defined only if
the components of the vector $\hat{\Psi}[\hat{\psi}(x)]$ have the same
physical dimension and, as a result, the elements of $\hat{J}$ have the same
physical dimension. Only commensurable quantities of the same physical
dimension can sensibly be treated as components of one vector or matrix, upon
which linear transformations may be performed. Consider a positive length
parameter

$l>0.$ (56)

Then all derivatives

$\hat{\bar{p}}^{n}\psi_{m}(x)=(-{\text{i}}l\partial_{x})^{n}\psi_{m}(x)$

of different orders $n$ with the dimensionless momentum operator

$\hat{\bar{p}}=l\hat{p}$ (57)

have the same physical dimension and may be joined into one vector
$\hat{\Psi}[\hat{\psi}(x)]$.

###  VII.1 Case of equal top-momentum orders $N_{m}=N$

We first consider the technically simplest case of all wave-function
components $\psi_{m}(x)$, $m=1,\ldots,M$, having the same $N_{m}=N$ top
momentum order in the Hamiltonian $\hat{H}(\hat{p})$ [Eq. (1)], in which case
$\mathcal{N}=NM$ [Eq. (24)]. We introduce the vector as

$\hat{\Psi}[\hat{\psi}(x)]=\left(\begin{array}[]{c}\hat{\psi}(x)\\\
\hat{\bar{p}}\hat{\psi}(x)\\\ \hat{\bar{p}}^{2}\hat{\psi}(x)\\\ \ldots\\\
\hat{\bar{p}}^{N-1}\hat{\psi}(x)\end{array}\right).$ (58)

The current is then presented in the form (25) with the hermitian current
matrix of the block-triangular structure

$\hat{J}=\left(\begin{array}[]{ccccc}\hat{J}_{1}&\hat{J}_{2}&\hat{J}_{3}&\ldots&\hat{J}_{N}\\\
\hat{J}_{2}&\hat{J}_{3}&\ldots&\hat{J}_{N}&\hat{0}\\\
\hat{J}_{3}&\ldots&\hat{J}_{N}&\hat{0}&\ldots\\\
\ldots&\hat{J}_{N}&\hat{0}&\ldots&\hat{0}\\\
\hat{J}_{N}&\hat{0}&\ldots&\hat{0}&\hat{0}\\\ \end{array}\right),\mbox{
}\hat{J}^{\dagger}=\hat{J},$ (59)

where

$\hat{J}_{n}=\frac{\hat{h}_{n}}{l^{n-1}},\mbox{
}\hat{J}_{n}^{\dagger}=\hat{J}_{n},\mbox{ }n=1,\ldots,N,$ (60)

and $\hat{0}$ is the $M\times M$ zero matrix. Due to the length scale $l$ in
$\hat{\Psi}[\hat{\psi}(x)]$ [Eq. (58)], the matrices $\hat{J}_{n}$, describing
the contributions to the current from each momentum order
$\hat{h}_{n}\hat{p}^{n}$ in the Hamiltonian $\hat{H}(\hat{p})$, contain the
factors $1/l^{n-1}$, which ensures that they all have the same physical
dimension of velocity. This way, the properly introduced current matrix
$\hat{J}=\hat{J}(\hat{h}_{1},\ldots,\hat{h}_{N};l)$ and its eigenvalues
necessarily depend on the fictitious length scale $l$, which may be changed at
will.

We point out that the zero-order part $\hat{h}_{0}$ of the Hamiltonian, the
energy matrix representing various energy-level shifts of and couplings
between the basis states of the wave function, never enters the current (20)
and hence does not affect at all the form (52) of the general BCs. The BCs are
the same for any $\hat{h}_{0}$, in particular, for absent
$\hat{h}_{0}=\hat{0}$.

The current matrix $\hat{J}$ [Eq. (59)] for equal top momentum orders
$N_{m}=N$ has a special block-triangular structure: there are no terms on one
side of the anti-diagonal and there is the same $\hat{J}_{n}$ matrix on each
sub-anti-diagonal on the other side. This is the consequence of the
symmetrized form (21) of the current. This structure leads to the key property
of the eigenvalues of the current matrix $\hat{J}$, which is crucial for the
BCs formalism to be well-defined. This property is based on the following
theorem.

Theorem: For equal top-momentum orders $N_{m}=N$, the current matrix $\hat{J}$
[Eq. (59)] is degenerate if and only if the top-momentum-order matrix
$\hat{h}_{N}$ in the Hamiltonian $\hat{H}(\hat{p})$ [Eq. (1)] is degenerate.

####  VII.1.1 Proof of the theorem

Proof of If $\hat{J}$ is degenerate, then $\hat{h}_{N}$ is degenerate.

Consider the matrix equation

$\hat{J}\hat{\Psi}=\hat{0}$ (61)

for the vector

$\hat{\Psi}=\left(\begin{array}[]{c}\hat{\Psi}_{0}\\\ \hat{\Psi}_{1}\\\
\ldots\\\ \hat{\Psi}_{N-1}\end{array}\right),$

consisting of the subvectors $\hat{\Psi}_{n}$ ($n=0,\ldots,N-1$) of size $M$.
In this section, $\hat{0}$ are null vectors of the respective sizes. Using the
structure (59) of the current matrix, Eq. (61) is equivalent to $N$ matrix
equations

$\hat{J}_{1}\hat{\Psi}_{0}+\ldots+\hat{J}_{N}\hat{\Psi}_{N-1}=\hat{0},$
$\ldots$ $\hat{J}_{N-1}\hat{\Psi}_{0}+\hat{J}_{N}\hat{\Psi}_{1}=\hat{0},$ (62)
$\hat{J}_{N}\hat{\Psi}_{0}=\hat{0}.$ (63)

Suppose $\hat{J}_{N}=\hat{h}_{N}/l^{N-1}$ is nondegenerate. It then follows
from Eq. (63) that $\hat{\Psi}_{0}=\hat{0}$. Eq. (62) then reduces to
$\hat{J}_{N}\hat{\Psi}_{1}=\hat{0}$, from which it follows that
$\hat{\Psi}_{1}=\hat{0}$. Repeating this procedure, we obtain that all
$\hat{\Psi}_{n}=\hat{0}$ ($n=0,\ldots,N-1$) vanish, and hence, the matrix
equation (61) has only a trivial solution $\hat{\Psi}=\hat{0}$. Therefore, the
current matrix $\hat{J}$ is nondegenerate, regardless of the values of the
lower-momentum-order matrices $\hat{J}_{n}=\hat{h}_{n}/l^{n-1}$,
$n=1,\ldots,N-1$. The statement is proven by contradiction.

Proof of If $\hat{h}_{N}$ is degenerate, then $\hat{J}$ is degenerate.

Suppose $\hat{\Psi}_{N-1}\neq\hat{0}$ is the eigenvector of $\hat{J}_{N}$ with
zero eigenvalue, $\hat{J}_{N}\hat{\Psi}_{N-1}=\hat{0}$. The nonzero vector

$\hat{\Psi}=\left(\begin{array}[]{c}\hat{0}\\\ ...\\\ \hat{0}\\\
\hat{\Psi}_{N-1}\end{array}\right)\neq\hat{0}$

then gives

$\hat{J}\hat{\Psi}=\hat{0},$

and hence, $\hat{J}$ is degenerate.

###  VII.2 Case of unequal top-momentum orders $N_{m}$

Here, we present a similar key theorem for the case of unequal top momentum
orders $N_{m}=N$ of the wave-function components $m=1,\ldots,M$. Not to
overburden the presentation, we consider the case of different $N_{m}$ for
$N=3$; generalization to higher $N$ is straightforward.

We first group the components of the wave function (3)

$\hat{\psi}(x)=\left(\begin{array}[]{c}\hat{\psi}_{1}(x)\\\
\hat{\psi}_{2}(x)\\\ \hat{\psi}_{3}(x)\end{array}\right)$

into subvectors $\hat{\psi}_{1,2,3}(x)$ that have the top momentum orders
$N_{m}=1,2,3$, respectively. We assume that for each $N_{m}$ there is at least
one component, i.e., the sizes $M_{1,2,3}>0$ of the vectors
$\hat{\psi}_{1,2,3}(x)$ are nonzero; $M_{1}+M_{2}+M_{3}=M$. In this block
basis, the most general form of the Hamiltonian (1) reads

$\hat{H}(\hat{p})=\hat{h}_{0}+\hat{h}_{1}\hat{p}+\hat{h}_{2}\hat{p}^{2}+\hat{h}_{3}\hat{p}^{3}=\hat{h}_{0}+\left(\begin{array}[]{ccc}\hat{h}_{1}^{11}&\hat{h}_{1}^{12}&\hat{h}_{1}^{13}\\\
\hat{h}_{1}^{21}&\hat{h}_{1}^{22}&\hat{h}_{1}^{23}\\\
\hat{h}_{1}^{31}&\hat{h}_{1}^{32}&\hat{h}_{1}^{33}\end{array}\right)\hat{p}+\left(\begin{array}[]{ccc}\hat{0}&\hat{0}&\hat{0}\\\
\hat{0}&\hat{h}_{2}^{22}&\hat{h}_{2}^{23}\\\
\hat{0}&\hat{h}_{2}^{32}&\hat{h}_{2}^{33}\end{array}\right)\hat{p}^{2}+\left(\begin{array}[]{ccc}\hat{0}&\hat{0}&\hat{0}\\\
\hat{0}&\hat{0}&\hat{0}\\\
\hat{0}&\hat{0}&\hat{h}_{3}^{33}\end{array}\right)\hat{p}^{3},$ (64)

with $(\hat{h}_{n}^{kk^{\prime}})^{\dagger}=\hat{h}_{n}^{k^{\prime}k}$,
$k,k^{\prime}=1,2,3$, and $\hat{0}$ being zero matrices of respective orders.

We introduce the vector as

$\hat{\Psi}[\hat{\psi}(x)]=\left(\begin{array}[]{c}\hat{\psi}_{1}(x)\\\
\hat{\psi}_{2}(x)\\\ \hat{\psi}_{3}(x)\\\ \hat{\bar{p}}\hat{\psi}_{2}(x)\\\
\hat{\bar{p}}\hat{\psi}_{3}(x)\\\
\hat{\bar{p}}^{2}\hat{\psi}_{3}(x)\end{array}\right).$

The current quadratic form $j[\hat{\psi}(x)]$ [Eq. (20)] can then be presented
in the matrix form (25) with this vector and

$\hat{J}=\left(\begin{array}[]{cccccc}\hat{J}_{1}^{11}&\hat{J}_{1}^{12}&\hat{J}_{1}^{13}&\hat{0}&\hat{0}&\hat{0}\\\
\hat{J}_{1}^{21}&\hat{J}_{1}^{22}&\hat{J}_{1}^{23}&\hat{J}_{2}^{22}&\hat{J}_{2}^{23}&\hat{0}\\\
\hat{J}_{1}^{31}&\hat{J}_{1}^{32}&\hat{J}_{1}^{33}&\hat{J}_{2}^{32}&\hat{J}_{2}^{33}&\hat{J}_{3}^{33}\\\
\hat{0}&\hat{J}_{2}^{22}&\hat{J}_{2}^{23}&\hat{0}&\hat{0}&\hat{0}\\\
\hat{0}&\hat{J}_{2}^{32}&\hat{J}_{2}^{33}&\hat{0}&\hat{J}_{3}^{33}&\hat{0}\\\
\hat{0}&\hat{0}&\hat{J}_{3}^{33}&\hat{0}&\hat{0}&\hat{0}\\\
\end{array}\right),\mbox{ }$ (65)
$\hat{J}_{n}^{kk^{\prime}}=\frac{\hat{h}_{n}^{kk^{\prime}}}{l^{n-1}},\mbox{
}k,k^{\prime}=1,2,3.$

In essence, the current matrix (65) can be obtained from the one (59) for
equal top momentum orders $N_{m}=3$ with $\hat{h}_{n}$ from Eq. (64) by
discarding the zero rows and columns in the latter, as well as discarding the
respective derivatives of the wave-function components in the vector
$\hat{\Psi}[\hat{\psi}(x)]$. This analogous structure of the current matrix
(65) leads to the key property of its eigenvalues, which follows from the
theorem.

Theorem: For unequal top-momentum orders $N_{m}=1,2,3$ for different wave-
function components, the current matrix $\hat{J}$ [Eq. (65)] is degenerate if
and only if at least one of the top-momentum-order matrices
$\hat{h}_{1}^{11}$, $\hat{h}_{2}^{22}$, or $\hat{h}_{3}^{33}$ in the
Hamiltonian (64) is degenerate.

The proof is analogous to the one for the case of equal $N_{m}=N$.

###  VII.3 Sign structure of the current-matrix eigenvalues is independent of
$l$

The degeneracy of the current matrix $\hat{J}$ as its parameters are spanned
signifies a change of the signs of one or more of its eigenvalues as they
cross zero. Therefore, it follows from the above theorems that the sign
structure of the eigenvalues $\mathcal{J}_{\nu}$, $\nu=1,\ldots,\mathcal{N}$,
of the current matrix $\hat{J}$ is determined solely by the top-momentum-order
matrices in the Hamiltonian: by $\hat{h}_{N}$ for equal $N_{m}=N$ and by
$\hat{h}_{n}^{nn}$ [$n=1,2,3$, Eq. (64)] for unequal $N_{m}$. Namely, the
signs of the eigenvalues $\mathcal{J}_{\nu}$ can never be changed by changing
any of the lower-order matrices or, most importantly, by changing the
fictitious length-scale parameter $l$ [Eqs. (56) and (57)], even though the
eigenvalues $\mathcal{J}_{\nu}$ themselves do depend on them. This justifies
the labelling of the eigenvalues according to their signs as

$\mathcal{J}_{+,\nu_{+}}=\mathcal{J}_{\nu}>0,\mbox{
}\nu_{+}=1,\ldots,\mathcal{N}_{+},$
$\mathcal{J}_{-,\nu_{-}}=\mathcal{J}_{\nu}<0,\mbox{
}\nu_{-}=1,\ldots,\mathcal{N}_{-},$

as well as the labelling of the respective eigenvectors
$\hat{s}_{\pm,\nu_{\pm}}$ and projections
$\Psi_{\pm,\nu_{\pm}}[\hat{\psi}(x)]$ [Eqs. (23) and (29)] of
$\hat{\Psi}[\hat{\psi}(x)]$ [Eq. (58)] onto these eigenvectors. Consequently,
the corresponding characterization of the projections
$\Psi_{\pm,\nu_{\pm}}[\hat{\psi}(x)]$ as right- and left-moving waves,
introduced in Sec. VI, is also determined solely by the top-momentum-order
matrices. An explicit illustration for the fourth-order-in-momentum ($N=4$)
Hamiltonian is provided in Appendix B as an example.

This property that the sign structure of the current-matrix eigenvalues is
independent of the fictitious length scale $l$ is extremely important as it
provides an essential mathematical justification for the procedure of deriving
the general BCs (52). Otherwise, the whole formalism of the general BCs would
be ill-defined for Hamiltonians with the top momentum orders $N>1$ higher than
linear.

The degeneracy of the top-momentum-order matrix $\hat{h}_{N}$ (or one of
$\hat{h}_{n}^{nn}$) as its parameters are spanned, when the current matrix
$\hat{J}$ does become degenerate and the sign changes of its eigenvalues
$\mathcal{J}_{\nu}$ do occur, signifies a qualitative change in the band
structure (which is understandable since the asymptotic band structure at
large momenta is determined by them). Therefore, it is sensible to treat the
sectors in the parameter space of $\hat{h}_{N}$ where it is nondegenerate
separately, as regions where the key qualitative properties of the band
structure are preserved. For example, for the quadratic-in-momentum ($N=2$)
Hamiltonian $\hat{H}(p)=h_{2}p^{2}$ for a one-component ($M=1$) wave function,
the current matrix

$\hat{J}=\frac{h_{2}}{l}\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$
(66)

with eigenvalues $\pm|h_{2}|/l$ is nondegenerate in both sectors $h_{2}>0$ and
$h_{2}<0$; in each of them, $\mathcal{N}_{+}=\mathcal{N}_{-}=1$ and the
boundary and general BC are well-defined, see Sec. XII below. But at $h_{2}=0$
the bulk spectrum becomes flat, as the change between the positive and
negative curvature occurs.

###  VII.4 Fictitious length scale $l$ is not an additional parameter of the
general boundary conditions

Although the fictitious length scale $l$ has to inevitably be introduced into
the formalism for top-momentum order $N>1$ and enters the general BCs via the
$\hat{\Psi}[\hat{\psi}(x);l]$ dependence, here we explain that it is not an
additional free parameter of the general BCs.

Let us rewrite the general BCs (52) with the dependence of the vectors
$\hat{\bar{\Psi}}_{\pm}[\hat{\psi}(x)]=\hat{\bar{\Psi}}_{\pm}[\hat{\psi}(x);l]$
on $l$ explicitly indicated:

$\hat{\bar{\Psi}}_{+}[\hat{\psi}(x=0);l]=\hat{U}\hat{\bar{\Psi}}_{-}[\hat{\psi}(x=0);l].$
(67)

For a fixed $l$, spanning all possible unitary matrices
$\hat{U}\in{\text{U}}(\mathcal{N}/2)$ will span the family of all possible
BCs.

If, for a fixed $\hat{U}$, $l$ is changed to some other value $l^{\prime}$,
one obtains the BCs

$\hat{\bar{\Psi}}_{+}[\hat{\psi}(x=0);l^{\prime}]=\hat{U}\hat{\bar{\Psi}}_{-}[\hat{\psi}(x=0);l^{\prime}]$

that are not equivalent to Eq. (67) (they specify a different Hilbert space
and cannot be obtained by equivalence transformations). However, one can
always find a unitary matrix
$\hat{U}^{\prime}=\hat{U}^{\prime}(l,l^{\prime};\hat{U})$, which would depend
on $\hat{U}$, $l$, and $l^{\prime}$, such that the latter BCs are equivalent
to

$\hat{\bar{\Psi}}_{+}[\hat{\psi}(x=0);l]=\hat{U}^{\prime}\hat{\bar{\Psi}}_{-}[\hat{\psi}(x=0);l]$

with the original $l$. This point will also be explicitly illustrated in Sec.
XII for the quadratic-in-momentum ($N=2$) one-component ($M=1$) model.

##  VIII Semi-analytical method of calculating bound states

Once the BCs have been derived, the CM of a system with a boundary is fully
specified by its Hamiltonian $\hat{H}(\hat{p})$ [Eq. (1)] and BCs (52) and one
can calculate the complete set of eigenstates and their spectrum. Bound states
are the eigenstates that decay into the bulk, with the wave function
$\hat{\psi}(x)\rightarrow\hat{0}$ as $x\rightarrow+\infty$ for the $x>0$ half-
infinite system. (The eigenstates of the continuum part of the spectrum for
the system with a boundary can, of course, also be calculated; these can be
formulated in terms of a scattering problem.)

Here we present the (semi)-analytical method of calculating the bound states
within CMs with BCs. This method has been used in some previous works,
especially for the simplest models; nonetheless, it seems worthwhile to
formulate it clearly and emphasize it, since other, less efficient methods,
like finite-size calculations, are still being commonly used. An analogous
method for lattice models has also been formulated [32].

The method follows directly from the theory of linear differential equations.
Bound states can exist only within the energy gaps of the bulk spectrum of the
respective infinite system, i.e., the spectrum of the matrix $\hat{H}(p)$ for
real $p$. One first constructs the general solution to the stationary
Schrödinger equation (44) at a given energy $\epsilon$ within the gaps that
decays into the bulk. Such general solution is a linear combination of
particular solutions of the form

$\hat{\chi}{\text{e}}^{{\text{i}}px},$ (68)

where the momentum $p$ satisfies the characteristic equation

$\det[\hat{H}(p)-\epsilon\hat{1}]=0$ (69)

and $\hat{\chi}$ is the corresponding nontrivial “eigenvector” solution to

$[\hat{H}(p)-\epsilon\hat{1}]\hat{\chi}=0.$

There are $\mathcal{N}$ [Eq. (24)] momentum solutions $p_{\alpha}(\epsilon)$
to Eq. (69). For the energy $\epsilon$ within the gap, there are, by
construction, no real momentum solutions; each momentum solution has a nonzero
imaginary part. Due to $\mathcal{N}_{+}=\mathcal{N}_{-}=\mathcal{N}/2$, there
always are $\mathcal{N}/2$ momentum solutions with positive and negative
imaginary parts. For the $x>0$ system, only the $\mathcal{N}/2$ momentum
solutions with a positive imaginary part $\text{Im}\,p_{\alpha}(\epsilon)>0$
are kept, labeled $\alpha=1,\ldots,\mathcal{N}/2$, to have decaying particular
solutions (68). The general solution at a given energy $\epsilon$, decaying
into the bulk, reads

$\hat{\psi}(x;\epsilon)=\sum_{\alpha=1}^{\mathcal{N}/2}c_{\alpha}\hat{\chi}_{\alpha}(\epsilon){\text{e}}^{{\text{i}}p_{\alpha}(\epsilon)x},$
(70)

where $c_{\alpha}$ are arbitrary coefficients. [It is assumed here that there
are indeed $\mathcal{N}$ linearly independent eigenvectors
$\hat{\chi}_{\alpha}(\epsilon)$ for $\mathcal{N}$ momentum solutions
$p_{\alpha}(\epsilon)$. In the case of degenerate momentum solutions
$p_{\alpha}(\epsilon)$, the number of eigenvectors may sometimes be less than
the multiplicity. In that case, enough particular solutions still exists, but
their coordinate dependence differs from that of Eq. (68). The adaptation to
this case is also straightforward and follows from the theory of differential
equations.]

This general solution is then inserted into the general BCs (52). Denoting

$\hat{\bar{\Psi}}_{\pm}[\hat{\chi}_{\alpha}(\epsilon){\text{e}}^{{\text{i}}p_{\alpha}(\epsilon)x}]=\hat{\bar{\Psi}}_{\pm}^{\alpha}(\epsilon){\text{e}}^{{\text{i}}p_{\alpha}(\epsilon)x}$

according to the action of the momentum operator $\hat{p}$ in
$\hat{\bar{\Psi}}_{\pm}[\hat{\psi}(x)]$, we obtain the homogeneous system

$\sum_{\alpha=1}^{\mathcal{N}/2}[\hat{\bar{\Psi}}_{+}^{\alpha}(\epsilon)-\hat{U}\hat{\bar{\Psi}}_{-}^{\alpha}(\epsilon)]c_{\alpha}=\hat{0}$

of $\mathcal{N}/2$ equations for $\mathcal{N}/2$ unknown variables
$c_{\alpha}$, where $\hat{0}$ is the null vector of size $\mathcal{N}/2$. One
can equivalently present this system in the matrix form

$\hat{X}(\epsilon)\hat{c}=\hat{0},$ (71)

where

$\hat{X}(\epsilon)=\left(\begin{array}[]{ccc}\hat{\bar{\Psi}}_{+}^{1}(\epsilon)-\hat{U}\hat{\bar{\Psi}}_{-}^{1}(\epsilon)&\ldots&\hat{\bar{\Psi}}_{+}^{\mathcal{N}/2}(\epsilon)-\hat{U}\hat{\bar{\Psi}}_{-}^{\mathcal{N}/2}(\epsilon)\end{array}\right)$

is an $\mathcal{N}/2\times\mathcal{N}/2$ matrix and

$\hat{c}=\left(\begin{array}[]{c}c_{1}\\\ \ldots\\\
c_{\mathcal{N}/2}\end{array}\right)$

is the vector of the coefficients.

The problem of finding bound states has been reduced to solving the system
(71) of equations for the coefficients of the general decaying solution (70).
Generally, for an arbitrary energy $\epsilon$, the matrix $\hat{X}(\epsilon)$
is nondegenerate and there are no nontrivial solutions for $\hat{c}$.
Nontrivial solutions appear at those energies $\epsilon$ at which the matrix
becomes degenerate, i.e.,

$\det\hat{X}(\epsilon)=0$

is the equation for the energy $\epsilon$ of possible bound states. The
solutions to this equation determine the bound-state energies and the
corresponding nontrivial solutions $\hat{c}$ to Eq. (71) yield their wave
functions according Eq. (70).

We consolidate the discussion of all the advantages that the formalism of
general BCs and this calculation method provide for the study of bound states
in Sec. X.

##  IX Physical meaning and realization of general boundary conditions

The general BCs (52) have been formally derived from the fundamental principle
of quantum mechanics, the norm conservation, and there is a legitimate
question about their physical meaning and what they represent for real
systems. We address this question now.

Consider a “microscopic” model [33] with a bulk Hamiltonian
$\hat{\mathcal{H}}(W_{\text{bulk}})$ characterized by the set
$W_{\text{bulk}}$ of parameters and a boundary characterized by the set
$W_{\text{boundary}}$ of parameters. Such microscopic model could be a lattice
model, in which case the boundary is described by a specific termination, or a
different CM with more degrees of freedom (more wave-function components
and/or higher momentum powers), with the boundary described by some specific
BCs. Assume that there is a well-defined low-energy limit of this microscopic
model. Let $\hat{H}(\hat{p};W_{\text{bulk}}^{\text{low}})$ be the continuum
Hamiltonian to which $\hat{\mathcal{H}}(W_{\text{bulk}})$ reduces in this
limit. Typically, among the parameters
$W_{\text{bulk}}=(W_{\text{bulk}}^{\text{low}},W_{\text{bulk}}^{\text{high}})$
of the microscopic Hamiltonian $\hat{\mathcal{H}}(W_{\text{bulk}})$ there are
some $W_{\text{bulk}}^{\text{high}}$ that do not enter the low-energy
Hamiltonian $\hat{H}(\hat{p};W_{\text{bulk}}^{\text{low}})$:
$W_{\text{bulk}}^{\text{high}}$ could be describing regions in the full
Brillouin zone away from the expansion points and/or higher momentum powers in
the expansion in the microscopic CM.

Since the family of general BCs (52) for the continuum bulk Hamiltonian
$\hat{H}(\hat{p};W_{\text{bulk}}^{\text{low}})$ is exhaustive (includes all
possible BCs), the specific boundary described by $W_{\text{boundary}}$ in the
microscopic model will necessarily be described in the low-energy model by the
BCs (52) with a specific instance
$\hat{U}(W_{\text{bulk}}^{\text{high}},W_{\text{boundary}})$ of the unitary
matrix determined by the set of parameters
$(W_{\text{bulk}}^{\text{high}},W_{\text{boundary}})$. [In the microscopic
model, the boundary must, of course, be defined in accord with the norm-
conservation principle; therefore, the latter will be automatically satisfied
in the low-energy model as well, and its BCs are presentable in the form
(52).] This matrix
$\hat{U}(W_{\text{bulk}}^{\text{high}},W_{\text{boundary}})$ will generally be
determined by both the structure of the boundary, described by the parameters
$W_{\text{boundary}}$, and the high-energy parameters
$W_{\text{bulk}}^{\text{high}}$ of the microscopic bulk Hamiltonian
$\hat{\mathcal{H}}(W_{\text{bulk}})$. This way, all necessary information
about the microscopic structure of the boundary and higher-energy part of the
bulk spectrum of the microscopic model is contained in these BCs. Looking at
this relation in the reverse order, any microscopic model of a system with a
boundary that has a given continuum model with specific Hamiltonian
$\hat{H}(\hat{p})$ and matrix $\hat{U}$ of the BCs as its low-energy limit,
can be regarded as a (microscopic) realization of the latter.

If the microscopic model
$\\{\hat{\mathcal{H}}(W_{\text{bulk}}^{\text{low}},W_{\text{bulk}}^{\text{high}}),W_{\text{boundary}}\\}$
allows for variation of the parameters
$(W_{\text{bulk}}^{\text{high}},W_{\text{boundary}})~{}\in~{}\mathcal{W}$
within some domain $\mathcal{W}$, then the corresponding set of BCs with the
matrices
$\\{\hat{U}(W_{\text{bulk}}^{\text{high}},W_{\text{boundary}}),(W_{\text{bulk}}^{\text{high}},W_{\text{boundary}})\in\mathcal{W}\\}$
can be realized for a fixed form of the Hamiltonian
$\hat{H}(\hat{p};W_{\text{bulk}}^{\text{low}})$ of the CM, specified by the
low-energy parameters $W_{\text{bulk}}^{\text{low}}$. For a given microscopic
model with a moderate number of variable parameters, it can be that only a
subset of the whole parameter space ${\text{U}}(\mathcal{N}/2)$ of possible
BCs is spanned this way. However, with several different microscopic models
that are represented by the same low-energy Hamiltonian
$\hat{H}(\hat{p};W_{\text{bulk}}^{\text{low}})$, the whole BCs parameter space
can always be spanned. Alternatively, it should always be possible to
construct a general enough microscopic model, within which the whole BCs
parameter space of the low-energy CM can be spanned.

Therefore, we conclude that the physical meaning of the family of general BCs
(52) satisfying the norm-conservation principle is that it asymptotically
describes all possible underlying microscopic models, with all possible
structures of their boundaries and all possible behaviors of their bulk
Hamiltonians in the Brillouin zone away from the expansion points, that are
represented by a given continuum Hamiltonian $\hat{H}(\hat{p})$ [Eq. (1)] in
the low-energy limit. The implications of this for the bound states are
discussed in the next section.

When the fully specified microscopic model
$\\{\hat{\mathcal{H}}(W_{\text{bulk}}^{\text{low}},W_{\text{bulk}}^{\text{high}}),W_{\text{boundary}}\\}$
of a system with a boundary is provided, both the Hamiltonian
$\hat{H}(\hat{p},W_{\text{bulk}}^{\text{low}})$ and the BCs, characterized by
$\hat{U}(W_{\text{bulk}}^{\text{high}},W_{\text{boundary}})$, of the CM can be
derived from this microscopic model. For Hamiltonians, such systematic low-
energy-expansion procedure that consistently eliminates higher-energy degrees
of freedom is well-known [34] and is part of the $k\cdot p$ method. It can
also always be adapted to include the derivation of the corresponding BCs. The
main step is presenting the microscopic wave function as an expansion in terms
of low-energy envelopes in the coordinate space, but, for the derivation of
the BCs, inclusion of the decaying short-spatial-scale particular solutions,
if present, is also necessary.

Here we list a few examples of the models, for which such procedure of
deriving the BCs has been performed. There are variants of this procedure,
depending on the type of the model, but usually it reduces to one of the
following two procedures (or their combination): (i) Eliminating high-energy
bands, thereby decreasing the size of the local (in momentum) Hilbert space.
In Sec. XIV, we derive the BC for the low-energy quadratic-in-momentum one-
component model ($N=1,M=2$), which describes the linear-in-momentum two-
component model ($N=2,M=1$) in the vicinity of the minimum of the upper band.
Here, the total number $\mathcal{N}=2$ of degrees of freedom stays the same,
but the wave-function components are effectively “converted” into the power of
momentum. In Ref. 7, BCs were derived for the quadratic-node Luttinger
semimetal model [35] from the Kane model with hard-wall BCs by eliminating the
states with the angular momentum $\frac{1}{2}$. This allowed us to demonstrate
that the Luttinger semimetal exhibits one or two bands of surface states. In
Refs. [36, 29], BCs were derived for the linear-in-momentum model of the 1D
superconductor from the quadratic-in-momentum model with the hard-wall BCs.

(ii) Eliminating higher-order momentum terms in CMs within the same local
Hilbert space. In Ref. 8, BCs were derived for the linear-in-momentum model of
a quantum anomalous Hall system from the quadratic-in-momentum model with
hard-wall BCs (one block of the Bernevig-Hughes-Zhang model [37]) in the
vicinity of the topological phase transition. A similar procedure arises when
the microscopic model is a lattice model. In Ref. 18, BCs for the linear-in-
momentum model of graphene were derived from the lattice model.

Note that when instances of the BCs for the low-energy model are derived this
way from the microscopic model, they will be delivered in a completely
arbitrary form; the structure of the standardized form (52) is not guaranteed
to be identified at all. Also, the current-nullification constraint is not
applied explicitly as a derivation principle, but rather will be automatically
satisfied, since it must already be satisfied in the microscopic model.

Concluding this part, we point out that, analogously to the situation with the
Hamiltonians in the $k\cdot p$ method, there are two “sides” of the formalism
of general BCs that should not be conflated: (i) derivation of the general BCs
from general principles (norm conservation and, if present, symmetries, see
the next Sec. XI) and exploring their whole parameter space; (ii) derivation
of instances of the general BCs from the underlying microscopic models.

##  X General boundary conditions deliver general bound-state structures

The main value of the formalism of general BCs for CMs lies in what it
provides for the study of bound states. As follows from the previous Sec. IX,
since the general BCs (52) describe all possible boundaries, the CM with these
BCs will capture all possible bound-state structures of all microscopic models
represented by the considered continuum Hamiltonian in the low-energy limit.
All properties of bound states that originate at low energies will be present
in such CM. The bound-state structures of the CM will asymptotically agree
with those of the microscopic models from which the CM originates, in the
region of momenta and energies where the CM is valid. According to the
nonredundant one-to-one parametrization of the family of general BCs by
unitary matrices (Sec. V.5), one instance of $\hat{U}$ provides one instance
of possible BCs, with one specific respective bound-state structure. Spanning
the parameter space ${\text{U}}(\mathcal{N}/2)$ of $\hat{U}$, all possible
bound-state structures for the CM Hamiltonian $\hat{H}(\hat{p})$ are obtained.
The evolution of bound-state structures as a function of $\hat{U}$ provides
their exhaustive characterization.

The semi-analytical method formulated in Sec. VIII allows one to fully exploit
the advantages of the formalism in the study of bound states. Owing to the
relatively small number $\mathcal{N}$ of relevant degrees of freedom (wave-
function components $M$ and orders of momentum $N_{m}$) sufficient to capture
the essential behavior in the low-energy limit, such models are usually still
simple enough that the whole parameter spaces of their Hamiltonian (see Sec.
XI) and general BCs can be fully explored and their general bound-state
structures can be found in a tractable and explicit manner. For the simplest
models, the general bound-state structure can be found entirely analytically.
For more complicated model, one can still carry out this procedure “semi-
analytically”, with minimal or modest computational resources. The latter may
be required for finding particular solutions (68) and solving the final
equation (71) for the bound-state energy.

Another apparent advantage of the semi-analytical method is that bound states
can be found for a truly half-infinite system; as a result, the low-energy
features (such as the vicinity of the nodes of semimetals in higher
dimensions) can be resolved with any desired accuracy. This appealing property
may be lacking in other methods, such as the commonly employed finite-size
numerical calculations, where resolution is limited by spatial quantization
effects.

##  XI Application of the formalism to families of bulk systems

Figure 3:  Schematic illustration of possible application schemes of the
formalism of general BCs. The space of parameters $h$ of the Hamiltonian
$\hat{H}(\hat{p};h)$ is represented by a 2D plane $h=(h_{a},h_{b})$; the
parameter space ${\text{U}}(\mathcal{N}/2)$ of the unitary matrices $\hat{U}$
of the general BCs (52) is represented by a circle (which is true for
$\mathcal{N}=1$). (a) The Hamiltonian $\hat{H}(\hat{p};h)$ is fixed ($h$ has a
fixed value represented by a point), the BCs are of the most general form,
with $\hat{U}\in{\text{U}}(\mathcal{N}/2)$ spanning the whole space. (b), (c),
(d) Applications schemes to families of bulk systems. (b) Both the Hamiltonian
$\hat{H}(\hat{p};h)$ and the BCs of are the most general forms, with $h$ and
$\hat{U}$ spanning their whole respective spaces. (c) Symmetry-constrained
family of the bulk Hamiltonians and general BCs. The Hamiltonian
$\hat{H}(\hat{p},h)$ with $h\in\mathcal{H}_{s}$ is of the most general form
satisfying certain symmetries; the symmetry-constrained parameter vector space
$\mathcal{H}_{s}$ is represented by a line. The BCs are of the most general
form, with $\hat{U}\in{\text{U}}(\mathcal{N}/2)$. The model describes a family
of bulk systems satisfying certain symmetries with all possible boundaries
(which generally break the bulk symmetries). (d) Symmetry-constrained families
of both the bulk Hamiltonians and BCs. The Hamiltonian $\hat{H}(\hat{p},h)$
with $h\in\mathcal{H}_{s}$ and BCs with
$\hat{U}\in\mathcal{U}_{s}\subset{\text{U}}(\mathcal{N}/2)$ are of the most
general forms satisfying certain symmetries; the subset $\mathcal{U}_{s}$ is
represented as consisting of two points, which is the case, e.g., for linear-
in-momentum models with chiral or charge-conjugations symmetries [7, 8, 9].
The model describes a family of systems with a boundary satisfying certain
symmetries (a symmetry class). The application scheme (d) is perfectly suited
for the study of bound states in topological systems; the bound states within
such models respect the symmetries and reflect bulk topology.

In this section, we outline possible application schemes of the formalism of
general BCs for CMs.

###  XI.1 Symmetry-constrained family of Hamiltonians, general boundary
conditions

The main focus of the previous sections was the derivation of all possible BCs
for a given Hamiltonian $\hat{H}(\hat{p})$, which was implied to be fixed.
According to Sec. X, this scheme describes all possible boundaries and
delivers an exhaustive characterization of all possible bound-state structures
for a fixed bulk system. This application scheme is very useful, e.g., for a
specific material, whose bulk structure is fixed, but different boundaries are
possible.

However, even more powerful application schemes arise (Fig. 3) when general
families of bulk systems are considered, when, rather than a single fixed
Hamiltonian $\hat{H}(\hat{p})$ [Fig. 3(a)], a family of bulk Hamiltonians
$\hat{H}(\hat{p};h)$, $h\in\mathcal{H}_{s}$, is considered, where the
collection of parameters $h$ spans some space $\mathcal{H}_{s}$. Oftentimes,
these are families of Hamiltonians satisfying certain symmetries. In this
case, the most general form of Hamiltonians $\hat{H}(\hat{p};h)$ can be
derived based solely on symmetries using the method of invariants: for
specified relevant degrees of freedom, the wave functions components
$\psi_{m}(x)$ (with their transformation properties) and order order
$\mathcal{N}_{m}$ of momentum, all invariant linearly independent matrix
functions of momentum are found. The most general form of the Hamiltonian
$\hat{H}(\hat{p},h)$ is then an arbitrary linear combination of these basis
functions and $h$ are the collection of its coefficients and $\mathcal{H}_{s}$
is their vector space. This method is commonly used for “conventional”
symmetries, such as spatial and time-reversal, and is known as the $k\cdot p$
method. However, it can also be readily applied to “more abstract” symmetries,
such as chiral and charge-conjugation, relevant to topological systems, see
Sec. XI.2.

The general BCs parameterized by all possible unitary matrices
$\hat{U}\in{\text{U}}(\mathcal{N}/2)$ then represent all possible boundaries
for the whole family of bulk systems described by the family of Hamiltonians
$\hat{H}(\hat{p};h)$, $h\in\mathcal{H}_{s}$ [Fig. 3(c)], Note that in the
construction of the general BCs (52), only the current operator and the
associated quantities, such as the vectors $\hat{\Psi}_{\pm}[\hat{\psi}(x);h]$
of right- and left-movers, depend on the bulk parameters $h$, whereas the
unitary matrices $\hat{U}$ specifying the BCs are independent parameters
specifying possible boundaries. Accordingly, this model will deliver the
general bound-state structure for the whole family of bulk systems.

The scope of this application scheme is very wide since one family of CMs with
a given symmetry can describe many different real systems with many different
lattice structures. For example, crystals with different space groups but the
same point group are described at the $\Gamma$ point by one family of CMs. The
bulk crystal and its band structure can be quite complicated, and possible
surfaces of such crystals – even more so. This approach allows one to obtain
the general bound-state structures of such systems bypassing all these
complications.

###  XI.2 Symmetry-constrained families of Hamiltonians and boundary
conditions

In the above application scheme for a symmetry-constrained family of bulk
systems, an arbitrary instance $\hat{U}\in{\text{U}}(\mathcal{N}/2)$ of the
BCs (52) generally breaks the assumed symmetry of the bulk infinite system, in
which case the system with a boundary does not have that symmetry. Symmetry of
the system with a boundary is particularly important for topological systems
and their bound states.

Topological systems are categorized according to the symmetry classes, with
different sets of symmetries that protect various types of bulk topology of
infinite systems. Certain characteristics of the bound states of topological
systems also rely on those symmetries (such as the zero-energy bound states in
1D for particle-hole-type chiral or charge-conjugation symmetries;
counterpropagating edge states of $\mathcal{T}_{-}$-symmetric 2D topological
insulators). In order for the bound states to reflect (via bulk-boundary
correspondence) the bulk topological properties stemming from a certain
symmetry, the system with a boundary must satisfy that symmetry.

Within the framework of CMs, the symmetry of the system with a boundary means
that not only the Hamiltonian, but also the BCs must also satisfy that
symmetry [7, 8, 29, 9]. This requirement introduces constraints on the allowed
form of $\hat{U}$. As a result, the most general form of BCs under certain
symmetries is parameterized by the matrices $\hat{U}\in\mathcal{U}_{s}$ that
will typically belong to a subset
$\mathcal{U}_{s}\subset{\text{U}}(\mathcal{N}/2)$ of all possible unitary
matrices [Fig. 3(d)].

Therefore, a CM obtained this way, in which both the bulk Hamiltonian and BCs
are of the most general forms satisfying the symmetries, represents a whole
symmetry class of systems with a boundary. As such, the model will deliver the
general bound-state structure of the whole symmetry class. This application
scheme is therefore perfectly suited for the study of bound states in
topological systems that rely on symmetries.

This application scheme has recently been formulated in Ref. 9 as a separate
(symmetry-incorporating) formalism of general continuum models with BCs and
applied therein to the two-component linear-in-momentum model in 1D, 2D, and
3D. Previously, it has also been applied [29] to 1D superconductors with the
charge-conjugation symmetry $\mathcal{C}_{+}$, such that
$\mathcal{C}_{+}^{2}=1$.

The general Hamiltonian $\hat{H}(\hat{p};h)$ [Eq. (1)] with no symmetry
constraints [Fig. 3(b)], whose arbitrary matrices $\hat{h}_{n}$ are seen as
the free parameters $h$, supplemented with the general BCs (52), describes a
family systems with the boundary of the symmetry class A, according to the
topological classification.

##  XII Quadratic-in-momentum ($N=2$) one-component ($M=1$) model

###  XII.1 Hamiltonian and general boundary condition

In this and the next sections, we demonstrate the application of the formalism
of general BCs (52) to the simplest cases with the minimal number
$\mathcal{N}=2$ of degrees of freedom necessary to have a well-defined
boundary: the models with one $\mathcal{N}_{+}=1$ right- and one
$\mathcal{N}_{-}=1$ left-moving wave. There is one BC, the most general for of
which reads

$\bar{\Psi}_{+}[\hat{\psi}(0)]=U\bar{\Psi}_{-}[\hat{\psi}(0)].$ (72)

The vectors $\bar{\Psi}_{\pm}[\hat{\psi}(x)]$ are scalars and the unitary
matrix

$U={\text{e}}^{-{\text{i}}\nu}\in{\text{U}}(1),\mbox{ }\nu\in[0,2\pi)\sim
S^{1},$ (73)

is scalar and can be parameterized by the angle $\nu$ on a unit circle.
According to the interpretation (Sec. VI) of the form (52) of the general BCs,
$\nu$ can be regarded as the phase-shift angle.

For $\mathcal{N}=2$, there are only two possible types of CMs in terms of the
makeup (24) of their degrees of freedom: $(N,M)=(2,1)$, which is a quadratic-
in-momentum one-component model, and $(N,M)=(1,2)$, which is a linear-in-
momentum two-component model. In this section, we consider the quadratic-in-
momentum Hamiltonian

$\hat{H}_{2}(\hat{p})=h_{2}\hat{p}^{2}$ (74)

for the one-component wave function $\psi(x)$. Considering an additional
linear term $h_{1}\hat{p}$ does not bring anything new, since it can be
absorbed by changing the momentum origin. This analysis is therefore valid for
the quadratic-in-momentum Hamiltonian of the most general form. We assume
$h_{2}>0$; the case of $h_{2}<0$ is analogous.

The vector (58) reads

$\hat{\Psi}[\psi(x)]=\left(\begin{array}[]{c}\psi(x)\\\
\hat{\bar{p}}\psi(x)\end{array}\right).$

The current matrix is given by Eq. (66). It is diagonalized by the
eigenvectors [Eq. (27)]

$\hat{s}_{\pm}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\ \pm
1\end{array}\right)$

with the eigenvalues $\pm h_{2}/l$, respectively. The corresponding
projections (29) and (30) are

$\Psi_{\pm}[\psi(x)]=\hat{s}_{\pm}^{\dagger}\hat{\Psi}[\psi(x)]=\frac{1}{\sqrt{2}}[\psi(x)\pm\hat{\bar{p}}\psi(x)],$
(75) $\bar{\Psi}_{\pm}[\psi(x)]=\sqrt{\frac{h_{2}}{l}}\Psi_{\pm}[\psi(x)].$
(76)

The family of general BCs (72) for the quadratic Hamiltonian (74), spelled out
in terms of the wave function, therefore reads

$\sqrt{\frac{h_{2}}{l}}[\psi(0)+l\hat{p}\psi(0)]={\text{e}}^{-{\text{i}}\nu}\sqrt{\frac{h_{2}}{l}}[\psi(0)-l\hat{p}\psi(0)].$
(77)

One can rewrite this BC in the equivalent form

$\psi(0)=l\cot\tfrac{\nu}{2}\partial_{x}\psi(0).$ (78)

All possible BCs are parameterized by the angle $\nu\in[0,2\pi)$ on the unit
circle $S^{1}$ [Eq. (73)]; each value of $\nu$ corresponds to a distinct BC.

As explained in Sec. VII, the fictitious length scale $l$ is explicitly
present in the BC (78), as is the case for any model with momentum order $N>1$
higher than linear. To better illustrate the roles of the fictitious length
scale $l$ and parameter $\nu$, we also derive the general BC for the
Hamiltonian (74) with the simpler “poor man’s” approach. The general linear
homogeneous relation between the wave function and its derivative at the
boundary has the form

$\psi(0)=L\partial_{x}\psi(0),$ (79)

with initially a complex parameter $L$ of the physical dimension of length.
Imposing the nullification constraint (19) on the current [Eq. (22)]

$j[\psi(x)]=h_{2}\\{\psi^{*}(x)\hat{p}\psi(x)+[\hat{p}\psi(x)]^{*}\psi(x)\\}$

of the Hamiltonian (74) constrains $L$ to real values. The two forms (78) and
(79) are equivalent upon the identification

$L=l\cot\tfrac{\nu}{2}$ (80)

and differ only in parametrization: all possible forms of the BC will be
spanned upon $L\in(-\infty,+\infty)$ spanning the whole real axis space and
upon $\nu\in[0,2\pi)$ spanning the unit circle for a fixed $l$.

This also explicitly demonstrates the important point explained in Sec. VII.4
that the fictitious length parameter $l$ is not an additional to $\hat{U}$
free parameter of the family of general BCs: $l$ can have any arbitrary, but
fixed value, when spanning all possible BCs by varying $\hat{U}$. For a
different value $l^{\prime}$ of the length scale, the BC remains the same so
long as $L$ remains the same, i.e., the angle
$\nu^{\prime}=\nu^{\prime}(\nu,l,l^{\prime})$, satisfying
$L=l\cot\frac{\nu}{2}=l^{\prime}\cot\frac{\nu^{\prime}}{2}$ will deliver the
same BC for $l^{\prime}$ as the BC (78) with $l$ and $\nu$.

###  XII.2 Bound state for the general boundary condition

Figure 4: (a) The bulk spectrum and (b) the bound state of the quadratic-in-
momentum single-component model with the Hamiltonian
$\hat{H}_{2}(\hat{p})=h_{2}\hat{p}^{2}$ [Eq. (74)] and general boundary
condition (78). A single bound state with the energy $\mathcal{E}_{2}(L(\nu))$
[Eqs. (80) and (81), red] exists in the half $-\pi<\nu<0$ of the
${\text{U}}(1)\sim S^{1}$ parameter space of the BC.

For $h_{2}>0$, the bulk states fill the energy range $\epsilon>0$ and bound
states can exist only in the range $\epsilon<0$. We find that for $L<0$
[equivalently, for $\nu\in(-\pi,0)$ on the half-circle in Eq. (78)], the model
of a system occupying the half-line $x\geq 0$ with the Hamiltonian
$\hat{H}_{2}(\hat{p})$ [Eq. (74)] and BC (79) contains one bound state with
the wave function

$\psi_{\text{bs}}(x)=\sqrt{\frac{2}{|L|}}{\text{e}}^{-x/|L|}$

and energy

$\mathcal{E}_{2}(L)=-\frac{h_{2}}{L^{2}},$ (81)

whereas for $L>0$ there are no bound states, Fig. 4. (This also shows that $L$
is perhaps a more convenient parameter than $\nu$ for the BC for this
Hamiltonian.)

###  XII.3 Realizations of the boundary condition by potentials

Figure 5:  Realization of the general boundary condition (78) by potentials
for the quadratic-in-momentum single-component model with the Hamiltonian
$\hat{H}_{2}(\hat{p})=h_{2}\hat{p}^{2}$. (a) Potential wall (83). (b)
Potential well (87).

We now consider one way the general BC (78) for the Hamiltonian
$\hat{H}_{2}(\hat{p})=h_{2}\hat{p}^{2}$ can physically arise in the low-energy
limit from a “more microscopic” model. This is one example of the larger point
of the realization of general BCs (52), discussed in Sec. IX. Another
realization of the BC (78) will be presented in Sec. XIV.

Consider the Hamiltonian

$\hat{H}_{2V}(\hat{p})=h_{2}\hat{p}^{2}+V(x)$ (82)

describing a Schödinger particle in the potential $V(x)$ with the one-
component wave function $\psi_{V}(x)$.

####  XII.3.1 Potential wall

First, consider the potential “wall”

$V(x)=\left\\{\begin{array}[]{ll}0,&x>0,\\\ V_{<}>0,&x<0,\end{array}\right.$
(83)

depicted in Fig. 5(a). The wave function $\psi_{V}(x)$ is defined on the whole
real axis $x\in(-\infty,+\infty)$; as such, there are no boundaries and no
BCs.

The model (82) and (83) is an example of a “more microscopic” model related to
the CM (74) and (78) of interest in the way described in Sec. III.2: in the
region $x\geq 0$, it has the latter as its low-energy limit, which in this
case is simply the same, and in the “inaccessible” region $x<0$ only “high-
energy” degrees of freedom can exist, not captured by the low-energy model.
The BC for the low-energy wave function $\psi(x)$, defined only in the region
$x\geq 0$, arises upon systematically excluding these high-energy degrees of
freedom. We demonstrate this procedure next.

Consider the stationary Schrödinger equation

$\hat{H}_{2V}(\hat{p})\psi_{V}(x)=\epsilon\psi_{V}(x)$ (84)

at energy $|\epsilon|\ll V_{<}$. At such low energies, the wave function
$\psi_{V}(x)$ will rapidly decay into the region $x<0$ over a scale
$\sqrt{h_{2}/V_{<}}$ much shorter than the typical length scale
$\sqrt{h_{2}/|\epsilon|}$ of its variation in the $x>0$ region. The system
becomes asymptotically equivalent to the one defined on the half-line $x\geq
0$, with the Hamiltonian $\hat{H}_{2}=h_{2}\hat{p}^{2}$, the wave function
$\psi(x)$, and an effective BC at $x=0$. To obtain this BC, it is justified to
completely neglect the energy $\epsilon$ in the Schrödinger equation (84) in
the region $x<0$, which leads to the energy-independent solution

$\psi_{V}(x)=B{\text{e}}^{\varkappa_{<}x},\mbox{ }x<0,$

with $\varkappa_{<}=\sqrt{{V_{<}}/{h_{2}}}$ and an arbitrary coefficient $B$.
The arbitrary wave function $\psi_{V}(x)$ at $x>0$ must match with this
solution at $x=0$ according to the differential properties of the Hamiltonian
$\hat{H}_{2V}(\hat{p})$ with the potential (83). [Importantly, this implies
that there is no “more microscopic” structure at the potential jump at $x=0$,
i.e., that Eqs. (82) and (83) is the “fully microscopic” Hamiltonian that
determines the matching of the wave function.] As follows from the stationary
Schrödinger equation (84), the wave function and its first derivative at $x=0$
on the right and left must be equal, which gives

$B=\psi_{V}(+0),\mbox{ }B\varkappa_{<}=\partial_{x}\psi_{V}(+0).$

Excluding the coefficient $B$ from these relations and equating
$\psi_{V}(x)=\psi(x)$ in the region $x>0$, we obtain the effective BC

$\psi(0)=\frac{1}{\varkappa_{<}}\partial_{x}\psi(0)$ (85)

of the form (79) for the wave function $\psi(x)$, defined only at $x>0$. Since
$\varkappa_{<}>0$, there are no bound states in the low-energy model defined
in the region $x>0$ with $\hat{H}_{2}=h_{2}\hat{p}^{2}$ and BC (85), in accord
with the fact that there are no bound states in the exact solution for the
Hamiltonian (82) with the potential (83).

Note also that, since the variation length scale $\sqrt{h_{2}/|\epsilon|}$ of
the wave function in the region $x>0$ is much larger than $1/\varkappa_{<}$
for $|\epsilon|\ll V_{<}$, it is also justified to neglect the right hand-side
in Eq. (85) completely, and the BC reduces to

$\psi(0)=0.$ (86)

This is known as the “hard-wall” BC. We observe that the hard-wall BC is
realized asymptotically for a potential wall to the leading order in
$|\epsilon|/V_{<}\rightarrow+0$.

####  XII.3.2 Potential well

Next, we consider the case of the potential well

$V(x)=\left\\{\begin{array}[]{ll}0,&x>0,\\\
-V_{0},&-x_{0}<x<0,\end{array}\right.$ (87)

of width $x_{0}$ and depth $V_{0}>0$, depicted in Fig. 5(b). To the left of
the well, we consider an “infinitely high” potential wall; according to above,
the wave function $\psi_{V}(x)$ of the Hamiltonian (82) satisfies the hard-
wall BC [Eq. (86)]

$\psi_{V}(-x_{0})=0.$ (88)

The wave function $\psi_{V}(x)$ of this system is defined in the half-infinite
region $x\in(-x_{0},+\infty)$. Consider the energy $|\epsilon|\ll V_{0}$ much
smaller than the depth of the well. Similarly, at such low energies, the
system becomes asymptotically equivalent to the system defined only in the
region $x>0$, with the Hamiltonian $\hat{H}_{2}(\hat{p})=h_{2}\hat{p}^{2}$,
the wave function $\psi(x)$, and an effective BC for the latter at $x=0$.
Although the region $-x_{0}<x<0$ of the well is classically accessible, the
excitation energy $\sim V_{0}$ there is much higher than $\epsilon$ and the
region cannot be described by the low-energy wave function $\psi(x)$, see also
the comment 31. One may neglect the energy $\epsilon$ in the Schrödinger
equation (84) in the region $-x_{0}<x<0$, to obtain the energy-independent
solution satisfying the BC (88),

$\psi_{V}(x)=A\sin k_{0}(x+x_{0}),\mbox{ }-x_{0}<x<0,$

with $k_{0}=\sqrt{V_{0}/h_{2}}$ and an arbitrary coefficient $A$. Analogously
to the above, the wave function $\psi_{V}(x)$ in the region $x>0$ must match
with this solution according to the differential properties of the Hamiltonian
$\hat{H}_{2V}(\hat{p})$ with the potential (87), which leads to the relations

$A\sin k_{0}x_{0}=\psi_{V}(+0),\mbox{ }k_{0}A\cos
k_{0}x_{0}=\partial_{x}\psi_{V}(+0).$

Excluding the coefficient $A$ and equating $\psi_{V}(x)=\psi(x)$ in the region
$x>0$, we obtain the effective BC

$\psi(0)=L_{0}\partial_{x}\psi(0)$ (89)

of the form (79) with

$L_{0}=\frac{1}{k_{0}}\frac{\sin k_{0}x_{0}}{\cos k_{0}x_{0}}.$ (90)

Let us compare the properties of the microscopic [Eqs. (82), (87), and (88)]
and low-energy [$\hat{H}_{2}(\hat{p})=h_{2}\hat{p}^{2}$, Eq. (89)] models. On
the one hand, according to Sec. XII.2, for the low-energy model with
$\hat{H}_{2}(\hat{p})=h_{2}\hat{p}^{2}$ and general BC (78), there exists one
bound state for $L<0$ with the energy (81) and no bound state for $L>0$. On
the other hand, depending on the value of the parameter $k_{0}x_{0}$, the
microscopic model can have an arbitrary number of bound states, whose energies
$\epsilon$ are determined from the equation

$\tan k_{0}(\epsilon)x_{0}=-\frac{k_{0}(\epsilon)}{\varkappa(\epsilon)},\mbox{
}k_{0}(\epsilon)=\sqrt{\frac{V_{0}+\epsilon}{h_{2}}},\mbox{
}\varkappa(\epsilon)=\sqrt{\frac{-\epsilon}{h_{2}}},$ (91)

obtained by solving the model exactly. As $k_{0}x_{0}$ is increased (i.e., the
well is made deeper and/or wider), new bound states emerge at $\epsilon=0$ and
move down in energy. From this exact solution, the zeros
$\alpha_{n}=\pi(n+\frac{1}{2})$, $n=0,1,\ldots$, of $\cos k_{0}x_{0}$
correspond precisely to the emergence of a new bound state; exactly at such
values of $k_{0}x_{0}$ there is a non-normalizable solution with
$\psi(x>0)=\text{const}$. For $k_{0}x_{0}$ slightly larger than each zero
$\alpha_{n}$, there is a shallow bound state in addition to those already
existing at $k_{0}x_{0}$ slightly below $\alpha_{n}$. It is this shallow bound
state that is captured and quantitatively accurately described in this limit
$|k_{0}x_{0}-\alpha_{n}|\ll 1$ by the low-energy model; the asymptotic
expansion of its exact energy obtained from Eq. (91) agrees with Eq. (81) with
$L=L_{0}$ [Eq. (90)] in this limit. When $k_{0}x_{0}$ is away from zeros
$\alpha_{n}$, the bound state is not shallow anymore (its energy is outside of
the validity range $|\epsilon|\ll V_{0}$) and is therefore no longer
accurately described by the low-energy model. Accordingly, away from zeros
$\alpha_{n}$, the length scale $L_{0}\lesssim 1/k_{0}$ is much shorter than
the characteristic length scale $\sqrt{h_{2}/|\epsilon|}$ of the low-energy
model. Only close to the zeros, $L_{0}$ becomes large,
$k_{0}L_{0}\rightarrow\pm\infty$ as $k_{0}x_{0}\rightarrow\alpha_{n}\pm 0$.
Note that this model illustrates that $\partial_{x}\psi(0)=0$ is legitimate
case of the general BC (79), with $L=\pm\infty$; it describes, e.g., the
microscopic system with a potential well at the boundary in the borderline
regime, when a bound state emerges right at zero energy.

##  XIII Linear-in-momentum ($N=1$) two-component ($M=2$) model

###  XIII.1 Hamiltonian and general boundary condition

The second option of the model with the minimal number $\mathcal{N}=2$ of the
degrees of freedom is the model with the linear-in-momentum ($N=1$)
Hamiltonian and the two-component ($M=2$) wave function. Previously, general
BCs for such model have been explored for specific forms of the Hamiltonian in
Refs. 12, 22, 24, 20, 21, 7, 8. The most general form of such model has been
explored recently in Ref. 9.

Here, to illustrate the key properties of the formalism of general BCs, we
also consider only the specific, simplest form of the Hamiltonian:

$\hat{H}_{1}(\hat{p})=\hat{\tau}_{z}v_{z}\hat{p}+\hat{\tau}_{x}d_{x},$ (92)

which is also a special case of the general linear-in-momentum Hamiltonian Eq.
(54) of Sec. VI. Here, $\hat{\tau}_{x,z}$ are the Pauli matrices and we assume
$d_{x}>0$ and $v_{z}>0$. The wave function is denoted as

$\hat{\psi}(x)=\left(\begin{array}[]{c}\psi_{+}(x)\\\
\psi_{-}(x)\end{array}\right).$

The current matrix (55)

$\hat{J}=\hat{\tau}_{z}v_{z}$ (93)

and the current

$j[\hat{\psi}(x)]=\hat{\psi}^{\dagger}(x)\hat{\tau}_{z}v_{z}\hat{\psi}(x)=v_{z}\psi_{+}^{*}(x)\psi_{+}(x)-v_{z}\psi_{-}^{*}(x)\psi_{-}(x)$

of $\hat{H}_{1}(\hat{p})$ are determined by the velocity matrix. Since the