# Computability of magnetic Schrödinger and Hartree equations on unbounded
domains

Simon Becker<EMAIL_ADDRESS>Department of Applied Mathematics
and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge,
CB3 0WA, United Kingdom.  ,  Jonathan Sewell<EMAIL_ADDRESS>Department of
Applied Mathematics and Theoretical Physics, University of Cambridge,
Wilberforce Road, Cambridge, CB3 0WA, United Kingdom.  and  Euan Tebbutt
<EMAIL_ADDRESS>Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom.

###### Abstract.

We study the computability of global solutions to linear Schrödinger equations
with magnetic fields and the Hartree equation on $\mathbb{R}^{3}$. We show
that the solution can always be globally computed with error control on the
entire space if there exist a priori decay estimates in generalized Sobolev
norms on the initial state. Using weighted Sobolev norm estimates, we show
that the solution can be computed with uniform computational runtime with
respect to initial states and potentials. We finally study applications in
optimal control theory and provide numerical examples.

##  1\. Introduction

In this article, we address for the first time the question under which
conditions the solution to linear Schrödinger equations with external magnetic
field or the Hartree equation can be reduced to an effective dynamics on a
bounded domain that can then be numerically computed. We show that in both
cases, this is possible if the initial state can be controlled in a
generalized Sobolev-type norm that controls both the regularity and the decay
of the initial state. The linear Schrödinger equation with external magnetic
field describes a single-particle system subject to an external magnetic
field. The Hartree equation naturally occurs as a mean-field limit of many-
particle systems and in quantum transport theory [IZL94, BM91]. Due to their
relevance in physics, we focus on magnetic Schrödinger equations or Hartree
equations with bilinear interaction potential on $\mathbb{R}^{3}$.

Most of the literature dealing with numerical approximation schemes on
unbounded domains, has been concerned with the existence of transparent or
absorbing boundary conditions, see e.g. [KG18, JG07] and references therein,
for electro-magnetic time-dependent Schrödinger equations. In [KG18] spatial
restrictions are imposed on the electromagnetic vector potential, to avoid
undesired boundary effects. In this work, we circumvent such undesired
restrictions by studying a somewhat different question: _Can we forget about
the unboundedness of the domain and just restrict the dynamics to a bounded
domain whose size is uniform in the input parameters (initial state,
potentials, control) of our problem?_

Similar to the above results for linear Schrödinger equations, a mode
decomposition for nonlinear Schrödinger equations has been thoroughly
addressed on bounded domain by Soffer and Stucchio in [SS07, SS09].

The linear Schrödinger equation is often numerically discretized by a simple
Crank-Nicholson method, as this one preserves the $L^{2}$-norm. To numerically
study the non-linear dynamics, there exist many results on splitting methods
for nonlinear Hartree equations (also called Schrödinger-Poisson
equation)[F04, L08] and even relativistic [BD11] and fractional Hartree
equations [ZWW19].

Thus, even though there exists a vast literature on convergent discretization
schemes, an implementation on a physical computer must not rely on infinitely
many variables. For instance, a convergent algorithm that assumes a uniform
discretization of the entire space $\mathbb{R}^{3}$ cannot be implemented in
practice.

In Section 5, we also discuss applications of our work to optimal control
problems.

We illustrate our results with some basic numerical examples in Section 6.

We emphasize at this point that all the constants appearing in this work could
be made fully explicit. But for the sake of presentation, we do not specify
them explicitly.

Let us now describe the precise set-up of the equations we consider, starting
with the magnetic linear Schrödinger equation:

###  1.1. The magnetic Schrödinger equation

Let $H_{0}$ be a self-adjoint magnetic Schrödinger operator
$H_{0}=(-i\nabla-A)^{2}+V:D(H_{0})\subset L^{2}(\mathbb{R}^{3})\rightarrow
L^{2}(\mathbb{R}^{3})$ where $V$ is a time-independent _pinning potential_
$V$. We consider the evolution of a particle under the influence of an
external _control potential_ $V_{\operatorname{con}}$ with control function
$u\in W^{1,1}_{\operatorname{pcw}}(0,T)$. Writing
$V_{\operatorname{TD}}(t):=u(t)V_{\operatorname{con}}$ for the full time-
dependent potential, we study time-dependent linear Schrödinger equations of
the form

$\begin{split}i\partial_{t}\psi(x,t)&=(H_{0}+V_{\operatorname{TD}}(t))\psi(x,t),\quad(x,t)\in\mathbb{R}^{3}\times(0,T)\\\
\psi(\bullet,0)&=\varphi_{0}.\end{split}$ (1.1)

The assumptions we impose on the potentials are as follows

######  Assumption 1.

We demand that the static pinning potential can be written as a sum of a
regular (possibly unbounded at infinity) and singular part (possible local
singularities) $V=W_{\operatorname{reg}}+W_{\operatorname{sing}}$. For these
two potentials and the control potential $V_{\operatorname{con}}$ with control
$u$, we impose the assumptions that

  * •

$\|\langle\bullet\rangle^{-2}V_{\operatorname{con}}\|_{\infty}<\infty$ and
$\|\langle\bullet\rangle^{-2}W_{\operatorname{reg}}\|_{\infty}<\infty,$

  * •

$W_{\operatorname{sing}}\in L^{2}({\mathbb{R}}^{3}),$ and

  * •

$u\in W^{1,1}_{\operatorname{pcw}}(0,T).$

For the magnetic vector potential we either assume a constant magnetic field
$B_{0}>0$ with associated vector potential $A=\frac{B_{0}}{2}(-x_{2},x_{1},0)$
or a vector potential $A\in W^{1,\infty}({\mathbb{R}}^{3};{\mathbb{R}}^{3})$.

######  Remark 1.

Our choice of singular and regular part allows us to treat standard physical
examples of potentials such as the Coulomb potential $x\mapsto 1/|x|$
(singular part) and the harmonic potential $x\mapsto|x|^{2}$ (regular part and
control potential).

The Schrödinger equation (1.1) with linear control potential appears naturally
in the study of static physical systems with Hamiltonian $H_{0}$, under the
influence of a time-dependent bilinear electric potential.

In this article, we build upon techniques, introduced in [B05, BKP05], to
prove existence of solutions to Schrödinger equations in certain weighted
Sobolev spaces that ensure additional spatial decay. These spaces are
essential in our study of global numerical algorithms that provide solutions
to (1.1) on unbounded domains.

###  1.2. Hartree equation

In the case of the Hartree equation, we consider a single particle described
by the Hartree equation with Schrödinger operator
$H_{0}=(-i\nabla-A)^{2}+V:D(H_{0})\subset L^{2}(\mathbb{R}^{3})\rightarrow
L^{2}(\mathbb{R}^{3})$ and static _pinning potential_ $V$ and _control
potential_ $V_{\operatorname{con}}$ with time-dependent control function $u\in
W^{1,1}(0,T)$ such that $V_{\operatorname{TD}}(t):=u(t)V_{\operatorname{con}}$

$\begin{split}i\partial_{t}\psi(x,t)&=(H_{0}+V_{\operatorname{TD}}(t))\psi(x,t)+\left(|\psi(\bullet,t)|^{2}*\frac{1}{|\bullet|}\right)(x)\psi(x,t),\quad(x,t)\in\mathbb{R}^{3}\times(0,T)\\\
\psi(\bullet,0)&=\varphi_{0}.\end{split}$

The potentials and magnetic fields are assumed to satisfy the conditions in
Assumption 1.

In this section, we shall fix some notation that we use throughout the
article:

Notation. We denote by $H^{n}$ the standard Sobolev space with respect to the
$L^{2}$ inner product. By $H^{k}_{n}$ we denote the weighted Sobolev spaces
with scalar product induced by

$\|f\|_{H^{k}_{n}}:=\sqrt{\|f\|_{H^{k}}^{2}+\||x|^{n}f\|^{2}_{2}}.$

We write $a\lesssim b$ to indicate that there is a constant $C>0$ such that
$a\leq Cb.$ The space $W^{1,1}_{\operatorname{pcw}}(0,T)$ is the space of
piecewise $W^{1,1}$ functions on $(0,T).$ We denote by $C^{k}_{b}$ the Banach
space of $C^{k}$ functions whose first $k$ derivatives and the function itself
are all globally bounded. We denote by $\mathcal{L}(X)$ the bounded linear
operator on some normed space $X.$

We denote by $C>0$ an arbitrary constant and $C_{\mu}>0$ an arbitrary constant
that depends on some parameter $\mu.$

We sometimes abuse the notation for spatial integrals in $\mathbb{R}^{3}$ to
simplify the notation and mean the same when writing the following
expressions, if there is no misunderstanding possible,

$\int_{\mathbb{R}^{3}}f(x)\ dx=\int_{\mathbb{R}^{3}}f\
dx=\int_{\mathbb{R}^{3}}f.$

In particular, we also abuse the notation by identifying the following
expressions and versions of that

$\int_{\mathbb{R}^{3}}(1+|x|^{2})f=\int_{\mathbb{R}^{3}}(1+|x|^{2})f(x)=\int_{\mathbb{R}^{3}}(1+|x|^{2})f(x)\
dx.$

We recall the following elementary bound which links control on weighted
Sobolev norms to spatial decay in the $L^{2}$-sense

######  Remark 2.

Let $\phi$ be a function in $H_{\eta}({\mathbb{R}}^{3})$ then it follows that
for all $R>0$

$||\phi\operatorname{1\hskip-2.75ptl}_{B_{R}(0)^{c}}||_{L^{2}({\mathbb{R}}^{3})}\leq||\phi||_{H_{\eta}({\mathbb{R}}^{3})}(1+R^{2})^{-\eta/2}\leq||\phi||_{H_{\eta}({\mathbb{R}}^{3})}R^{-\eta}.$

Outline of the article.

  * •

In Section 2, we derive estimates on weighted solutions to the magnetic linear
Schrödinger equation. Theorem 1 contains bounds on solutions to the linear
Schrödinger equation with bounded magnetic fields and Theorem 2 contains such
estimates for constant magnetic fields. In Theorem 3, we prove a quantitative
reduction to an auxiliary boundary value problem.

  * •

In Section 3, we derive weighted bounds on solutions to the Hartree equation,
see Theorem 4 and prove a quantitative reduction to a boundary value problem
in Theorem 5.

  * •

In Section 4, we introduce a convergent numerical discretization scheme and
prove the convergence with an explicit rate, cf. Theorem 6.

  * •

Section 5 discusses applications of our results to optimal control theory.

  * •

The final Section 6 illustrates numerical consequences of our analysis.

##  2\. The magnetic linear Schrödinger equation

We start by recalling the following classical Lemma [BH20, BKP05] that shows
that the free Schrödinger equation is well-posed in the weighted Sobolev space
$H^{2}_{2}$:

######  Lemma 2.1.

Let $\varphi_{0}\in H^{2}_{2}$. There exists a $\psi\in C([0,T];H^{2}_{2})$
solving

$\begin{split}i\partial_{t}\psi&=-\Delta\psi\text{ with initial condition
}\psi(x,0)=\varphi_{0}(x)\end{split}$

such that $\left\lVert\psi(\bullet,t)\right\rVert_{H^{2}_{2}}\leq
C_{T}\left\lVert\varphi_{0}(\bullet)\right\rVert_{H^{2}_{2}}.$

As a first step to study the magnetic linear Schrödinger equation, we show the
existence of solutions to a regularized version of the magnetic linear
Schrödinger equation with discrete derivatives, for some regularization
parameter $h>0,$

$(\nabla_{h}\psi)(x):=\left(\frac{\psi(x+he_{1})-\psi(x)}{h},..,\frac{\psi(x+he_{3})-\psi(x)}{h}\right)$
(2.1)

coupled to a bounded magnetic vector potential and regularized potentials.

######  Lemma 2.2.

Let $V\in L^{\infty}((0,T);C_{b}^{2}({\mathbb{R}}^{3}))$ and $A\in
W^{2,\infty}({\mathbb{R}}^{3};{\mathbb{R}}^{3})$. For initial states
$\varphi_{0}\in H^{2}_{2}({\mathbb{R}}^{3})$, the equation

$\begin{split}i\partial_{t}\psi(x,t)&=-\Delta\psi(x,t)+V(x,t)\psi(x,t)+iA(x)\cdot(\nabla_{h}\psi)(x,t)+i(\nabla_{-h}\cdot(A\psi))(x,t)\\\
\psi(x,0)&=\varphi_{0}(x)\end{split}$ (2.2)

has a unique mild solution $\psi\in C([0,T];H^{2}_{2}({\mathbb{R}}^{3}))$.
Furthermore, if $\rho=\left\lVert
V\right\rVert_{L^{\infty}(0,T;C_{b}^{2}({\mathbb{R}}^{3}))}$, $\mu=\left\lVert
A\right\rVert_{W^{2,\infty}}$, then there exists $C_{T,\rho,\mu,h}$ such that
for all $t\in[0,T]$, $\left\lVert\psi(\bullet,t)\right\rVert_{H^{2}_{2}}\leq
C_{T,\rho,\mu,h}\left\lVert\varphi_{0}(\bullet)\right\rVert_{H^{2}_{2}}.$

###### Proof.

Let $Y=C([0,T];H^{2}_{2}({\mathbb{R}}^{3}))$ with norm
$\lVert\psi\rVert_{Y}=\sup_{t\in[0,T]}e^{-\lambda
t}\left\lVert\psi(\bullet,t)\right\rVert_{H^{2}_{2}({\mathbb{R}}^{3})}$ for
some $\lambda>0$ to be specified later. In terms of the one-parameter group
$S(t)=e^{i\Delta t}$, solving the equation (2.2) is equivalent to solving the
fixed-point equation

$\psi(t)=S(t)\varphi_{0}-i\int_{0}^{t}S(t-s)\left(\psi(s)V(s)+(iA\cdot\nabla_{h}\psi)(s)+(i\nabla_{-h}\cdot(A\psi))(s)\right)ds.$

Let $B_{R}=\\{\psi\in Y:\lVert\psi\rVert_{Y}\leq R\\}$, for some $R$ to be
specified later, and define a map $\Phi:B_{R}\to B_{R}$ by

$\Phi(\psi(t))=S(t)\varphi_{0}-i\int_{0}^{t}S(t-s)\left(\psi(s)V(s)+(iA\cdot\nabla_{h}\psi)(s)+(i\nabla_{-h}\cdot(A\psi))(s)\right)ds.$

To see that $\Phi$ maps $B_{R}$ into itself, first notice that

$\begin{split}\left\lVert\psi(s)V(s)\right\rVert_{H^{2}_{2}}&\lesssim\left\lVert\Delta(\psi(s)V(s))\right\rVert_{2}+\left\lVert(1+|\bullet|^{2})\psi(\bullet,s)V(\bullet,s)\right\rVert_{2}\\\
&\lesssim\left\lVert\Delta\psi(s)V(s)+\nabla\psi(s)\cdot\nabla V(s)+\psi\Delta
V\right\rVert_{2}+\rho\lVert\psi(s)\rVert_{H_{2}}\leq
C_{\rho}\left\lVert\psi(s)\right\rVert_{H^{2}_{2}}.\end{split}$

We then use in addition that $[\nabla_{h},\Delta]=0$

$\begin{split}\left\lVert
A\cdot\nabla_{h}\psi(s)\right\rVert_{H^{2}_{2}}&\lesssim\left\lVert\Delta(A\cdot\nabla_{h}\psi(s))\right\rVert_{2}+\left\lVert(1+|\bullet|^{2})A(\bullet)\cdot\nabla_{h}\psi(\bullet,s)\right\rVert_{2}\\\
&\leq C_{\mu,h}\left\lVert\psi(s)\right\rVert_{H^{2}_{2}}\end{split}$

and

$\begin{split}\left\lVert\nabla_{-h}\cdot(A\cdot\psi)(s)\right\rVert_{H^{2}_{2}}&\lesssim\left\lVert\Delta(\nabla_{-h}\cdot(A\cdot\psi)(s))\right\rVert_{2}+\left\lVert(1+|\bullet|^{2})\nabla_{h}\cdot(A\cdot\psi)(s)\right\rVert_{2}\\\
&\leq
C_{h}(\left\lVert\Delta(A\cdot\psi)(s)\right\rVert_{2}+\mu\lVert\psi(s)\rVert_{H_{2}})\leq
C_{\mu,h}\left\lVert\psi(s)\right\rVert_{H^{2}_{2}}.\end{split}$

Let $C_{T}$ be the constant from Lemma 2.1, let
$2(C_{\rho}+C_{\mu,h})C_{T}\leq\lambda$ , and let $R\geq
2C_{T}\left\lVert\varphi_{0}\right\rVert_{H^{2}_{2}}.$ Hence, we find that,
$\Phi$ is indeed a contraction:

$\begin{split}\lVert\Phi(\psi_{1})-\Phi(\psi_{2})\rVert_{Y}&=\sup_{t}\left(e^{-\lambda
t}\left\lVert\int_{0}^{t}S(t-s)\left(V(s)+iA\cdot\nabla_{h}+i\nabla_{-h}\cdot
A\right)(\psi_{1}-\psi_{2})(s)ds\right\rVert_{H^{2}_{2}}\right)\\\ &\leq
C_{T}(C_{\rho}+C_{\mu,h})\sup_{t}\left(e^{-\lambda t}\int_{0}^{t}e^{\lambda
s}e^{-\lambda
s}\left\lVert(\psi_{1}-\psi_{2})(s)\right\rVert_{H^{2}_{2}}ds\right)\\\ &\leq
C_{T}(C_{\rho}+C_{\mu,h})\lVert\psi_{1}-\psi_{2}\rVert_{Y}\sup_{t}\left(e^{-\lambda
t}\int_{0}^{t}e^{\lambda
s}ds\right)\leq\frac{1}{2}\lVert\psi_{1}-\psi_{2}\rVert_{Y}.\end{split}$

Therefore $\Phi$ has a unique fixed point in $Y$.

The previous estimate immediately implies that
$\left\lVert\psi(\bullet,t)\right\rVert_{H^{2}_{2}}\leq
C_{T,\rho,\mu,h}\left\lVert\varphi_{0}(\bullet)\right\rVert_{H^{2}_{2}}$. ∎

######  Remark 3.

We want to mention at this point that both Lemma 2.1 and Lemma 2.2 hold in
$H^{1}_{1}$ as well by very similar arguments.

In the following Lemma, we get rid of the discrete differentiation of the
magnetic vector potential that we used in Lemma 2.2 to show the existence of
solutions.

######  Lemma 2.3.

Let $T>0$ and potentials as in Assumption 1 such that $\left\lVert
u\right\rVert_{W^{1,1}(0,T)}=\alpha$,
$\left\lVert|\bullet|^{-2}V_{\operatorname{con}}(x)\right\rVert_{L^{\infty}}\leq\rho,$
and
$\left\lVert|\bullet|^{-2}W_{\operatorname{reg}}(x)\right\rVert_{L^{\infty}}\leq\rho.$
For vector potentials $A\in W^{1,\infty}({\mathbb{R}}^{3};{\mathbb{R}}^{3})$
with $\lVert A\rVert_{W^{1,\infty}}=\mathcal{A}$, there exists a constant
$C_{T,\alpha,\rho}$ such that for any $\varphi_{0}\in H^{1}_{1}$, the equation

$\begin{split}i\partial_{t}\psi(x,t)&=(-i\nabla-A)^{2}\psi(x,t)+V\psi(x,t)\\\
\psi(x,0)&=\varphi_{0}(x)\end{split}$

has a unique solution $\psi\in L^{\infty}(0,T;H^{1}_{1})$, and $\psi$
satisfies $\left\lVert\psi\right\rVert_{L^{\infty}(0,T;H^{1}_{1})}\leq
C_{T,\alpha,\rho,\mathcal{A}}\lVert\varphi_{0}\rVert_{H^{1}_{1}}.$

###### Proof.

First, approximate the potentials by elements of
$L^{\infty}(0,T;C_{b}^{2}({\mathbb{R}}^{3}))$ in the following way:

Let $\chi\in C^{\infty}_{c}({\mathbb{R}}^{3})$, where $0\leq\chi$, chosen
symmetric $\chi(x)=\chi(-x)$ and normalized
$\int_{{\mathbb{R}}^{3}}\chi(x)dx=1$. We define the rescaled function
$\zeta_{\varepsilon}(x)=\frac{1}{\varepsilon^{3}}\chi\left(\frac{x}{\varepsilon}\right).$
Using the truncation map
$(\operatorname{Trunc}_{\varepsilon}f)(x)=\operatorname{sign}(f(x))\max\left(\frac{1}{\varepsilon},f(x)\right)$,
we define for $i\in\\{\operatorname{reg},\operatorname{con}\\}$

$V_{i}^{\varepsilon}(x):=(\zeta_{\varepsilon}*\operatorname{Trunc}_{\varepsilon}(V_{i}))(x)=\int_{{\mathbb{R}}^{3}}\operatorname{Trunc}_{\varepsilon}(V_{i}(x+\varepsilon
y))\chi(y)dy.$ (2.3)

Finally, let
$W_{\operatorname{sing}}^{\varepsilon}:=\zeta_{\varepsilon}*W_{\operatorname{sing}}.$

By properties of convolutions it can be shown that

$\begin{split}\left\lVert(1+|\bullet|^{2})^{-1}W_{\operatorname{reg}}^{\varepsilon}(\bullet)\right\rVert_{L^{\infty}}&\leq\rho\text{
and
}\left\lVert(1+|\bullet|^{2})^{-1}V_{\operatorname{con}}^{\varepsilon}(\bullet)\right\rVert_{L^{\infty}}\leq\rho.\end{split}$

For the discrete derivative

$D_{h}\psi=-i\nabla_{h}\psi,$ (2.4)

with $\nabla_{h}$ as in (2.1), we have by Lemma 2.2, that for all
$\varepsilon>0$ , there exists $\psi_{\varepsilon}\in
L^{\infty}(0,T;H^{2}_{2})$ satisfying

$\begin{split}i\partial_{t}\psi(x,t)_{\varepsilon}&=(H_{B}^{\varepsilon}+V^{\varepsilon})\psi_{\varepsilon}(x,t)\\\
\psi_{\varepsilon}(x,0)&=\varphi_{0}(x)\end{split}$

where $H_{B}^{\varepsilon}=-\Delta-AD_{\varepsilon}-D_{-\varepsilon}A+A\cdot
A$ is the regularized magnetic Schrödinger operator. We then intend to bound
the $H^{1}_{1}$ norm of $\psi_{\varepsilon}$ uniformly in $\varepsilon$.

We multiply the preceding equation by $(1+|x|^{2})\bar{\psi}_{\varepsilon}(x)$
and integrate over ${\mathbb{R}}^{3}$

$\int_{\mathbb{R}^{3}}(1+|x|^{2})(\overline{\psi_{\varepsilon}}i\partial_{t}\psi_{\varepsilon})(x,t)+(1+|x|^{2})(\overline{\psi_{\varepsilon}}H_{B}^{\varepsilon}\psi_{\varepsilon})(x,t)+(1+|x|^{2})V^{\varepsilon}(x)|\psi_{\varepsilon}(x,t)|^{2}\
dx=0.$

Thus by taking the imaginary part, we find

$\operatorname{Im}\int_{\mathbb{R}^{3}}(1+|x|^{2})(\overline{\psi_{\varepsilon}}i\partial_{t}\psi_{\varepsilon})(x,t)-(1+|x|^{2})(\psi_{\varepsilon}H_{B}^{\varepsilon}\psi_{\varepsilon})(x,t)\
dx=0.$

In the first term, we note that
$\operatorname{Im}(i\overline{\psi_{\varepsilon}}\partial_{t}\psi_{\varepsilon})=\operatorname{Re}(\overline{\psi_{\varepsilon}}\partial_{t}\psi_{\varepsilon})=\frac{1}{2}\partial_{t}|\psi_{\varepsilon}|^{2}.$
This implies that

$\begin{split}\partial_{t}\int_{\mathbb{R}^{3}}(1+|x|^{2})|\psi_{\varepsilon}(x,t)|^{2}\
dx&=2\operatorname{Im}\int_{{\mathbb{R}}^{3}}(1+|x|^{2})(\overline{\psi_{\varepsilon}}H_{B}^{\varepsilon}\psi_{\varepsilon})(x,t)\
dx\\\
&=2\operatorname{Im}\int_{{\mathbb{R}}^{3}}(1+|x|^{2})(\overline{\psi_{\varepsilon}}\left(-\Delta-
AD_{\varepsilon}-D_{-\varepsilon}A\right)\psi_{\varepsilon})(x,t)\
dx.\end{split}$

Integration by parts yields

$\begin{split}-2\operatorname{Im}\int_{{\mathbb{R}}^{3}}(1+|x|^{2})(\bar{\psi}_{\varepsilon}\Delta\psi_{\varepsilon})(x)\
dx&=2\operatorname{Im}\int_{{\mathbb{R}}^{3}}\underbrace{(1+|x|^{2})|\nabla\psi_{\varepsilon}(x)|^{2}}_{\in\mathbb{R}}\
dx+2\operatorname{Im}\int_{{\mathbb{R}}^{3}}2\bar{\psi}_{\varepsilon}(x)x\cdot\nabla\psi_{\varepsilon}(x)\
dx\\\
&=\operatorname{Im}\int_{{\mathbb{R}}^{3}}4\overline{\psi_{\varepsilon}(x)}x\cdot\nabla\psi_{\varepsilon}(x)\
dx\leq\int_{{\mathbb{R}}^{3}}4|\psi_{\varepsilon}(x)||x||\nabla\psi_{\varepsilon}(x)|\
dx.\end{split}$

Hence, using that $ab\leq\frac{a^{2}}{2}+\frac{b^{2}}{2}$, this implies

$\begin{split}-2\operatorname{Im}\int_{{\mathbb{R}}^{3}}(1+|x|^{2})\bar{\psi}_{\varepsilon}\Delta\psi_{\varepsilon}\
dx&\leq 2\left(\int_{{\mathbb{R}}^{3}}|x|^{2}|\psi_{\varepsilon}|^{2}\
dx+\int_{{\mathbb{R}}^{3}}|\nabla\psi_{\varepsilon}|^{2}dx\right).\end{split}$

In terms of $F(x)=(1+|x|^{2})A(x)$, we find

$\begin{split}\int_{{\mathbb{R}}^{3}}(1+|x|^{2})\overline{\psi(x)}((AD_{\varepsilon}+D_{-\varepsilon}A)\psi)(x)\
dx&=\int_{{\mathbb{R}}^{3}}\overline{\psi(x)}((FD_{\varepsilon}+D_{-\varepsilon}F)\psi)(x)\
dx\\\
&\quad+\int_{{\mathbb{R}}^{3}}\overline{\psi(x)}([(1+|x|^{2}),D_{-\varepsilon}]A\psi)(x)\
dx.\end{split}$

We now observe that
$\int_{{\mathbb{R}}^{3}}\overline{\psi(x)}((FD_{\varepsilon}+D_{-\varepsilon}F)\psi)(x)\
dx\in{\mathbb{R}}$ which is because $FD_{\varepsilon}+D_{-\varepsilon}F$ is
self-adjoint.

Observe then that for
$g(x)=(f(x-he_{1})(D_{-\varepsilon})_{1}x^{2},..,f(x-he_{1})(D_{-\varepsilon})_{3}x^{2})$,
where $(D_{-\varepsilon})_{i}$ is the discretized derivative, (2.4), in
direction $i$, we have

$\begin{split}([(1+x^{2}),D_{-\varepsilon}]f)(x)&=(1+x^{2})(D_{-\varepsilon}f)(x)-D_{-\varepsilon}((1+x^{2})f(x))\\\
&=(1+x^{2})D_{-\varepsilon}f(x)-(1+x^{2})D_{-\varepsilon}f(x)+g(x)=g(x).\end{split}$

This implies that

$\operatorname{Im}\int_{{\mathbb{R}}^{3}}(1+|x|^{2})\overline{\psi(x)}((AD_{\varepsilon}+D_{-\varepsilon}A)\psi)(x)\
dx=2\int_{{\mathbb{R}}^{3}}\overline{\psi(x)}\sum_{i=1}^{3}A_{i}\psi(x-he_{i})(D_{-\varepsilon})_{i}x^{2}\
dx.$

This last term can then be estimated by
$\int_{{\mathbb{R}}^{3}}(1+x^{2})|\psi(x)|^{2}$, since we assume in this Lemma
that $A$ is bounded. Therefore,

$\frac{d}{dt}\lVert\psi_{\varepsilon}\rVert_{H_{1}}^{2}\leq(2+2\mathcal{A})\lVert\psi_{\varepsilon}\rVert_{H_{1}}^{2}+2\lVert\psi_{\varepsilon}\rVert_{H^{1}}^{2}.$
(2.5)

Starting from
$i\partial_{t}\psi_{\varepsilon}=H_{B}^{\varepsilon}\psi_{\varepsilon}+V^{\varepsilon}\psi_{\varepsilon},$
we multiply this equation by $\partial_{t}\bar{\psi}_{\varepsilon}$, take the
real part, and integrate the expression over $\mathbb{R}^{3}$. This yields

$\begin{split}0&=\operatorname{Re}\int_{{\mathbb{R}}^{3}}-\Delta\psi_{\varepsilon}\partial_{t}\bar{\psi}_{\varepsilon}-(\partial_{t}\bar{\psi}_{\varepsilon})(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon}+A\cdot
A\psi_{\varepsilon}\partial_{t}\bar{\psi}_{\varepsilon}+V^{\varepsilon}\psi_{\varepsilon}\partial_{t}\bar{\psi}_{\varepsilon}\
dx.\end{split}$ (2.6)

To treat the second term, in the above integrand, notice that

$\begin{split}\int_{{\mathbb{R}}^{3}}\partial_{t}(\bar{\psi}_{\varepsilon}(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon})(x)\
dx=2\operatorname{Re}\int_{{\mathbb{R}}^{3}}(\partial_{t}\bar{\psi}_{\varepsilon})(x)(AD_{\varepsilon}+D_{-\varepsilon}A)(x)\psi_{\varepsilon}(x)\
dx\end{split}$

by self-adjointness. Therefore, the identify in (2.6) reduces to

$\begin{split}0&=\operatorname{Re}\int_{{\mathbb{R}}^{3}}\nabla\psi_{\varepsilon}\cdot\partial_{t}\nabla\bar{\psi}_{\varepsilon}-(\partial_{t}\bar{\psi}_{\varepsilon})(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon}+|A|^{2}\psi_{\varepsilon}\partial_{t}\bar{\psi}_{\varepsilon}+V^{\varepsilon}\psi_{\varepsilon}\partial_{t}\bar{\psi}_{\varepsilon}\
dx\\\
&=\frac{1}{2}\int_{{\mathbb{R}}^{3}}\partial_{t}|\nabla\psi_{\varepsilon}|^{2}-\partial_{t}(\bar{\psi}_{\varepsilon}(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon})+\partial_{t}|A\psi_{\varepsilon}|^{2}+V^{\varepsilon}\partial_{t}\left(|\psi_{\varepsilon}|^{2}\right)\
dx\end{split}$

and hence

$\begin{split}\frac{d}{dt}\int_{{\mathbb{R}}^{3}}|\nabla\psi_{\varepsilon}|^{2}\
dx&=\int_{{\mathbb{R}}^{3}}(\partial_{t}V^{\varepsilon})|\psi_{\varepsilon}|^{2}\
dx\\\
&\quad+\frac{d}{dt}\int_{{\mathbb{R}}^{3}}(\bar{\psi}_{\varepsilon}(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon})-|A\psi_{\varepsilon}|^{2}-V^{\varepsilon}|\psi_{\varepsilon}|^{2}\
dx.\end{split}$ (2.7)

In addition, we recall that

$\begin{split}\int_{{\mathbb{R}}^{3}}|\partial_{t}V^{\varepsilon}||\psi_{\varepsilon}|^{2}\
&\leq|u^{\prime}(t)|\left\lVert\frac{V_{\operatorname{con}}(\bullet)}{1+|\bullet|^{2}}\right\rVert_{L^{\infty}}\lVert\psi_{\varepsilon}(\bullet,t)\rVert_{H^{1}}^{2}.\end{split}$
(2.8)

We now consider an energy $E_{\lambda}^{\varepsilon}(t)$ defined for
$\lambda>0$ to be chosen later

$E_{\lambda}^{\varepsilon}(t)=\lVert\psi_{\varepsilon}(t)\rVert_{H^{1}}^{2}+\lambda\lVert\psi_{\varepsilon}(t)\rVert_{H_{1}}^{2}.$

Using the preceding estimates and identities (2.5), (2.7), and (2.8), we see
that

$\begin{split}\frac{d}{dt}E_{\lambda}^{\varepsilon}(t)&=\frac{d}{dt}\int_{{\mathbb{R}}^{3}}|\nabla\psi_{\varepsilon}|^{2}\
dx+\lambda\frac{d}{dt}\int_{{\mathbb{R}}^{3}}(1+|x|^{2})|\psi_{\varepsilon}|^{2}\
dx\\\
&\leq\frac{d}{dt}\int_{{\mathbb{R}}^{3}}(\bar{\psi}_{\varepsilon}(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon})-|A\psi_{\varepsilon}|^{2}-V^{\varepsilon}|\psi_{\varepsilon}|^{2}\
dx+|u^{\prime}(t)|\left\lVert\frac{V_{\operatorname{con}}^{\varepsilon}(\bullet)}{1+|\bullet|^{2}}\right\rVert_{L^{\infty}}\lVert\psi_{\varepsilon}\rVert_{H^{1}}^{2}\\\
&\quad+2\lambda\int_{{\mathbb{R}}^{3}}|x|^{2}|\psi_{\varepsilon}|^{2}\
dx+2\lambda\int_{{\mathbb{R}}^{3}}|\nabla\psi_{\varepsilon}|^{2}\ dx\\\
&\leq\frac{d}{dt}\int_{{\mathbb{R}}^{3}}(\overline{\psi_{\varepsilon}(t)}(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon}(t))-|A\psi_{\varepsilon}|^{2}-V^{\varepsilon}|\psi_{\varepsilon}|^{2}\
dx+C\left(1+|u^{\prime}(t)|\right)E_{\lambda}^{\varepsilon}(t)\end{split}$
(2.9)

where $C$ is a constant that depends only on $T$, $\rho,$ and $\lambda$. Let
$\beta(t)=1+|u^{\prime}(t)|$, integration of (2.9) from $0$ to $t$ shows that
using the Cauchy-Schwarz inequality and Lemma 3.1

$\begin{split}E_{\lambda}^{\varepsilon}(t)&\leq\int_{{\mathbb{R}}^{3}}(\overline{\psi_{\varepsilon}(t)}(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon}(t))-|A\psi_{\varepsilon}(t)|^{2}-V^{\varepsilon}(t)|\psi_{\varepsilon}(t)|^{2}\
\\\
&\quad+\left|\int_{{\mathbb{R}}^{3}}(\overline{\psi_{\varepsilon}(0)}(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon}(0))-|A\psi_{\varepsilon}(0)|^{2}-V^{\varepsilon}(0)|\psi_{\varepsilon}(0)|^{2}\
\right|\\\ &\quad+C\int_{0}^{t}\beta(s)E_{\lambda}^{\varepsilon}(s)\
ds+E_{\lambda}^{\varepsilon}(0).\end{split}$

Notice that by (2.3)

$\begin{split}\int_{{\mathbb{R}}^{3}}|W_{\operatorname{sing}}^{\varepsilon}(x)||\psi_{\varepsilon}(x,t)|^{2}\
dx&=\int_{{\mathbb{R}}^{3}}\int_{{\mathbb{R}}^{3}}|\psi_{\varepsilon}(x,t)^{2}W_{\operatorname{sing}}(x-\varepsilon
y)\chi(y)|\ dy\ dx\\\
&=\int_{{\mathbb{R}}^{3}}\chi(y)\int_{{\mathbb{R}}^{3}}|\psi_{\varepsilon}(x+\varepsilon
y,t)^{2}W_{\operatorname{sing}}(x)|\ dx\ dy\\\
&=\int_{{\mathbb{R}}^{3}}\chi(y)\left\langle|W_{\operatorname{sing}}(\bullet)|\psi_{\varepsilon}(\bullet+\varepsilon
y,t),\psi_{\varepsilon}(\bullet+\varepsilon y,t)\right\rangle_{L^{2}}\
dy.\end{split}$

Therefore, by the relative zero-boundedness of the singular potential with
respect to the Laplacian, [BH20, Remark 3.8], it follows that for all
$\delta>0$ , there exists $C_{\delta}$ such that for all $t\in[0,T]$ :

$\begin{split}\int_{{\mathbb{R}}^{3}}|W_{\operatorname{sing}}^{\varepsilon}(x)||\psi_{\varepsilon}(x,t)|^{2}\
dx&\leq\int_{{\mathbb{R}}^{3}}\left(\delta\lVert\nabla\psi_{\varepsilon}(\bullet+\varepsilon
y,t)\rVert_{2}^{2}+C_{\delta}\lVert\psi_{\varepsilon}(\bullet+\varepsilon
y,t)\rVert_{2}^{2}\right)\chi(y)\ dy\\\
&=\delta\lVert\nabla\psi_{\varepsilon}(\bullet,t)\rVert_{2}^{2}+C_{\delta}\lVert\psi_{\varepsilon}(\bullet,t)\rVert_{2}^{2}.\end{split}$

Notice also that straight from the assumptions on the magnetic vector
potential and potentials

$\begin{split}\int_{{\mathbb{R}}^{3}}|W_{\operatorname{reg}}^{\varepsilon}(x)||\psi_{\varepsilon}(x,t)|^{2}\
dx\leq\rho\lVert\psi_{\varepsilon}(t)\rVert_{H_{1}}^{2}\text{ and
}\quad\int_{{\mathbb{R}}^{3}}|A(x)|\psi_{\varepsilon}(x,t)|^{2}\ dx\leq\lVert
A\rVert^{2}_{\infty}\lVert\psi_{\varepsilon}(t)\rVert_{H_{1}}^{2}.\end{split}$

Then it follows that for all $\kappa>0$

$\begin{split}\left|\int_{{\mathbb{R}}^{3}}\bar{\psi}_{\varepsilon}(x,t)(A(x)D_{\varepsilon}+D_{-\varepsilon}A(x))\psi_{\varepsilon}(x,t))\
\right\rvert&\lesssim\frac{\|\psi_{\varepsilon}(t)\|^{2}_{2}}{\kappa}+\kappa\mathcal{A}\|\psi_{\varepsilon}(t)\|^{2}_{H^{1}}\end{split}$

since both $\|D_{\varepsilon}f\|_{\infty}\leq\|\nabla f\|_{\infty}$ and
$\|D_{\varepsilon}f\|_{2}\leq\|f\|_{H^{1}}$ for suitable $f.$

Putting all these together, picking $\delta,\kappa$ sufficiently small, and
evaluating $E_{\lambda}^{\varepsilon}(0)$, we get for all $\lambda$,

$E_{\lambda}^{\varepsilon}(t)\leq\frac{1}{2}\lVert\psi_{\varepsilon}(t)\rVert_{H^{1}}^{2}+C_{\alpha,\mathcal{A},\rho,T}\lVert\psi_{\varepsilon}\rVert_{H_{1}}^{2}+C_{\alpha,\mathcal{A},\rho,\lambda,T}\lVert\varphi_{0}\rVert_{H^{1}_{1}}^{2}+C\int_{0}^{t}\beta(s)E_{\lambda}^{\varepsilon}(s)\
ds.$

Pick $\lambda>2C_{\alpha,\mathcal{A},\rho,T}$, and we get

$\lVert\psi_{\varepsilon}(t)\rVert_{H^{1}_{1}}^{2}\lesssim\lVert\varphi_{0}\rVert_{H^{1}_{1}}^{2}+\int_{0}^{t}\beta(s)\lVert\psi_{\varepsilon}(t)\rVert_{H^{1}_{1}}^{2}\
ds.$

Then, by Gronwall’s inequality, we get
$\lVert\psi_{\varepsilon}(t)\rVert_{H^{1}_{1}}\lesssim\lVert\varphi_{0}\rVert_{H^{1}_{1}}.$
∎

As our first theorem shows, the solution even exists in $H^{2}_{2}$.

######  Theorem 1.

With the same notation and same assumptions as in Lemma 2.3, but now for any
initial state $\varphi_{0}\in H^{2}_{2}$, the equation

$\begin{split}i\partial_{t}\psi(x,t)&=(-i\nabla-A(x))^{2}\psi(x,t)+V(x,t)\psi(x,t)\\\
\psi(x,0)&=\varphi_{0}(x)\end{split}$

has a unique solution $\psi\in L^{\infty}(0,T;H^{2}_{2})$, and this $\psi$
satisfies $\left\lVert\psi\right\rVert_{L^{\infty}(0,T;H^{2}_{2})}\leq
C_{T,\alpha,\rho}\lVert\varphi_{0}\rVert_{H^{2}_{2}}.$

###### Proof.

We start by deriving a bound on $\lVert\psi_{\varepsilon}\rVert_{H^{2}}$ which
we will do in an extra Lemma:

######  Lemma 2.4.

There exists $C$ independent of $\varepsilon$ and $t$ such that

$\lVert\psi_{\varepsilon}\rVert_{H^{2}}\leq
C\lVert\psi_{\varepsilon}\rVert_{H_{2}}+C\lVert\partial_{t}\psi_{\varepsilon}\rVert_{L_{2}}$

Using some of the previously established results we see that

###### Proof.

Directly from the Schrödinger equation, we obtain

$\begin{split}\lVert\psi_{\varepsilon}\rVert_{H^{2}}&\leq\lVert\partial_{t}\psi_{\varepsilon}\rVert_{2}+2\mathcal{A}\lVert
D_{\varepsilon}\psi_{\varepsilon}\rVert_{2}+(\mathcal{A+A}^{2})\lVert\psi_{\varepsilon}\rVert_{2}+\rho(|u(t)|+1)\lVert\psi_{\varepsilon}\rVert_{H_{1}}+\lVert
W_{\operatorname{sing}}\psi_{\varepsilon}\rVert_{2}\\\
&\leq\lVert\partial_{t}\psi_{\varepsilon}\rVert_{2}+(2\mathcal{A}+\delta)\lVert\nabla\psi_{\varepsilon}\rVert_{2}+(\mathcal{A+A}^{2}+\rho|u(t)|+\rho+C_{\delta})\lVert\psi_{\varepsilon}\rVert_{H_{1}}.\end{split}$

However, from
$\lVert\nabla\psi_{\varepsilon}\rVert_{2}\leq\sqrt{\lVert\psi_{\varepsilon}\rVert_{2}\lVert\psi_{\varepsilon}\rVert_{H^{2}}}\leq\frac{1}{\kappa}\lVert\psi_{\varepsilon}\rVert_{2}+\kappa\lVert\psi_{\varepsilon}\rVert_{H^{2}}$,
we see that by choosing $\delta,\kappa$ to be sufficiently small, there exists
$C$ independent of $\varepsilon$ and $t$ such that
$\lVert\psi_{\varepsilon}\rVert_{H^{2}}\leq
C\lVert\psi_{\varepsilon}\rVert_{H_{2}}+C\lVert\partial_{t}\psi_{\varepsilon}\rVert_{L_{2}}.$

∎

We now continue with the energy estimates: Starting from
$i\partial_{t}\psi_{\varepsilon}=H_{B}^{\varepsilon}\psi_{\varepsilon}+V^{\varepsilon}\psi_{\varepsilon}$
, multiply by $|x|^{4}\overline{\psi_{\varepsilon}}$, integrate over
${\mathbb{R}}^{3}$ and take the imaginary part

$\begin{split}&\frac{1}{2}\frac{d}{dt}\int_{{\mathbb{R}}^{3}}|x|^{4}|\psi_{\varepsilon}|^{2}\
dx=\operatorname{Im}\int_{{\mathbb{R}}^{3}}(\nabla\psi_{\varepsilon})\cdot\nabla(|x|^{4}\psi_{\varepsilon})-|x|^{4}\overline{\psi_{\varepsilon}}(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon}\
dx\\\ &\leq 2\int_{{\mathbb{R}}^{3}}|x|^{2}|\nabla\psi_{\varepsilon}|^{2}\
dx+2\int_{{\mathbb{R}}^{3}}|x|^{4}|\psi_{\varepsilon}|^{2}\
dx-\operatorname{Im}\int_{{\mathbb{R}}^{3}}|x|^{4}\overline{\psi_{\varepsilon}}(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon}\
dx.\end{split}$

Note that

$\begin{split}\operatorname{Im}\int_{{\mathbb{R}}^{3}}|x|^{4}\overline{\psi_{\varepsilon}}(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon}\
dx&=\operatorname{Im}\int_{{\mathbb{R}}^{3}}\overline{\psi_{\varepsilon}}(|x|^{4}AD_{\varepsilon}+D_{-\varepsilon}|x|^{4}A)\psi_{\varepsilon}\
dx\\\
&\quad+\operatorname{Im}\int_{{\mathbb{R}}^{3}}\overline{\psi_{\varepsilon}}(|x|^{4}D_{-\varepsilon}A-D_{-\varepsilon}|x|^{4}A)\psi_{\varepsilon}\
dx.\end{split}$

The first term of the RHS is in ${\mathbb{R}}$ by self-adjointness. Therefore

$\begin{split}&\operatorname{Im}\int_{{\mathbb{R}}^{3}}|x|^{4}\overline{\psi_{\varepsilon}}(AD_{\varepsilon}+D_{-\varepsilon}A)\psi_{\varepsilon}=\operatorname{Im}\int_{{\mathbb{R}}^{3}}\overline{\psi_{\varepsilon}}(|x|^{4}D_{-\varepsilon}A-D_{-\varepsilon}|x|^{4}A)\psi_{\varepsilon}\
dx\\\ &\quad-\sum_{j}(|x-\varepsilon e_{j}|^{4}A_{j}(x-\varepsilon
e_{j})\psi_{\varepsilon}(x-\varepsilon
e_{j})-|x|^{4}A_{j}(x)\psi_{\varepsilon}(x))\Bigg{)}\ dx\\\
&=\operatorname{Im}\int_{{\mathbb{R}}^{3}}\overline{\psi_{\varepsilon}}\frac{i}{\varepsilon}\sum_{j}\left((|x-\varepsilon
e_{j}|^{4}-|x|^{4})A_{j}(x-\varepsilon e_{j})\psi_{\varepsilon}(x-\varepsilon
e_{j})\right)\
dx\lesssim\mathcal{A}\int_{{\mathbb{R}}^{3}}|x|^{4}|\psi_{\varepsilon}|^{2}\
dx\end{split}$

for some constant $C>0.$ Therefore, we have shown that

$\begin{split}\frac{d}{dt}\int_{{\mathbb{R}}^{3}}|x|^{4}|\psi_{\varepsilon}|^{2}\
dx&\leq 4\int_{{\mathbb{R}}^{3}}|x|^{2}|\nabla\psi_{\varepsilon}|^{2}\
dx+(2C\mathcal{A}+4)\int_{{\mathbb{R}}^{3}}|x|^{4}|\psi_{\varepsilon}|^{2}\
dx.\end{split}$

Now notice that

$\begin{split}\int_{{\mathbb{R}}^{3}}|x|^{2}|\nabla\psi_{\varepsilon}|^{2}\
dx&=-\int_{{\mathbb{R}}^{3}}\overline{\psi_{\varepsilon}}\left(|x|^{2}\Delta\psi_{\varepsilon}+2x\cdot\nabla\psi_{\varepsilon}\right)\
dx\\\
&\leq\int_{{\mathbb{R}}^{3}}|x|^{2}|\psi_{\varepsilon}|^{2}+|\nabla\psi_{\varepsilon}|^{2}+|x|^{4}|\psi_{\varepsilon}|^{2}+|\partial_{t}\psi_{\varepsilon}|^{2}+|x|^{2}|\psi_{\varepsilon}|^{2}A\cdot
A\ dx\\\
&\quad+\rho(|u(t)|+1)\int_{{\mathbb{R}}^{3}}(|x|^{2}+|x|^{4})|\psi_{\varepsilon}|^{2}\
dx+\int_{{\mathbb{R}}^{3}}|x|^{4}|\psi_{\varepsilon}|^{2}+|\psi_{\varepsilon}|^{2}|W_{\operatorname{sing}}|\
dx\\\ &\quad+\lVert|\bullet|^{2}\psi_{\varepsilon}\rVert_{2}\lVert
AD_{\varepsilon}\psi_{\varepsilon}\rVert_{2}+\lVert|\bullet|^{2}\psi_{\varepsilon}\rVert_{2}\lVert
D_{-\varepsilon}A\psi_{\varepsilon}\rVert_{2}\\\
&\lesssim\lVert\psi_{\varepsilon}\rVert_{H_{2}}^{2}+\lVert\psi_{\varepsilon}\rVert_{H^{1}}^{2}+\lVert\partial_{t}\psi_{\varepsilon}\rVert_{2}^{2}.\end{split}$

By Lemma 2.4 and the conservation of $\lVert\psi_{\varepsilon}\rVert_{2},$ we
conclude that independent of $t$ and $\varepsilon$

$\frac{d}{dt}\int_{{\mathbb{R}}^{3}}(1+|x|^{4})|\psi_{\varepsilon}|^{2}\
dx\lesssim\lVert\psi_{\varepsilon}\rVert_{H_{2}}^{2}+\lVert\partial_{t}\psi_{\varepsilon}\rVert_{2}^{2}.$

Let
$E_{\varepsilon}(t)=\lVert\psi_{\varepsilon}\rVert_{H_{2}}^{2}+\lVert\partial_{t}\psi_{\varepsilon}\rVert_{2}^{2}$.
By integrating the previous line from 0 to $t$, we see that

$\int_{{\mathbb{R}}^{3}}(1+|x|^{4})|\psi_{\varepsilon}(t)|^{2}\
dx\lesssim\int_{0}^{t}E_{\varepsilon}(t)dt+\int_{{\mathbb{R}}^{3}}(1+|x|^{4})|\varphi_{0}|^{2}\
dx.$

Therefore,
$\lVert\psi_{\varepsilon}(t)\rVert_{H_{2}}^{2}\lesssim\int_{0}^{t}E_{\varepsilon}(t)dt+\lVert\varphi_{0}\rVert_{H_{2}}^{2}.$
We now aim to find a bound on
$\lVert\partial_{t}\psi_{\varepsilon}\rVert_{2}^{2}$ .

Let $\theta=\partial_{t}\psi_{\varepsilon}$ then from
$i\partial_{t}\theta=H_{B}^{\varepsilon}\theta+(W_{\operatorname{reg}}^{\varepsilon}+W_{\operatorname{sing}}^{\varepsilon})\theta+u^{\prime}(t)V_{\operatorname{con}}^{\varepsilon}\psi_{\varepsilon}+u(t)V_{\operatorname{con}}^{\varepsilon}\theta,$
we find that

$\frac{d}{dt}\lVert\theta\rVert_{2}^{2}\leq|u^{\prime}(t)|\rho\lVert\psi_{\varepsilon}\rVert_{H_{2}}^{2}+|u^{\prime}(t)|\lVert\theta\rVert_{2}^{2}.$

Therefore there exists $C_{\rho}$ depending only on $\rho$ such that

$\begin{split}\lVert\theta(t)\rVert_{2}^{2}&\leq\lVert\theta(0)\rVert_{2}^{2}+C_{\rho}\int_{0}^{t}|u^{\prime}(s)|E_{\varepsilon}(s)ds\end{split}$

and $\lVert\theta(0)\rVert_{2}=\lVert
H_{B}^{\varepsilon}\varphi_{0}+V^{\varepsilon}\varphi_{0}\rVert_{2}\lesssim\lVert\varphi_{0}\rVert_{H^{2}_{2}}.$

By combining these estimates, we get that there exists $C$ independent of $t$
and $\varepsilon$ such that

$E_{\varepsilon}(t)\lesssim\lVert\varphi_{0}\rVert_{H^{2}_{2}}^{2}+C\int_{0}^{t}(1+|u^{\prime}(s)|)E_{\varepsilon}(s)\
ds.$

Therefore by Gronwall’s inequality $E_{\varepsilon}(t)\lesssim
e^{\alpha+T}\lVert\varphi_{0}\rVert_{H^{2}_{2}}^{2}.$ Using Lemma 2.4 we see
we can bound $\lVert\psi_{\varepsilon}(t)\rVert_{H^{2}_{2}}$ by a multiple of
$\sqrt{E_{\varepsilon}(t)}$ , therefore there exists $C$ independent of $t$
and $\varepsilon$ such that
$\lVert\psi_{\varepsilon}(t)\rVert_{H^{2}_{2}}\lesssim\lVert\varphi_{0}\rVert_{H^{2}_{2}}.$
∎

We now treat the case of a homogeneous magnetic field:

######  Theorem 2.

With the same notation and same assumptions as in Lemma 2.3 but for
homogeneous magnetic fields with vector potential
$A(x)=\frac{B_{0}}{2}(-x_{2},x_{1},0),$ there exists a constant
$C_{T,\alpha,\rho}$ such that for any $\varphi_{0}\in H^{2}_{2}$, the equation

$\begin{split}i\partial_{t}\psi(x,t)&=(-i\nabla-A)^{2}\psi(x,t)+V\psi(x,t)\text{
with }\psi(\bullet,0)=\varphi_{0}\end{split}$

has a unique solution $\psi\in L^{\infty}(0,T;H^{2}_{2})$ satisfying
$\left\lVert\psi\right\rVert_{L^{\infty}(0,T;H^{2}_{2})}\leq
C_{T,\alpha,\rho}\lVert\varphi_{0}\rVert_{H^{2}_{2}}.$

###### Proof.

Let $A_{\varepsilon}(x):=\frac{B_{0}}{2\varepsilon}(-\tanh(\varepsilon
x_{2}),\tanh(\varepsilon x_{1}),0).$ We know from Theorem 1 that for all
$\varphi_{0}$ there exists $\psi_{\varepsilon}\in H^{2}_{2}$ solving

$\begin{split}i\partial_{t}\psi_{\varepsilon}(x,t)&=(-i\nabla-
A_{\varepsilon})^{2}\psi_{\varepsilon}(x,t)+V\psi_{\varepsilon}(x,t)\text{
with }\psi_{\varepsilon}(x,0)=\varphi_{0}(x).\end{split}$

Lemma 2.4 still holds for $\psi_{\varepsilon}$ since

$\begin{split}\lVert\Delta\psi_{\varepsilon}\rVert_{2}&\leq\lVert\partial_{t}\psi_{\varepsilon}\rVert_{2}+2\lVert
A_{\varepsilon}\nabla\psi_{\varepsilon}\rVert_{2}+\lVert A_{\varepsilon}\cdot
A_{\varepsilon}\psi_{\varepsilon}\rVert_{2}+\lVert
V\psi_{\varepsilon}\rVert_{2}\end{split}$ (2.10)

and integration by parts implies for $\kappa>0$ arbitrary

$\begin{split}\lVert
A_{\varepsilon}\nabla\psi_{\varepsilon}\rVert_{2}^{2}&\leq\int_{{\mathbb{R}}^{3}}\Big{(}|\nabla(A_{\varepsilon}\cdot
A_{\varepsilon})||\psi_{\varepsilon}||\nabla\psi_{\varepsilon}|+A_{\varepsilon}\cdot
A_{\varepsilon}|\psi_{\varepsilon}||\Delta\psi_{\varepsilon}|\Big{)}(x)dx\\\
&\leq\frac{B_{0}^{4}}{4\kappa}\lVert\psi_{\varepsilon}\rVert_{H_{1}}^{2}+\kappa\lVert\psi_{\varepsilon}\rVert_{H^{1}}^{2}+\frac{B_{0}^{2}}{4\kappa}\lVert\psi_{\varepsilon}\rVert_{H_{2}}^{2}+\kappa\lVert\psi_{\varepsilon}\rVert_{H^{2}}^{2}.\end{split}$

By picking $\kappa$ to be sufficiently small, it is clear in (2.10) that
$\lVert\psi_{\varepsilon}\rVert_{H^{2}}\lesssim
C\lVert\psi_{\varepsilon}\rVert_{H_{2}}+C\lVert\partial_{t}\psi_{\varepsilon}\rVert_{L_{2}}.$

Therefore there exists $C$ independent of $\varepsilon$ and $t$ such that
$\lVert\psi_{\varepsilon}\rVert_{H^{2}_{2}}\leq C\sqrt{E_{\varepsilon}(t)}$
where
$E_{\varepsilon}(t)=\lVert\psi_{\varepsilon}\rVert_{H_{2}}^{2}+\lVert\partial_{t}\psi_{\varepsilon}\rVert_{L_{2}}^{2}.$

To bound $\lVert\psi_{\varepsilon}\rVert_{H_{2}}$, we start from
$i\partial_{t}\psi_{\varepsilon}=(-i\nabla-
A_{\varepsilon})^{2}\psi_{\varepsilon}+V\psi_{\varepsilon}$ and multiply by
$|x|^{4}\overline{\psi_{\varepsilon}}$. Taking the imaginary part and
integrating over ${\mathbb{R}}^{3}$, we see that

$\begin{split}\operatorname{Re}\int_{{\mathbb{R}}^{3}}|x|^{4}(\overline{\psi_{\varepsilon}}\partial_{t}\psi_{\varepsilon})(x)\
dx&=\operatorname{Im}\int_{{\mathbb{R}}^{3}}-|x|^{4}(\overline{\psi_{\varepsilon}}\Delta\psi_{\varepsilon})(x)-2i|x|^{4}(\overline{\psi_{\varepsilon}}A_{\varepsilon}\cdot\nabla\psi_{\varepsilon})(x)\
dx.\end{split}$

Integration by parts yields

$\begin{split}\operatorname{Im}\int_{{\mathbb{R}}^{3}}2i|x|^{4}(\overline{\psi_{\varepsilon}}A_{\varepsilon}\cdot\nabla\psi_{\varepsilon})(x)\
dx&=\operatorname{Re}\int_{{\mathbb{R}}^{3}}-2\psi_{\varepsilon}(x)\nabla\cdot(|x|^{4}\overline{\psi_{\varepsilon}}(x)A_{\varepsilon}(x))\
dx\\\
&=\operatorname{Im}\int_{{\mathbb{R}}^{3}}-2i|\psi_{\varepsilon}(x)|^{2}A_{\varepsilon}(x)\cdot\nabla(|x|^{4})-2i\overline{\psi_{\varepsilon}(x)}|x|^{4}A_{\varepsilon}(x)\cdot\nabla\psi_{\varepsilon}(x)\
dx\end{split}$

which shows that

$\operatorname{Im}\int_{{\mathbb{R}}^{3}}2i|x|^{4}(\overline{\psi_{\varepsilon}}A_{\varepsilon}\cdot\nabla\psi_{\varepsilon})(x)\
dx=\operatorname{Im}\int_{{\mathbb{R}}^{3}}-i|\psi_{\varepsilon}(x)|^{2}A_{\varepsilon}(x)\cdot\nabla(|x|^{4})\
dx.$

Hence, this implies

$\begin{split}\frac{1}{2}\frac{d}{dt}\int_{{\mathbb{R}}^{3}}|x|^{4}|\psi_{\varepsilon}(x)|^{2}\
dx&=\operatorname{Im}\int_{{\mathbb{R}}^{3}}\nabla(|x|^{4}\overline{\psi_{\varepsilon}(x)})\cdot(\nabla\psi_{\varepsilon}(x))+i|\psi_{\varepsilon}(x)|^{2}A_{\varepsilon}(x)\cdot\nabla(|x|^{4})\
dx\\\ &\leq
4\int_{{\mathbb{R}}^{3}}|\psi_{\varepsilon}(x)||x|^{3}|\nabla\psi_{\varepsilon}(x)|+|\psi_{\varepsilon}(x)|^{2}|A_{\varepsilon}(x)||x|^{3}\
dx\\\
&\lesssim(|B_{0}|+1)\lVert\psi_{\varepsilon}\rVert_{H_{2}}^{2}+\lVert\psi_{\varepsilon}\rVert_{H^{2}}^{2}.\end{split}$

Therefore we obtain the estimate
$\lVert\psi_{\varepsilon}\rVert_{H_{2}}^{2}\lesssim\lVert\varphi_{0}\rVert_{H_{2}}^{2}+\int_{0}^{t}E_{\varepsilon}(s)\
ds.$

By exactly the same argument as last time we see that there exists $C_{\rho}$
depending only on $\rho$ such that

$\begin{split}\lVert\partial_{t}\psi_{\varepsilon}(t)\rVert_{2}^{2}&\leq\lVert\partial_{t}\psi_{\varepsilon}(0)\rVert_{2}^{2}+C_{\rho}\int_{0}^{t}|u^{\prime}(s)|E_{\varepsilon}(s)ds.\end{split}$

By combining these estimates, we get that there exists $C$ independent of $t$
and $\varepsilon$ such that

$E_{\varepsilon}(t)\lesssim\lVert\varphi_{0}\rVert_{H^{2}_{2}}^{2}+C\int_{0}^{t}(1+|u^{\prime}(s)|)E_{\varepsilon}(s)\
ds.$

Therefore by Gronwall’s inequality $E_{\varepsilon}(t)\lesssim
e^{\alpha+T}\lVert\varphi_{0}\rVert_{H^{2}_{2}}^{2}$ and thus
$\lVert\psi_{\varepsilon}(t)\rVert_{H^{2}_{2}}\lesssim\lVert\varphi_{0}\rVert_{H^{2}_{2}}.$
∎

Our next Lemma will be used to obtain an explicit decay estimate on the full
solution to the Schrödinger equation with magnetic field on a bounded domain:

######  Lemma 2.5.

There exists $C>0$ such that for all $R\geq r>2$ and $f\in
H^{2}_{2}(B_{R}(0))\cap H^{1}_{0}(B_{R}(0))$,

$\left|\int_{\partial B_{r}(0)}\overline{f(x)}\nabla_{n}f(x)\
dS(x)\right|\lesssim\frac{\lVert f\rVert_{H^{2}_{2}}^{2}}{r^{2}}.$

###### Proof.

We introduce the notation $A_{r_{1},r_{2}}:=B_{r_{2}}(0)\setminus
B_{r_{1}}(0)$ and $A_{r}:=A_{r,\infty}$. For all $R\geq r>2$:

$\begin{split}\inf_{r\in[s-1,s]}\int_{\partial B_{r}(0)}|f(x)\nabla_{n}f(x)|\
dS(x)&\leq\int_{s-1}^{s}\int_{\partial B_{r}(0)}|f(x)\nabla_{n}f(x)|\ dS(x)\
dr\\\ &\leq\int_{A_{s-1,s}}|f(x)\nabla f(x)|\ dx\\\
&\leq\left(\int_{A_{s-1,s}}|f(x)|^{2}\ dx\int_{A_{s-1,s}}|\nabla f(x)|^{2}\
dx\right)^{\frac{1}{2}}\\\ &\leq\frac{\lVert f\rVert_{H_{2}(B_{s})}\lVert
f\rVert_{H^{1}(B_{s})}}{(s-1)^{2}}.\end{split}$ (2.11)

Since $h(s)=\int_{\partial B_{s}(0)}|f(x)\nabla_{n}f(x)|\ dS(x)$ is
continuous, there exists $r_{s}\in[s-1,s]$ at which this infimum is attained.
Now notice that

$\begin{split}\left|\int_{\partial B_{s}(0)}\overline{f(x)}\nabla_{n}f(x)\
dS(x)\right|&\leq\left|\int_{A_{r_{s},s}}(\bar{f}\Delta f)(x)\
dx\right|+\left|\int_{A_{r_{s},s}}|\nabla f(x)|^{2}\ dx\right|\\\
&\qquad+\left|\int_{\partial B_{r_{s}}(0)}\bar{f}(x)\nabla_{n}f(x)\
dS(x)\right|\\\ &\leq\int_{A_{r_{s},s}}\left|(\bar{f}\Delta f)(x)\right|\
dx+\int_{A_{r_{s},s}}|\nabla f(x)|^{2}\ dx\\\ &+(s-1)^{-2}\lVert
f\rVert_{H_{2}(B_{s})}\lVert f\rVert_{H^{1}(B_{s})}.\end{split}$ (2.12)

It now remains to estimate the two last integrals in (2.12). We see that

$\begin{split}\left|\int_{A_{r_{s},s}}(\bar{f}\Delta f)(x)\
dx\right|&\leq\left(\int_{A_{r_{s},s}}|f(x)|^{2}\ dx\int_{A_{r_{s},s}}|\Delta
f(x)|^{2}\ dx\right)^{\frac{1}{2}}\\\
&\leq\frac{1}{(s-1)^{2}}\int_{A_{r_{s},s}}(s-1)^{4}|f(x)|^{2}\
dx+\frac{1}{(s-1)^{2}}\int_{A_{r_{s},s}}|\Delta f(x)|^{2}\ dx\\\
&\leq\frac{1}{(s-1)^{2}}\lVert f\rVert^{2}_{H^{2}_{2}(B_{s})}.\end{split}$
(2.13)

Here, we used that $f$ vanishes on $B_{R}(0)$ by the Dirichlet boundary
condition and estimates (2.12) and (2.13). Therefore, we find since
$n(x)=\frac{x}{|x|}$

$\begin{split}\left|\int_{\partial B_{r}(0)}(|\psi|^{2}A\cdot n)(x)\
dS(x)\right|&\leq\frac{\rho}{r}\int_{\partial B_{r}(0)}|\psi(x)|^{2}x\cdot n\
dS(x)=\frac{\rho}{r}\int_{B_{r}(0)}\nabla\cdot(|\psi(x)|^{2}x)\ dx\\\
&\leq\frac{\rho}{r}\left(3r^{-4}\int_{B_{r}(0)}|x|^{4}|\psi(x)|^{2}\
dx+2\int_{B_{r}(0)}|x||\psi(x)||\nabla\psi(x)|\ dx\right)\\\
&\leq\frac{\rho}{r}\Bigg{(}3r^{-4}\int_{B_{r}(0)}|x|^{4}|\psi(x)|^{2}\
dx+r^{-1}\int_{B_{r}(0)}|x|^{4}|\psi(x)|^{2}\ dx\\\
&+r^{-1}\int_{B_{r}(0)}|\nabla\psi(x)|^{2}\ dx\Bigg{)}\leq
r^{-2}C\lVert\psi\rVert^{2}_{H^{2}_{2}}.\end{split}$ (2.14)

Therefore there exists $C>0$ such that for all $R\geq s>2$, $f\in H^{2}_{2}$

$\begin{split}\left|\int_{\partial B_{s}(0)}\bar{f}(x)\nabla_{n}f(x)\
dS(x)\right|\leq\frac{\frac{1}{4}C}{(s-1)^{2}}\lVert
f\rVert^{2}_{H^{2}_{2}(B_{s})}\leq\frac{C}{s^{2}}\lVert
f\rVert^{2}_{H^{2}_{2}(B_{s})}\end{split}$ (2.15)

as required. ∎

We now want to compare the solution $\psi$, where $\psi$ solves the equation
$i\partial_{t}\psi=(H_{0}^{B}+V_{\operatorname{TD}})\psi$ on
${\mathbb{R}}^{3}$ and has initial condition $\varphi_{0}\in H^{2}_{2}$, to a
solution $\psi^{D}$, where $\psi^{D}$ solves the same equation on a bounded
domain $\Omega$ with Dirichlet boundary conditions and initial condition
$\varphi^{D}_{0}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$.

It is easy to check that the existence of such a solution on the bounded
domain $\Omega$ follows as in Theorem 1. The essential ingredient is the
existence of a self-adjoint magnetic Dirichlet Laplacian with domain
$H^{2}_{2}(\Omega)\cap H^{1}_{0}(\Omega)$ defined by the quadratic form

$q:H^{1}_{0}(\Omega)^{2}\rightarrow{\mathbb{R}}\text{ with
}q(\psi,\psi)=\langle(-i\nabla-A)\psi,(-i\nabla-A)\psi\rangle_{L^{2}}.$

######  Lemma 2.6.

With the same notation and constants as in Theorem 1, but now for any initial
state $\varphi_{0}\in H^{2}_{2}(\Omega)\cap H^{1}_{0}(\Omega)$ on $\Omega$
with Dirichlet boundary conditions, the equation

$\begin{split}i\partial_{t}\psi(x,t)&=(-i\nabla-A)^{2}\psi(x,t)+V\psi(x,t)\\\
\psi(x,0)&=\varphi_{0}(x)\end{split}$ (2.16)

has a unique solution $\psi\in L^{\infty}(0,T;H^{2}_{2})$, and this $\psi$
satisfies $\left\lVert\psi\right\rVert_{L^{\infty}(0,T;H^{2}_{2})}\leq
C_{T,\alpha,\rho}\lVert\varphi_{0}\rVert_{H^{2}_{2}}.$

Similarly, under the same notation and constants as in Lemma 2.3, we have for
the above equation
$\left\lVert\psi\right\rVert_{L^{\infty}(0,T;H^{2}_{2})}\leq
C_{T,\alpha,\rho,\mathcal{A}}\lVert\varphi_{0}\rVert_{H^{2}_{2}}.$

The difference $\xi=\psi-\psi^{D}$ on $\Omega$ solves
$i\partial_{t}\xi=(H_{0}^{B}+V_{\operatorname{TD}})\xi.$

Multiplying by $\bar{\xi}$, taking imaginary parts and integration by parts
gives for $n$ the unit normal

$\begin{split}\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\xi(x)|^{2}\
dx&=-\operatorname{Im}\left(\int_{\partial\Omega}(\overline{\xi}\nabla_{n}\xi)(x)\
dS\right)+\int_{\partial\Omega}|\psi(x)|^{2}(A\cdot n)(x)\ dS.\end{split}$

If $\Omega=B_{R}(0)$ and $A=\frac{B_{0}}{2}(-x_{2},x_{1},0),$ the potential of
a homogeneous field, then $A\cdot n=0.$ While instead if $A$ is a bounded
function with $\lVert A\rVert_{\infty}\leq\rho$, then

$\begin{split}\left|\int_{\partial B_{R}(0)}|\psi(x)|^{2}(A\cdot n)(x)\
dS(x)\right|&=\left\lvert\int_{B_{R}(0)}\nabla\cdot(|\psi|^{2}A)(x)\
dx\right\rvert\leq\left\lvert\rho\int_{B_{R}(0)}|\psi(x)|^{2}+2|(\psi\nabla\psi)(x)|\
dx\right\rvert\\\
&\leq\frac{\rho}{R}\left(3R^{-4}\int_{B_{R}(0)}|x|^{4}|\psi(x)|^{2}\
dx+2\int_{B_{R}(0)}|x||\psi(x)||\nabla\psi(x)|\ dx\right)\\\
&\leq\frac{\rho}{R}\Bigg{(}3R^{-4}\int_{B_{R}(0)}|x|^{4}|\psi(x)|^{2}\
dx+R^{-1}\int_{B_{R}(0)}|x|^{4}|\psi(x)|^{2}\ dx\\\
&+R^{-1}\int_{B_{R}(0)}|\nabla\psi(x)|^{2}\ dx\Bigg{)}\lesssim
R^{-2}\lVert\psi\rVert^{2}_{H^{2}_{2}}.\end{split}$

Now notice by Lemma 2.5

$\begin{split}\int_{\partial B_{R}(0)}\overline{\xi(x)}\nabla_{n}\xi(x)\
dS(x)&\lesssim\frac{\lVert\psi-\psi^{D}\rVert^{2}_{H^{2}_{2}}}{R^{2}}\lesssim\frac{\lVert\varphi_{0}\rVert^{2}_{H^{2}_{2}}+\lVert\varphi_{0}^{D}\rVert^{2}_{H^{2}_{2}}}{R^{2}}.\end{split}$

Thus, we have proven the following Theorem

######  Theorem 3.

Let $\psi$ be the solution to the linear Schrödinger equation with magnetic
field (1.1) with potentials and magnetic vector potentials satisfying
Assumption 1. Then, there is some absolute constant $C_{\rho,T}$ such that for
any $R$ sufficiently large, where $\psi^{D}$ is the solution to the Dirichlet
problem (2.16) on $B_{R}(0),$

$\|\psi-\psi^{D}\|_{2}\leq C_{\rho,T}/R^{2}.$

##  3\. Hartree equation

We now turn to the evolution of an effective single particle described by the
Hartree equation with Schrödinger operator
$H_{0}=(-i\nabla-A)^{2}+V:D(H_{0})\subset L^{2}(\mathbb{R}^{3})\rightarrow
L^{2}(\mathbb{R}^{3})$, static _pinning potential_ $V$, _control potential_
$V_{\operatorname{con}},$ and time-dependent control function $u\in
W^{1,1}(0,T)$ such that
$V_{\operatorname{TD}}(t):=u(t)V_{\operatorname{con}}:$

$\begin{split}i\partial_{t}\psi(x,t)&=(H_{0}+V_{\operatorname{TD}}(t))\psi(x,t)+\left(|\psi(\bullet,t)|^{2}*\frac{1}{|\bullet|}\right)(x)\psi(x,t),\quad(x,t)\in\mathbb{R}^{3}\times(0,T)\\\
\psi(\bullet,0)&=\varphi_{0}.\end{split}$ (3.1)

under Assumption 1 on the potentials.

######  Remark 4.

Since we discussed the magnetic field and its complications already in the
first part of this article, we shall neglect it in the treatment of the
Hartree equation in the sequel to simplify the presentation.

###  3.1. Local Existence of Solutions to the Hartree equation

We first need to show that equation (3.1) has a unique solution on
${\mathbb{R}}^{3}\times[0,T]$. To do this we start by showing that a unique
local solution exists and use an energy estimate to show the local solution
can be extended into a unique global solution.

To show the existence of the local solution we collect in the following Lemma
some basic estimates on the Hartree non-linearity, as in [B05, Lemma $2.3$],
that we shall frequently use throughout this section:

######  Lemma 3.1.

Let $\Omega\subset\mathbb{R}^{3}$ be a domain and $\psi\in H^{1}(\Omega)$, we
then define $F_{\Omega}(\psi)(x)=\int_{\Omega}\frac{|\psi(x)|^{2}}{|x-y|}\ dy\
\psi(x)$ and just write $F:=F_{\mathbb{R}^{3}}.$ Then, there are constants
$c,C_{F}>0$ independent of $\Omega$ such that

  1. (1)

For all $\psi,\phi\in H^{1}(\Omega)$ we have
$||F_{\Omega}(\psi)-F_{\Omega}(\phi)||_{2}\leq
c(||\psi||^{2}_{H^{1}}+||\phi||^{2}_{H^{1}})||\psi-\phi||_{2}.$

  2. (2)

There is $C_{F}>0$ such that for all $\psi,\phi\in H^{2}_{2}(\Omega)$

$\begin{split}||F_{\Omega}(\psi)-F_{\Omega}(\phi)||_{H^{2}_{2}}&\leq
C_{F}(||\psi||^{2}_{H^{1}}+||\phi||^{2}_{H^{2}_{2}})||\psi-\phi||_{H^{2}_{2}}\text{
and }\\\ ||F_{\Omega}(\psi)||_{H^{2}_{2}}&\leq
C_{F}||\psi||^{2}_{H^{1}}||\psi||_{H^{2}_{2}}.\end{split}$

For the linear part of the nonlinear evolution equation (3.1), Theorem 1 and
Lemma 2.6 guarantee the existence of evolution operators associated with the
time-dependent Hamiltonian

$H(t)=(-i\nabla-A)^{2}+W_{\operatorname{reg}}+W_{\operatorname{sing}}+u(t)V_{\operatorname{con}}$

which is a family $\\{U(t,s):t,s\in[0,T]\\}$ on $H^{2}_{2}(\Omega)\cap
H^{1}_{0}(\Omega)$ such that for all $\psi_{0}\in H^{2}_{2}(\Omega)\cap
H^{1}_{0}(\Omega)$ the following properties hold

  1. (1)

$U(t,s)U(s,r)\psi_{0}=U(t,r)\psi_{0}$ and $U(t,t)\psi_{0}=\psi_{0},$

  2. (2)

$(t,s)\mapsto U(t,s)\psi_{0}$ is strongly continuous in $L^{2}$ on $[0,T]^{2}$
and $U(t,s)$ is an isometry on $L^{2}(\Omega),$

  3. (3)

For all $s,t\in[0,T],U(t,s)\in\mathcal{L}(H^{2}_{2})$ and $(t,s)\mapsto
U(t,s)\psi_{0}$ is weakly continuous from $[0,T]^{2}$ to $H^{2}_{2}(\Omega)$.
Moreover there is

$M=M(T,||u||_{W}^{1,1},||W_{\operatorname{sing}}||_{L^{p}},||\langle\bullet\rangle^{-2}W_{\operatorname{reg}}||_{L^{\infty}},||\langle\bullet\rangle^{-2}V_{\operatorname{con}}||_{L^{\infty}})$

such that $||U(t,s)\psi||_{H^{2}_{2}}\leq M||\psi||_{H^{2}_{2}}.$

  4. (4)

In the $L^{2}(\Omega)$-sense, we have
$i\partial_{t}U(t,s)\psi_{0}=H(t)U(t,s)\psi_{0}$ and
$i\partial_{s}U(t,s)\psi_{0}=U(t,s)H(t)\psi_{0}.$

The evolution operators exist by Theorem 1.

Now we can prove the existence and uniqueness of a local solution using a
simple contraction argument on $\mathbb{R}^{3}:$

######  Lemma 3.2.

For $T_{0}>0$ small enough, there exists a unique solution in
$H^{2}_{2}(\Omega)$ to

$\begin{split}i\partial_{t}\psi(x,t)&=(H_{0}+V_{\operatorname{TD}}(t))\psi(x,t)+\left(|\psi(\bullet,t)|^{2}*\frac{1}{|\bullet|}\right)(x)\psi(x,t),\quad(x,t)\in\mathbb{R}^{3}\times(0,T_{0})\\\
\psi(\bullet,0)&=\varphi_{0}.\end{split}$

###### Proof.

Consider the functional $\Lambda:\psi\mapsto
U(\bullet,0)\psi_{0}-i\int_{0}^{\bullet}U(\bullet,s)F(\psi(s))ds$

and the set $B=\\{\psi\in
L^{\infty}((0,\tau);H^{2}_{2}),||\psi||_{L^{\infty}(0,\tau;H^{2}_{2})}\leq
2M||\psi_{0}||_{H^{2}_{2}}\\}.$

We want $\tau$ to be sufficiently small that $\Lambda$ is a contraction on
$B$, i.e. for $\psi\in B$ and Lemma 3.1

$\displaystyle||\Lambda(\psi)(t)||_{H^{2}_{2}}$ $\displaystyle=\left\lVert
U(t,0)\psi_{0}-i\int_{0}^{t}U(t,s)F(\psi(s))ds\right\rVert_{H^{2}_{2}}$
$\displaystyle\leq M||\psi_{0}||_{H^{2}_{2}}+8\tau
M^{4}C_{F}||\psi_{0}||_{H^{2}_{2}}.$

So we need to choose $\tau>0$ sufficiently small that $8\tau
C_{F}M^{3}||\psi_{0}||^{2}_{H^{2}_{2}}<1$, and then

$\begin{split}||\Lambda(\psi)||_{L^{\infty}(0,\tau;H^{2}_{2})}\leq
2M||\psi_{0}||_{H^{2}_{2}}.\end{split}$

So $\Lambda(\psi)\in B$, and thus $\Lambda$ maps $B$ into itself. Now, for
$\psi,\phi\in B$ then for all times $t\in[0,\tau]$, using again Lemma 3.1

$\displaystyle||\Lambda(\psi)(t)-\Lambda(\phi)(t)||_{H^{2}_{2}}$
$\displaystyle=\left\lVert\int_{0}^{t}U(t,s)(F(\psi(s))-F(\phi(s)))\
ds\right\rVert_{H^{2}_{2}}$ $\displaystyle\leq 8\tau
C_{F}M^{3}||\psi_{0}||^{2}_{H^{2}_{2}}||\psi-\phi||_{L^{\infty}([0,\tau];H^{2}_{2})}.$

So as $8\tau C_{F}M^{3}||\psi_{0}||^{2}_{H^{2}_{2}}<1$, $\Lambda$ is a
contraction on $B$ and so there is a unique solution to

$\begin{split}\psi(t)=U(t,0)\psi_{0}-i\int_{0}^{t}U(t,s)F(\psi(s))ds.\end{split}$

So if we apply $\partial_{t}$ to this equation, and using that
$i\partial_{t}U(t,s)\psi_{0}=H(t)U(t,s)\psi_{0}$ we get

$\begin{split}\partial_{t}\psi&=i\Delta\psi-iW_{\operatorname{reg}}\psi-
iW_{\operatorname{sing}}\psi-iu(t)V_{\operatorname{con}}\psi-
iF(\psi(t))\end{split}$

and so $\psi\in L^{\infty}((0,\tau);H^{2}_{2})$ means that $F(\psi)\in
L^{\infty}((0,\tau);H^{2}_{2})$ and $\Delta\psi\in
L^{\infty}((0,\tau);L^{2})$. We then recall the elementary bounds, which
follow from Assumption 1,

$\begin{split}||W_{\operatorname{reg}}\psi(t)||_{2}&\leq\left\lVert\frac{W_{\operatorname{reg}}}{1+|\bullet|^{2}}\right\rVert_{L^{\infty}}||\psi(t)||_{H_{2}}\text{
and
}||u(t)V_{\operatorname{con}}\psi(t)||_{2}\leq\left\lVert\frac{u(t)V_{\operatorname{con}}}{1+|\bullet|^{2}}\right\rVert_{L^{\infty}}||\psi(t)||_{H_{2}}.\end{split}$

Now using [BH20, Remark 3.8] $W_{\operatorname{sing}}\in
L^{2}(\mathbb{R}^{3})$ is infinitesimally bounded with respect to the negative
Laplacian. Thus, for any $\varepsilon>0$ there is $C_{\varepsilon}>0$ such
that

$\begin{split}||W_{\operatorname{sing}}\psi(t)||_{2}\leq\varepsilon||-\Delta\psi(t)||_{2}+C_{\varepsilon}||\psi(t)||_{2}.\end{split}$

This implies that
$||W_{\operatorname{sing}}\psi(t)||_{2}\lesssim||\psi(t)||_{H^{2}_{2}}.$ By
combining the previous bounds, we find $||\partial_{t}\psi(t)||_{2}\leq
C_{2}||\psi_{0}||_{H^{2}_{2}}.$

So it remains to prove that the solution is unique. Let $\psi$ and
$\psi^{\prime}$ be two solutions and $\phi=\psi-\psi^{\prime}$. Then
$\phi(0)=0$ and by subtracting the Schrödinger equation for each of the
solutions, multiplying by $\overline{\phi}$, integrating over
${\mathbb{R}}^{3}$ and taking the imaginary part we get

$\begin{split}\operatorname{Im}\int_{{\mathbb{R}}^{3}}i(\partial_{t}\phi)\overline{\phi}\
dx&=\operatorname{Im}\int_{{\mathbb{R}}^{3}}\left((-\Delta+W_{\operatorname{sing}}+W_{\operatorname{reg}}+u(t)V_{\operatorname{con}})\phi+F(\psi)-F(\psi^{\prime})\right)\overline{\phi}\
dx.\end{split}$

This shows that

$\begin{split}\frac{1}{2}\frac{d}{dt}||\phi||^{2}_{2}&\lesssim||\phi||_{2}(||\psi||^{2}_{H^{1}}+||\psi^{\prime}||^{2}_{H^{1}})||\psi-\psi^{\prime}||_{2}\lesssim||\phi||^{2}_{2}.\end{split}$

Then by Gronwall’s inequality $||\phi||^{2}_{2}=0$ so $\psi=\psi^{\prime}$ in
$L^{2}$ and so the solution is unique. ∎

###  3.2. Energy Estimate

To extend our local solution to a global one we need an energy estimate of the
solution of the Hartree equation for any arbitrary time $T$ and show that the
energy is bounded on our interval $[0,T]$.

Multiplying the equation by $\partial_{t}\overline{\psi}$, integrating over
$\mathbb{R}^{3}$, and taking the real part shows

$\operatorname{Re}\frac{d}{dt}\int_{\mathbb{R}^{3}}|\nabla\psi|^{2}\
dx+\operatorname{Re}\int_{\mathbb{R}^{3}}\frac{1}{2}(W_{\operatorname{sing}}+W_{\operatorname{reg}}+u(t)V_{\operatorname{con}})\partial_{t}(|\psi|^{2})+F(\psi)\partial_{t}\overline{\psi}\
dx=0.$

Now to deal with the last term, we calculate

$\begin{split}\frac{d}{dt}\int_{{\mathbb{R}}^{3}}F(\psi)\overline{\psi}\
dx&=\int_{{\mathbb{R}}^{3}}\left(|\psi|^{2}*\frac{1}{|\bullet|}\right)(y)\partial_{t}|\psi(y,t)|^{2}\
dy+\int_{{\mathbb{R}}^{3}}\left(|\psi|^{2}*\frac{1}{|\bullet|}\right)(x)\partial_{t}|\psi(x,t)|^{2}\
dx\\\
&=4\operatorname{Re}\int_{{\mathbb{R}}^{3}}\left(|\psi|^{2}*\frac{1}{|\bullet|}\right)(x)\psi(x,t)\partial_{t}\overline{\psi(x,t)}\
dx.\end{split}$

From this we conclude that

$\begin{split}2\frac{d}{dt}\int_{\mathbb{R}^{3}}|\nabla\psi(x)|^{2}\
dx+2\int_{\mathbb{R}^{3}}(W_{\operatorname{sing}}+W_{\operatorname{reg}}+u(t)V_{\operatorname{con}})\partial_{t}(|\psi|^{2})\
dx+\frac{d}{dt}\int_{\mathbb{R}^{3}}F(\psi)\overline{\psi}\ dx=0.\end{split}$

This in turn leads to

$\begin{split}\frac{d}{dt}\int_{\mathbb{R}^{3}}\left(|\nabla\psi|^{2}+(W_{\operatorname{sing}}+W_{\operatorname{reg}}+u(t)V_{\operatorname{con}})|\psi|^{2}+\frac{1}{2}F(\psi)\overline{\psi}\right)\
dx&=\int_{\mathbb{R}^{3}}u^{\prime}(t)V_{\operatorname{con}}|\psi|^{2}\ dx\\\
&\leq\left\|\frac{u^{\prime}(t)V_{\operatorname{con}}}{1+|\bullet|^{2}}\right\rVert_{\infty}||\psi||_{H_{1}}^{2}.\end{split}$

Now by taking (3.1) and multiplying it by $(1+|x|^{2})\overline{\psi}$, we can
take the imaginary part to get

$\begin{split}\operatorname{Im}i\partial_{t}\psi(1+|x|^{2})\overline{\psi}&=-\operatorname{Im}(1+|x|^{2})\overline{\psi}\Delta\psi\text{
and
}(1+|x|^{2})\partial_{t}(|\psi|^{2})=-\operatorname{Im}(1+|x|^{2})\overline{\psi}\Delta\psi.\end{split}$

Integrating over $\mathbb{R}^{3}$, we thus get

$\begin{split}\frac{d}{dt}\int_{\mathbb{R}^{3}}(1+|x|^{2})|\psi(x)|^{2}\
dx&\lesssim\int_{\mathbb{R}^{3}}|\nabla\psi(x)|^{2}\
dx+\int_{\mathbb{R}^{3}}|x|^{2}|\psi(x)|^{2}\ dx.\end{split}$

So now we can define $E$ at time $t\in[0,T]$ by

$\begin{split}E(t):=\int_{\mathbb{R}^{3}}|\nabla\psi(x,t)|^{2}+\lambda(1+|x|^{2})|\psi(x,t)|^{2}+\frac{1}{2}\left(|\psi(t)|^{2}*\frac{1}{|\bullet|}\right)(x)|\psi(x,t)|^{2}\
dx.\end{split}$

Here, $\lambda>0$ is a constant to be determined later in such a way to allow
$E(t)$ to be bounded

$\begin{split}E^{\prime}(t)&\leq\left\lVert\frac{u^{\prime}(t)V_{\operatorname{con}}}{1+|\bullet|^{2}}\right\rVert_{\infty}||\psi||_{H_{1}}^{2}+\frac{d}{dt}\int_{\mathbb{R}^{3}}(W_{\operatorname{sing}}+W_{\operatorname{reg}}+u(t)V_{\operatorname{con}})(x)|\psi(x,t)|^{2}\
dx\\\ &\quad+\lambda C\left(\int_{\mathbb{R}^{3}}|\nabla\psi(x,t)|^{2}\
dx+\int_{\mathbb{R}^{3}}|x|^{2}|\psi(x,t)|^{2}\ dx\right)\\\
&\leq\left(\left\lVert\frac{u^{\prime}(t)V_{\operatorname{con}}}{1+|\bullet|^{2}}\right\rVert_{\infty}+\lambda
C\right)E(t)+\frac{d}{dt}\int_{\mathbb{R}^{3}}(W_{\operatorname{sing}}+W_{\operatorname{reg}}+u(t)V_{\operatorname{con}})(x)|\psi(x,t)|^{2}\
dx.\end{split}$

Integrating over $(0,t)$ we get

$\begin{split}E(t)&\leq\int_{0}^{t}\left(\left\lVert\frac{u^{\prime}(s)V_{\operatorname{con}}}{1+|\bullet|^{2}}\right\rVert_{\infty}+\lambda
C\right)E(s)\ ds\\\
&+\int_{\mathbb{R}^{3}}(W_{\operatorname{sing}}(x)+W_{\operatorname{reg}}(x)+u(t)V_{\operatorname{con}}(x)|\psi(x,t)|^{2}-u(0)V_{\operatorname{con}}(x)|\psi(x,0)|^{2})\
dx.\end{split}$ (3.2)

Now we use again the infinitesimal boundedness with respect to the negative
Laplacian, together with [RS75, Theorem X.18] to find that for all
$\varepsilon>0$

$\begin{split}|\langle\psi,W_{\operatorname{sing}}\psi\rangle_{2}|&\leq\varepsilon|\langle\psi,\Delta\psi\rangle_{2}|+C_{\varepsilon}\langle\psi,\psi\rangle_{2}=\varepsilon||\nabla\psi||_{2}^{2}+C_{\varepsilon}||\psi||_{2}^{2}.\end{split}$

From this and (3.2), we obtain the estimate

$\begin{split}E(t)&\leq\int_{0}^{t}\left(\left\lVert\frac{u^{\prime}(s)V_{\operatorname{con}}}{1+|\bullet|^{2}}\right\rVert_{\infty}+\lambda
C\right)E(s)\
ds+\varepsilon||\nabla\psi(0)||_{2}^{2}+C_{\varepsilon}||\psi(0)||_{2}^{2}\\\
&+\left\lVert\frac{W_{\operatorname{reg}}}{1+|\bullet|^{2}}\right\rVert_{L^{\infty}}||\psi(0)||_{H_{1}}^{2}+\left\lVert\frac{u(0)V_{\operatorname{con}}}{1+|\bullet|^{2}}\right\rVert_{L^{\infty}}||\psi(0)||_{H_{1}}^{2}+\varepsilon||\nabla\psi(t)||_{2}^{2}\\\
&+C_{\varepsilon}||\psi(t)||_{2}^{2}+\left\lVert\frac{W_{\operatorname{reg}}}{1+|\bullet|^{2}}\right\rVert_{L^{\infty}}||\psi(t)||_{H_{1}}^{2}+\left\|\frac{u(t)V_{\operatorname{con}}}{1+|\bullet|^{2}}\right\rVert_{L^{\infty}}||\psi(t)||_{H_{1}}^{2}.\end{split}$

Taking $\varepsilon=\frac{1}{2}$ we find by the preservation of the $L^{2}$
norm of $\psi$, and since $u\in W_{\operatorname{pcw}}^{1,1}[0,T]$ that

$\begin{split}E(t)&\leq
C+\int_{0}^{t}\left(\left\lVert\frac{u^{\prime}(s)V_{\operatorname{con}}}{1+|\bullet|^{2}}\right\rVert_{\infty}+\lambda
C\right)E(s)\
ds+\frac{||\nabla\psi(t)||_{2}^{2}}{2}+C^{\prime}||\psi(t)||_{H_{1}}^{2}.\end{split}$
(3.3)

Noting that $E(0)\lesssim||\psi_{0}||^{2}_{H^{1}\cap
H_{1}}+||\psi_{0}||_{H^{1}}||\psi_{0}||_{2}^{3},$ we define $G$ for times
$t\in[0,T]$ by

$\begin{split}G(t)&:=\int_{\mathbb{R}^{3}}|\nabla\psi(t,x)|^{2}dx+\int_{\mathbb{R}^{3}}(1+|x|^{2})|\psi(t,x)|^{2}dx+\int_{\mathbb{R}^{3}}\left(|\psi(t,\bullet)|^{2}*\frac{1}{|\bullet|}\right)(x)|\psi(t,x)|^{2}dx\end{split}$

then taking $\lambda$ to be $C^{\prime}+\frac{1}{2}$ we can subtract
$\frac{1}{2}||\nabla\psi(t)||_{2}^{2}+C^{\prime}||\psi(t)||_{H_{1}}^{2}$ from
(3.3) to see that

$\begin{split}G(t)\leq
C+2\int_{0}^{t}\left(\left\lVert\frac{u^{\prime}(s)V_{\operatorname{con}}}{1+|\bullet|^{2}}\right\rVert_{\infty}+\lambda
C\right)G(s)\ ds.\end{split}$

Therefore, by Gronwall’s inequality
$G(t)\lesssim\exp\left(\int_{0}^{t}\beta(s)ds\right),$ where
$\beta(t)=\left\lVert\frac{u^{\prime}(t)V_{\operatorname{con}}}{1+|\bullet|^{2}}\right\rVert_{\infty},$
and so there exists $C_{T,\psi_{0}}>0$ such that for all $t\in[0,T]$:

$\begin{split}||\psi(t)||^{2}_{H^{1}\cap
H_{1}}+\int_{\mathbb{R}^{3}}\left(|\psi(x,t)|^{2}*\frac{1}{|\bullet|}\right)|\psi(x,t)|^{2}\
dx\leq C_{T,\psi_{0}}.\end{split}$

###  3.3. Global Existence

Now that we have an estimate for the energy we can use that the equation is
equivalent to the integral equation

$\begin{split}\psi(t)=U(t,0)\psi_{0}-i\int_{0}^{t}U(t,s)F(\psi(s))\
ds.\end{split}$

Thus, the claim follows from Grönwall’s inequality applied to

$\begin{split}||\psi(t)||_{H^{2}_{2}}&\lesssim||\psi_{0}||_{H^{2}_{2}}+\int_{0}^{t}||F(\psi(s))||_{H^{2}_{2}}\
ds\\\
&\lesssim||\psi_{0}||_{H^{2}_{2}}+\int_{0}^{t}||\psi(s)||_{H^{1}}^{2}||\psi(s)||_{H^{2}_{2}}\
ds\lesssim 1+\int_{0}^{t}||\psi(s)||_{H^{2}_{2}}\ ds.\end{split}$

Thus, we have shown that

######  Theorem 4.

With the same notation and same assumptions as in Lemma 2.3 there exists a
constant $C_{T}$ such that for any $\varphi_{0}\in H^{2}_{2}$, equation (3.1)

$\begin{split}i\partial_{t}\psi(x,t)&=(-i\nabla-A)^{2}\psi(x,t)+V\psi(x,t)\text{
with }\psi(\bullet,0)=\varphi_{0}\end{split}$

has a unique solution $\psi\in L^{\infty}(0,T;H^{2}_{2})$ satisfying
$\left\lVert\psi\right\rVert_{L^{\infty}(0,T;H^{2}_{2})}\leq
C_{T,\alpha,\rho}\lVert\varphi_{0}\rVert_{H^{2}_{2}}.$

###  3.4. Existence and uniqueness of a solution on a ball

We now look on a bounded open domain with piecewise smooth boundary
$\Omega\subset{\mathbb{R}}^{3}.$

To approximate the global dynamics on a bounded domain, we consider now the
operator $H_{0}=-\Delta+V:D(H_{0})\subset L^{2}(\Omega)\rightarrow
L^{2}(\Omega)$ and take potentials as before, but now restricted to the domain
$\Omega$ with Dirichlet boundary condition. We then study the equation

$\begin{split}i\partial_{t}u(x,t)&=(H_{0}+V_{\operatorname{TD}}(t))u(x,t)+\int_{\Omega}\frac{|u(y,t)|^{2}}{|x-y|}\
dy\ u(x,t),\quad(x,t)\in\Omega\times(0,T)\\\
u(x,t)&=0\quad(x,t)\in\partial\Omega\times(0,T)\text{ such that
}u(\bullet,0)=u_{0}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega).\end{split}$ (3.4)

It is easy to see that Theorem 4 holds true with the same constant
$C_{T,\alpha,\rho}$ for the Dirichlet problem.

###  3.5. Reduction to bounded domains

Now we need to show that the solution on the bounded domain tends in
$L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$ to the solution on
${\mathbb{R}}^{3}$ as the radius of the ball tends to $\infty$. So first we
prove the equivalent of [BH20, Lemma 7.1] for our version of the Schrödinger
equation.

######  Lemma 3.3.

(Perturbation of singular potentials & initial states). Let
$\widetilde{W}_{\operatorname{sing}},W_{\operatorname{sing}},W_{\operatorname{reg}},V_{\operatorname{con}},u$
satisfy the assumptions in Assumption 1, and $\widetilde{\psi_{0}},\psi_{0}\in
H_{2}^{2}({\mathbb{R}}^{3})$ be two initial states. Then the solution to

$\begin{split}i\partial_{t}\widetilde{\psi}(x,t)&=(-\Delta+\widetilde{W}_{\operatorname{sing}}+\widetilde{V}_{\operatorname{TD}}(t))\widetilde{\psi}(x,t)+\left(|\widetilde{\psi}(\bullet,t)|^{2}*\frac{1}{|\bullet|}\right)(x)\widetilde{\psi}(x,t)\\\
\widetilde{\psi}(\bullet,0)&=\widetilde{\psi_{0}}.\end{split}$

converges in $L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$ to the solution of

$\begin{split}i\partial_{t}{\psi}(x,t)&=(-\Delta+{W_{\operatorname{sing}}}+V_{\operatorname{TD}}(t)){\psi}(x,t)+\left(|{\psi}(\bullet,t)|^{2}*\frac{1}{|\bullet|}\right)(x){\psi}(x,t)\\\
{\psi}(\bullet,0)&=\psi_{0}\end{split}$

as $\widetilde{W}_{\operatorname{sing}}\to_{2}W_{\operatorname{sing}}$,
$\widetilde{V}_{\operatorname{con}}\to_{2}V_{\operatorname{con}}$, and
$\widetilde{\psi_{0}}\to_{2}\psi_{0}$. Furthermore there is

$\begin{split}C=C(T,||\psi||_{L^{\infty}((0,T),H^{2}({\mathbb{R}}^{d}))},||\widetilde{\psi}||_{L^{\infty}((0,T),H^{2}({\mathbb{R}}^{d}))})\end{split}$

such that

$\begin{split}||\psi-\widetilde{\psi}||_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}\leq
C\left(||W_{\operatorname{sing}}-\widetilde{W}_{\operatorname{sing}}||_{L^{2}({\mathbb{R}}^{3})}+||V_{\operatorname{con}}-\widetilde{V}_{\operatorname{con}}||_{2}+||\psi_{0}-\widetilde{\psi_{0}}||_{L^{2}({\mathbb{R}}^{3})}\right).\end{split}$

###### Proof.

Let $\xi=\psi-\widetilde{\psi}$ and subtract the two equations to get

$\begin{split}i\partial_{t}\xi=\left(-\Delta+W_{\operatorname{reg}}+\widetilde{W}_{\operatorname{sing}}+V_{\operatorname{TD}}(t)\right)\xi+(W_{\operatorname{sing}}-\widetilde{W}_{\operatorname{sing}})\psi+(F(\psi)-F(\widetilde{\psi}))\end{split}$

with initial condition $\xi(0)=\psi_{0}-\widetilde{\psi_{0}}$. Multiply by
$\overline{\xi}$, integrate over ${\mathbb{R}}^{3}$ and take the imaginary
part to get

$\begin{split}\frac{1}{2}\frac{d}{dt}\int_{{\mathbb{R}}^{3}}|\xi|^{2}&=\operatorname{Im}\int_{{\mathbb{R}}^{3}}\left((W_{\operatorname{sing}}-\widetilde{W}_{\operatorname{sing}})\psi+F(\psi)-F(\widetilde{\psi})\right)\overline{\xi}\\\
&\leq||\xi||_{L^{2}({\mathbb{R}}^{3})}\left(\left\lVert(W_{\operatorname{sing}}-\widetilde{W}_{\operatorname{sing}})\psi\right\rVert_{L^{2}({\mathbb{R}}^{3})}+\left\lVert
F(\psi)-F(\widetilde{\psi})\right\rVert_{L^{2}({\mathbb{R}}^{3})}\right).\end{split}$

Now note that

$\begin{split}\left\lVert(W_{\operatorname{sing}}-\widetilde{W}_{\operatorname{sing}})\psi\right\rVert_{2}\lVert\xi\rVert_{2}&\leq\left\lVert
W_{\operatorname{sing}}-\widetilde{W}_{\operatorname{sing}}\right\rVert_{2}^{2}\lVert\psi\rVert^{2}_{H^{2}}/2+\lVert\xi\rVert_{2}^{2}/2.\end{split}$

Integrating in time and using that $\psi\in
L^{\infty}((0,T),H^{2}({\mathbb{R}}^{3}))$, Lemma 3.1 yields

$\begin{split}||\xi(t)||^{2}_{L^{2}({\mathbb{R}}^{3})}&\leq||\xi(0)||^{2}_{L^{2}({\mathbb{R}}^{3})}+\left\lVert
W_{\operatorname{sing}}-\widetilde{W}_{\operatorname{sing}}\right\rVert^{2}_{2}\int_{0}^{t}||\psi(s)||^{2}_{H^{2}}ds+\int_{0}^{t}C||\xi(s)||_{2}^{2}ds\\\
&\leq||\xi(0)||^{2}_{L^{2}({\mathbb{R}}^{3})}+Ct\left\lVert
W_{\operatorname{sing}}-\widetilde{W}_{\operatorname{sing}}\right\rVert^{2}_{2}+\int_{0}^{t}C||\xi(s)||_{2}^{2}ds\end{split}$

where the constants depend on
$||\psi||_{L^{\infty}((0,T),H^{2}({\mathbb{R}}^{d}))},||\widetilde{\psi}||_{L^{\infty}((0,T),H^{2})}$.

Now Grönwall’s inequality implies that

$\begin{split}||\xi||_{L^{\infty}((0,T),L^{2}({\mathbb{R}}^{3}))}\lesssim\left\lVert
W_{\operatorname{sing}}-\widetilde{W}_{\operatorname{sing}}\right\rVert^{2}_{2}+||\psi_{0}-\widetilde{\psi_{0}}||_{L^{2}({\mathbb{R}}^{3})}.\end{split}$

The control potential can be included in a similar way. ∎

Since $u\in W^{1,1}_{\operatorname{pcw}}$ it suffices to treat from now on the
case that $u$ is constant, since $u$ can be approximated by piecewise constant
functions.

Now, we are ready to prove the reduction to a boundary value problem on a
bounded domain.

######  Theorem 5.

Let $\psi_{0}\in H^{2}_{2}(B_{R}(0))\cap H^{1}_{0}(B_{R}(0))$ be an initial
state, $T\in(0,\infty)$ and $V,V_{\operatorname{con}},u$ as in Assumption 1.
Then the equation with solution $\psi$ on $\mathbb{R}^{3}$ can be approximated
by $\psi^{D}$, the solution to the Dirichlet boundary value problem on
$B_{R}(0)$ for $R>0$. In particular, the difference $\xi=\psi-\psi^{D}$
satisfies, for some constant

$C=C(T,||u||_{W^{1,1}_{\operatorname{pcw}}(0,T)},||W_{\operatorname{sing}}||_{2},||\langle\bullet\rangle^{-2}W_{\operatorname{reg}}||_{L^{\infty}},||\langle\bullet\rangle^{-2}V_{\operatorname{con}}||_{L^{\infty}})$

we have

$\begin{split}||\xi||^{2}_{L^{\infty}((0,T),L^{2}({\mathbb{R}}^{3}))}\leq
C\left|\sup_{t\in(0,T)}\int_{\partial B_{R}(0)}\xi\nabla\bar{\xi}\
dS\right|\lesssim\frac{||\psi_{0}||_{H^{2}_{2}({\mathbb{R}}^{3})}}{R^{2}}.\end{split}$

###### Proof.

First we use Lemma 3.3 to reduce the initial state to an initial state with
support in $B_{R}(0)$. So we separate the solution $\psi$ into its behaviour
outside the ball $B_{R}(0)$, where the solution to the boundary value problem
is zero, and it’s behaviour inside $B_{R}(0)$.

To control the difference inside the ball, take the difference
$\xi=\psi-\psi^{D}$ such that

$\begin{split}i\partial_{t}\xi(t)&=(-\Delta+V+V_{\operatorname{TD}}(t))\xi(t)+F(\psi(t))-F(\psi^{D}(t))\text{
with }\xi(0)=0.\end{split}$

Multiplying by $\overline{\xi}$, integrate over $B_{R}(0)$, and taking the
imaginary part yields

$\begin{split}\frac{1}{2}\frac{d}{dt}&\int_{B_{R}(0)}|\xi(x,t)|^{2}dx=\operatorname{Im}\left(\int_{B_{R}(0)}-\overline{\xi}\Delta\xi+(F(\psi)-F(\psi^{D}))\overline{\xi}\
dx\right)\\\ &\leq\left|\int_{\partial B_{R}(0)}\overline{\xi}\nabla\xi\
dS\right|+\int_{B_{R}(0)}|F(\psi)-F(\psi^{D})||\overline{\xi}|\ dx\\\
&\lesssim\left|\int_{\partial B_{R}(0)}\overline{\xi}\nabla\xi\
dS\right|+||\xi||_{L^{2}(B_{R}(0))}^{2}.\end{split}$

Grönwall’s inequality implies then that
$||\xi(t)||^{2}_{L^{2}(B_{R}(0))}\lesssim C\left|\int_{\partial
B_{R}(0)}\overline{\xi}\nabla\xi\ dS\right|.$ Combining the two bounds above
with Lemma 2.5 yields then the bound of this Lemma. ∎

###  3.6. Approximation of Coulomb kernel

Next we aim to replace the Coulomb kernel, $1/|x|,$ with $f\in
C^{\infty}(\Omega)\cap L^{2}(\Omega)$ as we need smooth functions in order to
compute the solution. It is easy to see that Theorem 4 holds true for $1/|x|$
replaced by $f.$

######  Lemma 3.4.

Let $\psi$ be the solution to the Hartree equation and $\widetilde{\psi}$ be
the solution with $1/|x|$ replaced by $f\in C^{\infty}(\Omega)\cap
L^{2}(\Omega)$.

The difference of the two solutions $\xi_{:}=\widetilde{\psi}-\psi$ then
satisfies the PDE

$\begin{split}i\partial_{t}\xi=(H_{0}+V_{\operatorname{TD}})\xi+\int_{\Omega}f(\bullet-y)|\psi_{n}(y)|^{2}dy\
\widetilde{\psi}-\int_{\Omega}\frac{1}{|x-y|}|\psi(y)|^{2}dy\ \psi\end{split}$
(3.5)

and has the property that $\lVert\xi\rVert^{2}_{L^{\infty}(0,T;L^{2})}\leq
C_{T}\left\lVert f-\frac{1}{|\bullet|}\right\rVert_{1}$.

###### Proof.

Now multiply (3.5) by $\overline{\xi}$, take the imaginary part and integrate
over $\Omega$ to get

$\begin{split}\frac{1}{2}\frac{d}{dt}\|\xi\|^{2}_{2}&=\operatorname{Im}\int_{\Omega}\int_{\Omega}\left(f(x-y)|\widetilde{\psi}(y)|^{2}-\frac{1}{|x-y|}|\psi(y)|^{2}\right)dy\
\psi(x)\overline{\xi}(x)dx\\\
&\leq\left\lVert\int_{\Omega}f(\bullet-y)|\widetilde{\psi}(y)|^{2}-\frac{1}{|\bullet-y|}|\psi(y)|^{2}dy\right\rVert_{2}\|\psi\|_{L^{\infty}}\|\xi\|_{2}\\\
&=\left\lVert\int_{\Omega}\left(f(\bullet-y)-\frac{1}{|\bullet-y|}\right)|\widetilde{\psi}(y)|^{2}+\frac{1}{|\bullet-y|}\left(|\widetilde{\psi}(y)|^{2}-|\psi(y)|^{2}\right)dy\right\rVert_{2}\|\psi\|_{L^{\infty}}\|\xi\|_{2}.\end{split}$

We can bound the first term in the preceding equation by Young’s convolution
inequality

$\begin{split}&\left\lVert\
\left(f-\frac{1}{|\bullet|}\right)*|\widetilde{\psi}(\bullet)|^{2}\right\rVert_{2}\leq\left\lVert
f-\frac{1}{|\bullet|}\right\rVert_{1}\lVert|\widetilde{\psi}|^{2}\rVert_{2}\lesssim\left\lVert
f-\frac{1}{|\bullet|}\right\rVert_{1}\lVert\widetilde{\psi}\rVert_{H^{2}_{2}}^{2}\end{split}$

and similarly for the second term

$\begin{split}\left\lVert\frac{1}{|\bullet|}*\left(|\widetilde{\psi}(\bullet)|^{2}-|\psi(\bullet)|^{2}\right)\right\rVert_{2}&\leq\left\lVert\frac{1}{|\bullet|}\right\rVert_{2}\left\lVert|\widetilde{\psi}(\bullet)|^{2}-|\psi(\bullet)|^{2}\right\rVert_{1}\\\
&=C\int_{\Omega}|\psi(x)+\xi(x)|^{2}-|\psi(x)|^{2}\ dx\\\
&\lesssim\int_{\Omega}|\psi(x)||\xi(x)|+|\widetilde{\psi}(x)||\xi(x)|\ dx\\\
&\lesssim\left(\lVert\psi\rVert_{2}+\lVert\widetilde{\psi}\rVert_{2}\right)\lVert\xi\rVert_{2}.\end{split}$

Now using these bounds and that
$\lVert\psi\rVert_{H^{2}_{2}},\lVert\widetilde{\psi}\rVert_{H^{2}_{2}}\lesssim\lVert\psi_{0}\rVert_{H^{2}_{2}}$

$\begin{split}\frac{d}{dt}\lVert\xi\rVert^{2}_{2}&\lesssim\left(\left\lVert
f-\frac{1}{|\bullet|}\right\rVert_{1}\lVert\psi_{0}\rVert_{H^{2}_{2}}^{2}+\lVert\psi_{0}\rVert_{H^{2}_{2}}\lVert\xi\rVert_{2}\right)\lVert\psi_{0}\rVert_{H^{2}_{2}}\lVert\xi\rVert_{2}\\\
&\lesssim\left\lVert
f-\frac{1}{|\bullet|}\right\rVert_{1}\lVert\psi_{0}\rVert_{H^{2}_{2}}^{3}+C\lVert\psi_{0}\rVert_{H^{2}_{2}}^{2}\lVert\xi\rVert_{2}^{2}.\end{split}$

Integrating with respect to time gives and an application of Grönwall’s
inequality yields

$\begin{split}\lVert\xi(t)\rVert_{2}^{2}\lesssim\left\lVert
f-\frac{1}{|\bullet|}\right\rVert_{1}t\exp{Ct}.\end{split}$

Hence, $\lVert\xi\rVert^{2}_{L^{\infty}(0,T;L^{2})}\leq C_{T}\left\lVert
f-\frac{1}{|\bullet|}\right\rVert_{1}$. ∎

######  Example 1.

We can therefore choose the smooth function
$f_{\varepsilon}(x):=\frac{1}{(|x|^{2}+\varepsilon^{2})^{1/2}}$ which implies
that on any bounded domain $\Omega$

$\|f_{\varepsilon}-1/|\bullet|\|_{L^{1}}=\mathcal{O}_{\Omega}(\varepsilon).$

##  4\. Numerical methods

We have shown so far that we can replace both the Hartree equation, in Theorem
5, and linear magnetic Schrödinger equation, in Theorem 3, with the same
equation but now on a bounded domain.

###  4.1. Continuous Strang splitting scheme

For our numerical approximation, we consider the following continuous _Strang
splitting scheme_ of the solution on the bounded domain $\Omega:$ We now want
to approximate the solution $\psi(t_{n})$ with $t_{n}=n\tau$ for a step size
$\tau>0$ by functions $\psi_{n}$

$\begin{split}\psi_{n+1/2}^{-}&=e^{-\frac{i}{2}\tau(-i\nabla-A)^{2}}\psi_{n},\\\
\psi_{n+1/2}^{+}&=e^{-i\tau\mathcal{V}[\psi_{n+1/2}^{-}]}\psi_{n+1/2}^{-},\text{
and }\\\
\psi_{n+1}&=e^{-\frac{i}{2}\tau(-i\nabla-A)^{2}}\psi_{n+1/2}^{+}.\end{split}$
(4.1)

Here, $\mathcal{V}[\psi]:=f*|\psi|^{2}+V$ where $V$ includes all contributions
to the potential and $e^{-\frac{i}{2}\tau(-i\nabla-A)^{2}}\psi$ is the
solution to the linear Schrödinger equation

$\begin{split}i\partial_{t}u(x,t)&=(-i\nabla-A)^{2}u(x,t),\quad(x,t)\in\Omega\times(0,T)\\\
u(x,t)&=0\quad(x,t)\in\partial\Omega\times(0,T)\\\ u(\bullet,0)&=\psi\in
H^{2}(\Omega)\cap H^{1}_{0}(\Omega).\end{split}$

evaluated at time $t=\tau/2.$

####  4.1.1. Perturbation of the potential

######  Lemma 4.1.

Let $V,U\in L^{2}(\Omega)$ be potentials and denote by $\psi_{V},\psi_{U},$
two solutions to the Hartree equation (3.4) with the respective time-
independent potential with Dirichlet boundary conditions and some initial
state $\varphi_{0}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$, then for some
constant $C(T)>0$

$\|\psi_{V}-\psi_{U}\|_{2}\leq C(T,\varphi_{0})\|V-U\|_{2}.$

###### Proof.

Looking at the difference solution $\xi=\psi_{V}-\psi_{U}$, we find using
Lemma 3.1

$\begin{split}\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\xi(x,t)|^{2}\
dx&=\operatorname{Im}\left(\int_{\Omega}((V-U)\psi_{V}\bar{\xi})(x)+F(\psi_{U}(x))-F(\psi_{V}(x))\
dx\right)\\\
&\leq\|V-U\|_{2}\|\psi_{V}\|_{\infty}\|\xi\|_{2}+C_{F}(\|\psi_{U}\|_{H^{2}_{2}}+\|\psi_{V}\|_{H^{2}_{2}})\|\xi\|^{2}_{2}\\\
&\leq\|V-U\|_{2}^{2}\|\psi_{V}\|_{\infty}^{2}+(1+C_{F}(\|\psi_{U}\|_{H^{2}_{2}}+\|\psi_{V}\|_{H^{2}_{2}}))\|\xi\|_{2}^{2}\end{split}$

which after a straightforward application of Gronwalls lemma implies the
claim. ∎

######  Remark 5 (Control functions).

A very similar argument, as in the proof of Lemma 4.1 shows that the solution
to the magnetic Schrödinger or Hartree equation is Lipschitz continuous with
respect to the control function $u$ in the $L^{1}$-norm. We may therefore
assume without loss of generality that the control function is piecewise
constant in time and neglect the explicit time-dependence of the control
function in our discretization scheme.

####  4.1.2. Bounding the convolution

To estimate the convolution in the Hartree nonlinearity, we need the following
auxiliary lemma whose proof follows from elementary estimates that we shall
leave to the reader:

######  Lemma 4.2.

Let $\Omega$ be a domain. For $u,v,w\in L^{2}(\Omega)$, and $f\in
L^{\infty}(\Omega)$

$\begin{split}\lVert(f*(uv))w\rVert_{2}\leq\|f\|_{\infty}\|u\|_{2}\|v\|_{2}\|w\|_{2}.\end{split}$

For $u,v,w\in H^{1}(\Omega)$ we have

$\begin{split}\lVert(f*(uv))w\rVert_{H^{1}}\lesssim\|f\|_{\infty}(\|u\|_{H^{1}}\|v\|_{2}\|w\|_{2}+\|u\|_{2}\|v\|_{H^{1}}\|w\|_{2}+\|u\|_{2}\|v\|_{2}\|w\|_{H^{1}})\end{split}$

and for $u,v,w\in H^{2}(\Omega)$

$\begin{split}\lVert(f*(uv))w\rVert_{H^{2}}\lesssim\|f\|_{\infty}\sum_{\begin{subarray}{c}k,l,m\in\mathbb{N}_{0}\\\
k+l+m=2\end{subarray}}\|u\|_{H^{k}}\|v\|_{H^{l}}\|w\|_{H^{m}}.\end{split}$

Our global convergence proof now follows from combining a stability estimate
in Subsection 4.1.3, together with error propagation estimate in Subsection
4.1.4. This will lead us to (4.6) which shows the convergence of full solution
to a time-discretized Splitting scheme. We then study in Subsection 4.2 an
additional space-discretization to obtain a complete numerical discretization
scheme. The final result is then summarized in Theorem 6.

####  4.1.3. Stability of the splitting scheme

For the time-evolution in the following Lemma, we introduce the magnetic
Sobolev norm

$\|\psi\|_{H^{2}_{B}}:=\|\psi\|_{L^{2}}+\|(-i\nabla-A)^{2}\psi\|_{L^{2}}$

which is clearly equivalent to the usual $H^{2}$ norm on bounded domains for
magnetic vector potentials as specified in Assumption 1.

######  Lemma 4.3 ($L^{2}$ and $H^{2}$ stability).

We consider the time-discrete strang splitting scheme on some bounded domain
$\Omega$ for initial states $\psi_{0}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$
with a magnetic field as in Assumption 1

$\begin{split}\psi_{n+1/2}^{-}&=e^{-\frac{i}{2}\tau(-i\nabla-A)^{2}}\psi_{n}\\\
\psi_{n+1/2}^{+}&=e^{-i\tau\left(\left(f*|\psi_{n+1/2}^{-}|^{2}\right)+V\right)}\psi_{n+1/2}^{-}\\\
\psi_{n+1}&=e^{-\frac{i}{2}\tau(-i\nabla-A)^{2}}\psi_{n+1/2}^{+}.\end{split}$

We then introduce the time-evolution operator by
$\Phi_{\tau}\psi_{n}:=\psi_{n+1}.$

Let $\psi,\phi\in H^{2}$ be two states with
$\|\psi\|_{H^{2}},\|\phi\|_{H^{2}}\leq M_{1}$, then

$\begin{split}\|\Phi_{\tau}(\psi)-\Phi_{\tau}(\phi)\|_{2}&\leq
e^{c_{0}\tau}\|\psi-\phi\|_{2}\text{ and
}\|\Phi_{\tau}(\psi)-\Phi_{\tau}(\phi)\|_{H^{2}_{B}}\leq
e^{c_{1}\tau}\|\psi-\phi\|_{H^{2}_{B}}.\end{split}$

###### Proof.

As $e^{-\frac{i}{2}\tau(-i\nabla-A)^{2}}$ preserves the $L^{2}$ and
$H^{2}_{B}$ norm, we only need to consider the evolution by
$e^{-i\tau\left(\left(f*|\psi|^{2}\right)+V\right)}\psi.$ We shall continue
with the $L^{2}$ bound and observe that
$e^{-i\tau\left(\left(f*|\psi|^{2}\right)+V\right)}\psi=\theta(\tau)$ and
$e^{-i\tau\left(\left(f*|\phi|^{2}\right)+V\right)}\phi=\eta(\tau)$ where
$\theta$ and $\eta$ are solutions to

$\begin{split}i\dot{\theta}&=(f*|\psi|^{2}+V)\theta,\quad\theta(0)=\psi\\\
i\dot{\eta}&=(f*|\psi|^{2}+V)\eta,\quad\eta(0)=\phi.\end{split}$

So taking the difference

$\begin{split}i(\dot{\theta}-\dot{\eta})&=(f*|\psi|^{2}+V)\theta-(f*|\phi|^{2}+V)\eta\\\
&=(f*(\psi-\phi)\overline{\psi})\theta+(f*(\overline{\psi}-\overline{\phi})\phi)\theta+(f*|\phi|^{2})(\theta-\eta)+V(\theta-\eta).\end{split}$

Integrating this identity in time, we find by the unit $L^{2}$ norm of $\phi$
and $\psi$, and the initial state of $\theta(0)=\psi$, $\eta(0)=\phi$ that

$\begin{split}\lVert\theta(t)-\eta(t)\rVert_{2}\leq\lVert\psi-\phi\rVert_{2}+K_{0}\lVert\psi-\phi\rVert_{2}(\lVert\psi\rVert_{H^{1}}+\lVert\phi\rVert_{H^{1}})t\\\
\quad+\left(K_{0}\rVert\phi\rVert_{H^{1}}+\lVert
V\rVert_{W^{1,\infty}}\right)\int_{0}^{t}\lVert\theta(s)-\eta(s)\rVert_{2}\
ds.\end{split}$

By Grönwall’s inequality

$\begin{split}\lVert\theta(t)-\eta(t)\rVert_{2}\leq\|\psi-\phi\|_{2}(1+K_{0}(\|\psi\|_{H^{1}}+\|\phi\|_{H^{1}})t)\exp\left(\left(K_{0}\|\phi\|_{H^{1}}+\lVert
V\rVert_{W^{1,\infty}}\right)t\right)\end{split}$

and thus, writing $\mathcal{V}[\psi]:=f*|\psi|^{2}+V$,

$\begin{split}\|e^{-i\tau\mathcal{V}[\psi]}\psi-e^{-i\tau\mathcal{V}[\phi]}\phi\|_{2}&=\|\theta(\tau)-\eta(\tau)\|_{2}\\\
&\leq\lVert\psi-\phi\rVert_{2}(1+2K_{0}M_{1}\tau)\exp((K_{0}M_{1}+\lVert
V\rVert_{W^{1,\infty}})\tau)\\\
&\leq\lVert\psi-\phi\rVert_{2}e^{c_{0}\tau}.\end{split}$

The proof of the $H^{2}_{B}$ estimate follows in a similar fashion. ∎

####  4.1.4. Error propagation

Using a Lie algebraic description, that is largely inspired by the analysis in
[L08], we want to bound the error in one step in the splitting scheme. First
we let $\hat{T},\hat{V}$ be vector fields defined by

$\begin{split}\hat{T}(\psi)&=-i(-i\nabla-A)^{2}\psi=i\Delta\psi+i\langle\nabla,A\psi\rangle+i\langle
A,\nabla\psi\rangle+i|A|^{2}\psi,\text{ and }\\\
\hat{V}(\psi)&=-i(f*|\psi|^{2}+V)\psi,\end{split}$ (4.2)

then their Lie commutator is

$\begin{split}[\hat{T},\hat{V}](\psi)=&\hat{T}^{\prime}|_{\psi}(\hat{V}(\psi))-\hat{V}^{\prime}|_{\psi}(\hat{T}(\psi))\\\
=&i\Delta(-i(f*\psi\overline{\psi}-V)\psi)+i(f*\psi\overline{\psi}-V)i\Delta\psi+i(f*i\Delta\psi\overline{\psi})\psi+i(f*\psi\overline{i\Delta\psi})\psi\\\
&+i\langle\nabla,-iA(f*\psi\overline{\psi}-V)\rangle\psi+i\langle
A,-i\nabla(f*\psi\overline{\psi}-V)\rangle\psi+2(f*|A|^{2}|\psi|^{2})\psi\\\
&+if*(\langle\nabla,-iA\psi\rangle\overline{\psi}+\overline{\langle\nabla,-iA\psi\rangle}\psi)\psi+if*(\langle
A,-i\nabla\psi\rangle\overline{\psi}+\overline{\langle
A,-i\nabla\psi\rangle}\psi)\psi\\\ =&(\langle
2f*\nabla\psi,\nabla\overline{\psi}\rangle+2f*\psi\Delta\overline{\psi}+\Delta
V)\psi+2\langle(f*\nabla\psi\ \overline{\psi}+f*\psi\
\nabla\overline{\psi}+\nabla V),\nabla\psi\rangle\\\
&+i\langle\nabla,-iA(f*\psi\overline{\psi}-V)\rangle\psi+i\langle
A,-i\nabla(f*\psi\overline{\psi}-V)\rangle\psi+2(f*|A|^{2}|\psi|^{2})\psi\\\
&+if*(\langle\nabla,-iA\psi\rangle\overline{\psi}+\overline{\langle\nabla,-iA\psi\rangle}\psi)\psi+if*(\langle
A,-i\nabla\psi\rangle\overline{\psi}+\overline{\langle
A,-i\nabla\psi\rangle}\psi)\psi.\end{split}$

We thus have for the $L^{2}$ norm of the commutator that

$\begin{split}\lVert[\hat{T},\hat{V}](\psi)\rVert_{2}&\leq
C\lVert\psi\rVert_{H^{2}_{2}}\lVert\psi\rVert_{H^{1}_{1}}\lVert\psi\rVert_{2}+C\lVert
V\rVert_{W^{2,\infty}}\lVert\psi\rVert_{H^{1}_{1}}.\end{split}$ (4.3)

In the following, we write $\psi(\tau)=:\operatorname{exp}(\tau
D_{K})\psi_{0}$ to denote the solution to the equation

$\psi^{\prime}(t)=K(\psi(t)),\quad\psi(0)=\psi_{0}.$

By applying the variation of constants formula twice, we find for the solution

$\begin{split}\psi(\tau)&=\exp(\tau D_{H})\operatorname{id}(\psi_{0})\\\
&=\exp(\tau
D_{\hat{T}})\operatorname{id}(\psi_{0})+\int_{0}^{\tau}\exp((\tau-s)D_{H})D_{\hat{V}}\exp(sD_{\hat{T}})\operatorname{id}(\psi_{0})\
ds\\\ &=\exp(\tau
D_{\hat{T}})\operatorname{id}(\psi_{0})+\int_{0}^{\tau}\exp((\tau-s)D_{\hat{T}})D_{\hat{V}}\exp(sD_{\hat{T}})\operatorname{id}(\psi_{0})\
ds+r_{1}(\psi_{0}),\end{split}$

with remainder term

$\begin{split}r_{1}(\psi_{0})=\int_{0}^{\tau}\int_{0}^{\tau-s}\exp((\tau-s-\sigma)D_{H})D_{\hat{V}}\exp(\sigma
D_{\hat{T}})D_{\hat{V}}\exp(sD_{\hat{T}})\operatorname{id}(\psi_{0})\ d\sigma\
ds.\end{split}$

We can also write the time-discretization of the Strang splitting (4.1) on
continuous space $\Omega$ as

$\begin{split}\psi_{1}=\exp(\tfrac{\tau}{2}D_{\hat{T}})\exp(\tau
D_{\hat{V}})\exp(\tfrac{\tau}{2}D_{\hat{T}})\operatorname{id}(\psi_{0}).\end{split}$

By Taylor expansion, we have $\exp(\tau D_{\hat{V}})=I+\tau
D_{\hat{V}}+\tau^{2}\int_{0}^{1}(1-\theta)\exp(\theta\tau
D_{\hat{V}})D_{\hat{V}}^{2}d\theta$

$\begin{split}\psi_{1}=\exp(\tau
D_{\hat{T}})\operatorname{id}(\psi_{0})+\tau\exp(\tfrac{\tau}{2}D_{\hat{T}})D_{\hat{V}}\exp(\tfrac{\tau}{2}D_{\hat{T}})\operatorname{id}(\psi_{0})+r_{2}(\psi_{0})\end{split}$

with remainder
$r_{2}(\psi_{0})=\tau^{2}\int_{0}^{1}(1-\theta)\exp(\frac{\tau}{2}D_{\hat{T}})\exp(\theta\tau
D_{\hat{V}})D_{\hat{V}}^{2}\exp(\frac{\tau}{2}D_{\hat{T}})\operatorname{id}(\psi_{0})d\theta.$

Therefore, the error between the actual solution and the time-discretization
becomes

$\begin{split}\psi_{1}-\psi(\tau)&=\tau\exp(\tfrac{\tau}{2}D_{\hat{T}})D_{\hat{V}}\exp(\tfrac{\tau}{2}D_{\hat{T}})\operatorname{id}(\psi_{0})\\\
&-\int_{0}^{\tau}\exp((\tau-s)D_{\hat{T}})D_{\hat{V}}\exp(sD_{\hat{T}})\operatorname{id}(\psi_{0})ds+(r_{2}-r_{1})(\psi_{0}).\end{split}$

Let
$f(s):=\exp((\tau-s)D_{\hat{T}})D_{\hat{V}}\exp(sD_{\hat{T}})\operatorname{id}(\psi_{0})$,
then we have

$\psi_{1}-\psi(\tau)=\int_{0}^{\tau}f(\tau/2)-f(s)\
ds+(r_{2}-r_{1})(\psi_{0}).$ (4.4)

Thus, apart from the two remainder expressions, we are left to understand the
operator associated with the mid-point rule $L(g)=\tau
g(\frac{1}{2}\tau)-\int_{0}^{\tau}g(s)ds$. Using Peano’s kernel theorem with
the Peano kernel

$\begin{split}k_{1}(\theta)&=L[(\bullet-\theta)\operatorname{1\hskip-2.75ptl}_{\bullet\geq\theta}]=\tau\left(\theta-\frac{\tau}{2}\right)\operatorname{1\hskip-2.75ptl}_{\tau/2\leq\theta}-\theta^{2}/2.\end{split}$

we are then left to study an operator, which by integration by parts, since
$k_{1}(0)=k_{1}(\tau)=0$, satisfies

$\begin{split}Lf&=\int_{0}^{\tau}k_{1}(\theta)f^{(2)}(\theta)d\theta=-\int_{0}^{\tau}k_{1}^{\prime}(\theta)f^{\prime}(\theta)d\theta.\end{split}$

Now, clearly
$|k_{1}^{\prime}(\theta)|=|\tau\operatorname{1\hskip-2.75ptl}_{\tau/2\leq\theta}-\theta|\leq\tau$
and

$\begin{split}f^{\prime}(s)&=-\exp((\tau-s)D_{\hat{T}})[D_{\hat{T}},D_{\hat{V}}]\exp(sD_{\hat{T}})\operatorname{id}(\psi_{0})\\\
&=\exp((\tau-s)D_{\hat{T}})D_{[\hat{T},\hat{V}]}\exp(sD_{\hat{T}})\operatorname{id}(\psi_{0})\\\
&=e^{-is(-i\nabla-A)^{2}}[\hat{T},\hat{V}](e^{-i(\tau-s)(-i\nabla-A)^{2}}\psi_{0}).\end{split}$

which we can bound, using (4.3), as

$\lVert f^{\prime}(s)\rVert_{2}\leq
C(\lVert\psi\rVert_{H^{2}_{2}}\lVert\psi\rVert_{H^{1}_{1}}\lVert\psi\rVert_{2}+\lVert
V\rVert_{W^{2,\infty}}\lVert\psi\rVert_{H^{1}_{1}}).$

This implies that

$\lVert Lf\rVert_{2}\leq
C\tau^{2}(\lVert\psi\rVert_{H^{2}_{2}}\lVert\psi\rVert_{H^{1}_{1}}\lVert\psi\rVert_{2}+\lVert
V\rVert_{W^{2,\infty}}\lVert\psi\rVert_{H^{1}_{1}}).$ (4.5)

So now estimating the remainder terms in (4.4) using

$\begin{split}\lVert\hat{V}(\psi)\rVert_{2}&\leq
C\|f\|_{\infty}\lVert\psi\rVert_{2}^{3}+\lVert
V\rVert_{W^{\infty}}\lVert\psi\rVert_{2}\\\
\lVert\hat{V}^{\prime}(\psi)\phi\rVert_{2}&\leq
C\|f\|_{\infty}\lVert\psi\rVert_{2}^{2}\|\phi\|_{2}+\lVert
V\rVert_{W^{\infty}}\lVert\phi\rVert_{2},\end{split}$

we infer that $\lVert r_{1}\rVert_{2}+\lVert r_{2}\rVert_{2}\leq
C_{2}\tau^{2}$ and hence by combining the remainder estimates with (4.5)
$\lVert\psi_{1}-\psi(\tau)\rVert_{2}\leq C_{4}\tau^{2}.$

By the usual Lady Windermere’s fan argument, this implies that

$\lVert\psi_{N}-\psi(N\tau)\rVert_{2}\leq C(T)\tau.$ (4.6)

This shows the explicit global convergence of our time-discretized but space-
continuous splitting scheme. Our next goal is to obtain a space-discretized
version of (4.6).

###  4.2. Space-Discrete Splitting scheme

To define a convergent numerical discretization scheme, we employ a _cubic
discretization_.

Cubic Discretization: Consider a lattice of side length $h$ with lattice
points $(x_{j})_{j\in{\mathbb{Z}}^{3}}$ and a family of cubes
$Q_{x_{j}}=\times_{i=1}^{3}[x_{j}^{i}-\tfrac{h}{2},x_{j}^{i}+\tfrac{h}{2})$
with $j\in{\mathbb{Z}}^{3}$ that form a disjoint decomposition of
${\mathbb{R}}^{3}$ up to a set of measure zero. The cubic approximation of a
function $f\in L_{loc}^{1}({\mathbb{R}}^{3},\mathbb{C})$ is defined by

$\begin{split}f_{Q}(x):=\sum_{j\in{\mathbb{Z}}^{3}}\frac{1}{\operatorname{vol}(Q_{x_{j}})}\int_{Q_{x_{j}}}f(s)\
ds\operatorname{1\hskip-2.75ptl}_{Q_{x_{j}}}(x).\end{split}$

Now we need to introduce the discrete derivative, so let
$(\tau_{h}^{i}f)(x)=f(x-h\hat{e_{i}})$ be the translation by $h$ and
$\delta^{i}_{h}=(\tau^{i}_{-h}-\tau^{i}_{h})/(2h)$ the discretized symmetric
derivative in direction $i$ with step $h>0$. Then we can define the
discretized Laplacian $\Delta^{h}=\sum_{i=1}^{3}(\delta^{i}_{h})^{2}$ and
discretized gradient $\nabla^{h}=(\delta_{h}^{i})_{i}$.

Then it is easy to verify that $(\delta^{i}_{h})^{*}=-\delta_{h}^{i}$ and also
the product rule holds

$(\delta_{h}fg)(x)=f(x+h)(\delta_{h}g)(x)+g(x-h)(\delta_{h}f)(x).$

Moreover, by [BH20, Prop. 8.9], for $n\in{\mathbb{Z}}_{+},k\in{1,2,3},f\in
W^{n,p}({\mathbb{R}}^{d})$ and $p\in[1,\infty)$ we have

$\begin{split}\lVert((\delta_{h}^{k})^{n}-\partial_{k}^{n})f\rVert_{L^{p}({\mathbb{R}}^{3})}=o(1)\operatorname{as}h\to
0\end{split}$

and for $f\in W^{n+1,p}({\mathbb{R}}^{3})$ and $p\in[1,\infty]$ there is a
constant $C>0$ independent of $f$ and $h$ such that

$\begin{split}\lVert(\delta_{h}^{k})^{n}f_{Q}-\partial^{n}_{k}f\rVert_{L^{p}({\mathbb{R}}^{3})}\leq
C\lVert f\rVert_{W^{n+1,p}({\mathbb{R}}^{3})}h\ \operatorname{and}\
\lVert(\delta_{h}^{k})^{n}f\rVert_{L^{p}}\leq C\lVert
f\rVert_{W^{n,p}({\mathbb{R}}^{d})}.\end{split}$

In addition, for $p\in(1,\infty)$, $f\in
W^{n+\varepsilon,p}({\mathbb{R}}^{3})$, and any $\varepsilon\in(0,1]$ it
follows that

$\lVert(\delta_{h}^{k})^{n}f_{Q}-\partial_{k}^{n}f\rVert_{L^{p}({\mathbb{R}}^{3})}=\mathcal{O}(\lVert
f\rVert_{W^{n+\varepsilon,p}({\mathbb{R}}^{3})}h^{\varepsilon}).$

We can now directly use [BH20, Lemma 8.10] that for our Schrödinger equation
using the discretized derivative there is a constant $C=C(T)$ such that
$\lVert\psi^{h}\rVert_{L^{\infty}((0,T),H^{2}(\Omega))}\leq
C\lVert\phi_{0}\rVert_{H^{2}(\Omega)}.$

Now we need the equivalent of [BH20, Lemma 8.11], whose proof is a simple
adaptation of the proof given in [BH20], but with a $H^{2}_{h}$ norm and a
magnetic Schrödinger operator, instead.

######  Lemma 4.4.

For an initial state $\phi_{0}\in H^{2+\varepsilon}(\Omega)$ the difference
between $\psi$, the solution to the Schrödinger equation
$i\partial_{t}\psi=(-i\nabla-A)^{2}\psi$ with initial state $\phi_{0},$ and
the solution $\psi^{h}$, to the discretized Schrödinger equation
$i\partial_{t}\psi^{h}=H^{h}_{\operatorname{lin}}\psi^{h}$ with initial state
$\psi^{h}(0)=(\phi_{0})_{Q}$ and Hamiltonian
$H^{h}_{\operatorname{lin}}:=(-i\nabla^{h}-A^{h})^{2}$ satisfies for some
constant $C=C(T,\lVert\phi_{0}\rVert_{H^{2+\varepsilon}})$

$\begin{split}\lVert\psi-\psi^{h}\rVert_{L^{\infty}((0,T),L^{2})}\leq
Ch^{\varepsilon}.\end{split}$

To analyze the second step of the splitting scheme, where we ought to compare
the dynamics induced by
$V[\psi]=f*|\psi|^{2}+V=\int_{\Omega}f(x-y)|\psi(y)|^{2}\ dy+V$ with the one
in which $f$ is replaced by $f_{Q}$ and $V$ by $V_{Q}$

######  Lemma 4.5.

For an initial state $\phi_{0}\in H^{2}(\Omega)$, potential $V\in
W^{3,\infty}(\Omega)$ and $f\in C^{\infty}(\Omega)$ then there is a constant
$C=C(\lVert\phi_{0}\rVert_{H^{2}},\lVert V\rVert_{2},T)$ such that if $\psi$
and $\psi^{h}$ are respectively the solutions to

$\begin{split}i\partial_{t}\psi&=(f*|\phi_{0}|^{2}+V)\psi\text{ with
}\psi(0)=\phi_{0}\text{ and }\\\
i\partial_{t}\psi^{h}&=(f_{Q}*|\phi_{0}|^{2}_{Q}+V_{Q})\psi^{h}\text{ with
}\psi^{h}_{0}=(\phi_{0})_{Q},\end{split}$

then $\lVert\psi-\psi^{h}\rVert_{L^{\infty}((0,T),L^{2})}\leq Ch.$

###### Proof.

Let $\xi=\psi-\psi^{h}$. Then due to

$\begin{split}\partial_{t}\xi&=(f*|\phi_{0}|^{2}+V-f_{Q}*|\phi_{0}|_{Q}^{2}-V_{Q})\psi+(f_{Q}*|\phi_{0}|_{Q}^{2}+V_{Q})\xi\end{split}$

we find that

$\begin{split}\frac{d}{dt}\lVert\xi\rVert_{2}&\leq\lVert(f*|\phi_{0}|^{2}+V-f_{Q}*|\phi_{0}^{h}|^{2}-V_{Q})\psi\rVert_{2}+\lVert(f_{Q}*(|\phi_{0}|^{2})_{Q}+V_{Q})\xi\rVert_{2}\\\
&\leq\lVert
f*|\phi_{0}|^{2}+V-f_{Q}*(|\phi_{0}|^{2})_{Q}-V_{Q}\rVert_{\infty}\lVert\psi\rVert_{2}+\lVert
f_{Q}*(|\phi_{0}|^{2})_{Q}+V_{Q}\rVert_{\infty}\lVert\xi\rVert_{2}\end{split}$

and hence by Gronwall’s inequality

$\begin{split}\sup_{t\in(0,T)}\lVert\xi(t)\rVert_{2}&\leq\left(\lVert\phi_{0}-\phi_{0}^{h}\rVert_{2}+\lVert
f*|\phi_{0}|^{2}+V-f_{Q}*(|\phi_{0}|^{2})_{Q}-V_{Q}\rVert_{\infty}\lVert\psi\rVert_{L^{\infty}((0,T),L^{2})}T\right)\\\
&\quad\times\exp\left(\lVert
f_{Q}*(|\phi_{0}|^{2})_{Q}-V_{Q}\rVert_{\infty}T\right).\end{split}$

Now we need to use [BH20, Proposition 8.9] and the mean value theorem to have,
for constants $C$

$\lVert\phi_{0}-\phi_{0}^{h}\rVert_{2}\leq
C\lVert\phi_{0}\rVert_{H^{1}}h\text{ and }\lVert V-V_{Q}\rVert_{\infty}\leq
C\lVert V\rVert_{W^{1,\infty}}h.$

Finally for the convolution $\lVert
f*|\phi_{0}|^{2}-f_{Q}*(|\phi_{0}|^{2})_{Q}\rVert_{\infty}\leq C\lVert
f\rVert_{\infty}\lVert\phi_{0}\rVert^{2}_{2}h.$ Hence, there is
$C=C(\lVert\phi_{0}\rVert_{H^{1}},\lVert V\rVert_{W^{1,\infty}},\lVert
f\rVert_{W^{1,\infty}},T)$ such that

$\sup_{t\in(0,T)}\lVert\xi(t)\rVert_{2}\leq Ch.$

∎

Now we want to put all this together to get a scheme for calculating an
approximation to the solution.

As an intermediate step, we use a continuous (in space) Strang splitting
scheme

$\begin{split}\psi^{-}_{k+\frac{1}{2}}&=e^{-\frac{i\tau(-i\nabla-A)^{2}}{2}}\psi_{S}(t_{k})\\\
\psi^{+}_{k+\frac{1}{2}}&=\exp\left(-i\tau\mathcal{V}(\psi^{-}_{k+\frac{1}{2}})\right)\psi^{-}_{k+\frac{1}{2}}\\\
\psi_{S}(t_{k+1})&=e^{-\frac{i\tau(-i\nabla-A)^{2}}{2}}\psi^{+}_{k+\frac{1}{2}}\end{split}$
(4.7)

where $\psi_{S}(0)=\phi_{0}$ and $\mathcal{V}(\psi):=f*|\psi|^{2}+V$. Finally,
we want to approximate this scheme using the cubic discretization and a Crank-
Nicholson method for the propagation of the linear kinetic operator
$(-i\nabla-A)^{2}.$ In particular,

$\mathcal{C}^{\tau}_{h}\phi:=\left(1-\frac{i\tau}{2}H^{h}_{\operatorname{lin}}\right)\left(1+\frac{i\tau}{2}H^{h}_{\operatorname{lin}}\right)^{-1}\phi.$

Recall that for $\lambda\in\mathbb{R}$ and $t\in[0,T]$ there is $C_{T}>0$ such
that

$\left|e^{-it\lambda}-\left(\frac{1-i\frac{t}{2}\lambda}{1+i\frac{t}{2}\lambda}\right)\right|\leq
C_{T}t^{3}|\lambda|^{3},$

we see, using functional calculus that there is (Taylor expansion) some
constant $C_{T}>0$ independent of $h$ and $\tau\in[0,T]$ such that

$\begin{split}\left\lVert\left(e^{-iH_{\operatorname{lin}}^{h}\tau}-\mathcal{C}^{\tau}_{h}\right)\psi^{h}(t_{k})\right\rVert_{L^{2}(\Omega)}\leq
C_{T}\tau^{3}/h^{6}.\end{split}$ (4.8)

This yields a Strang splitting scheme

$\begin{split}\phi^{-}_{k+\frac{1}{2}}&=\mathcal{C}_{h}^{\tau/2}\phi_{S}(t_{k})\\\
\phi^{+}_{k+\frac{1}{2}}&=\exp_{K}\left(-i\tau\mathcal{V}_{Q}(\phi^{-}_{k+\frac{1}{2}})\right)\phi^{-}_{k+\frac{1}{2}}\\\
\phi_{S}(t_{k+1})&=\mathcal{C}_{h}^{\tau/2}\phi^{+}_{k+\frac{1}{2}}\end{split}$
(4.9)

where $\psi_{S}(0)=(\phi_{0})_{Q}$ and
$\mathcal{V}_{Q}(\psi_{Q}):=f_{Q}*(|\psi|^{2})_{Q}+V_{Q}$.

######  Theorem 6.

Given an initial state $\phi_{0}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ and
$V,f\in W^{1,\infty}(\Omega)$. There is a constant
$C=C(T,\tau,\lVert\phi_{0}\rVert_{H^{2}},\lVert V\rVert_{W^{1,\infty}},\lVert
f\rVert_{W^{1,\infty}})$ such that the solution $\phi_{S}$ obtained from the
Strang splitting scheme (4.9) and the full solution $\psi$ satisfy

$\max_{k;0\leq k\leq\lfloor T/\tau\rfloor}\rVert\psi(\tau k)-\psi_{S}(\tau
k)\rVert_{2}\leq C\lVert\phi_{0}-(\phi_{0})_{Q}\rVert_{2}.$ (4.10)

###### Proof.

We have already shown the convergence to the continuous Strang splitting
scheme (4.7) in Lemmas 4.4 and 4.5. It therefore suffices to approximate the
continuous splitting scheme by the discrete one . The $L^{2}$-convergence to
the discrete approximation by the Crank-Nicholson method in the discrete
Strang splitting scheme (4.9), follows already from (4.8). Thus, it only
remains to prove the convergence of the propagation of the non-linearity:

$\begin{split}\lVert&\left(e^{-i\tau\mathcal{V}(\psi^{-}_{k+1/2})}-e^{-i\tau\mathcal{V}_{Q}(\phi^{-}_{k+1/2})}\right)\phi^{-}_{k+1/2}\rVert_{2}\\\
&\leq\left\lVert\psi^{-}_{k+1/2}-\phi^{-}_{k+1/2}\right\rVert_{2}+\left\lVert\left(e^{-i\tau\mathcal{V}(\psi^{-}_{k+1/2})}-e^{-i\tau\mathcal{V}_{Q}(\phi^{-}_{k+1/2})}\right)\phi^{-}_{k+1/2}\right\rVert_{2}\\\
&\leq\left\lVert\psi^{-}_{k+1/2}-\phi^{-}_{k+1/2}\right\rVert_{2}+\tau\|f*|\psi^{-}_{k+1/2}|^{2}-f_{Q}*|\phi^{-}_{k+1/2}|^{2}\|_{\infty}\\\
&\leq\left\lVert\psi^{-}_{k+1/2}-\phi^{-}_{k+1/2}\right\rVert_{2}+\tau\|f*(|\psi^{-}_{k+1/2}|^{2}-|\phi^{-}_{k+1/2}|^{2})\|_{\infty}+\tau\left\lVert
f-f_{Q}\right\rVert_{\infty}\\\
&\leq(1+2\tau\|f\|_{\infty})\left\lVert\psi^{-}_{k+1/2}-\phi^{-}_{k+1/2}\right\rVert_{2}+\tau\left\lVert
f-f_{Q}\right\rVert_{\infty}.\end{split}$

So we have enough for global Lipschitz estimate (4.10) of the discrete Strang
splitting scheme.

∎

##  5\. Optimal control theory

In this section we study an abstract optimal control problem (OCP) for the
linear Schrödinger equation with magnetic field (1.1). We consider an energy
functional

$\mathcal{I}(u):=\left\langle\psi(T),S\psi(T)\right\rangle+\kappa\left\lVert
u\right\rVert_{H^{1}_{0}(0,T)}^{2}$ (5.1)

for a positive operator111it is easy to see that same arguments holds if $S$
is only assumed to be semi-bounded from below $S:D(S)\subset
L^{2}(\mathbb{R}^{3})\rightarrow L^{2}(\mathbb{R}^{3})$ with form domain
$D(\sqrt{S})$ continuously embedded in $H^{1}_{1}(\mathbb{R}^{3})$ and
parameter $\kappa>0$. Here, $\psi$ is the solution to the linear or nonlinear
Schrödinger equation with initial value $\varphi_{0}\in
H^{2}_{2}(\mathbb{R}^{3})$ and parameters as specified in Assumption 1, cf.
Theorem 1 and Theorem 2.

Examples of such optimal control problems are numerous [WG07] and the
motivation to study such a problem in quantum mechanics is often to engineer a
control function in order to reach a designated target state $\zeta\in
L^{2}(\mathbb{R}^{3})$. In this case, we can choose $S$ to be the projection
$S=\operatorname{id}-\langle\bullet,\zeta\rangle\zeta$. Other common examples
of penalization operators $S$ include the minimization/maximization of some
physical observable such as kinetic energy
$\left\langle\psi(T),-\Delta\psi(T)\right\rangle$. The term $\kappa\left\lVert
u\right\rVert_{H^{1}_{0}(0,T)}^{2}$, appearing in the control functional
(5.1), penalizes control functions of high energy.

######  Lemma 5.1.

The functional $\mathcal{I}$ has a minimizer in $H^{1}_{0}(0,T)$.

###### Proof.

The Banach-Alaoglu theorem implies the existence of a $H^{1}_{0}(0,T)$-weakly
convergent subsequence $u_{n}\overset{}{\rightharpoonup}u$ such that
$\lim_{n\rightarrow\infty}\mathcal{I}(u_{n})=\inf_{w\in
H^{1}_{0}(0,T)}\mathcal{I}(w).$ Since $u_{n}$ is weakly convergent, the
sequence is bounded. Hence, the sequence $\psi_{n}$ of solutions to (1.1) with
the above controls $u_{n}$ and joint initial value $\varphi_{0}\in
H^{2}_{2}(\mathbb{R}^{3})$ is bounded in

$W:=\left\\{\varphi\in
L^{\infty}((0,T),H^{2}_{2}(\mathbb{R}^{3})),\varphi^{\prime}\in
L^{\infty}((0,T),L^{2}(\mathbb{R}^{3}))\right\\}.$

As $H^{2}_{2}(\mathbb{R}^{3})$ is compact in
$H_{2-\varepsilon}^{2-\varepsilon}(\mathbb{R}^{3})$ the Aubin-Lions lemma
implies that for yet another subsequence of $(\psi_{n})$

$\lim_{n\rightarrow\infty}\left\lVert\psi_{n}-\psi\right\rVert_{C((0,T);H_{2-\varepsilon}^{2-\varepsilon}(\mathbb{R}^{3}))}=0$

where $\psi$ is the solution to the Schrödinger equation with initial value
$\varphi_{0}$ and control $u.$ Weak lower semicontinuity of the
$H^{1}_{0}(0,T)$ norm and $H_{2-\varepsilon}^{2-\varepsilon}$ convergence of
$(\psi_{n})$ implies that $\mathcal{I}(u)=\inf_{w\in
H^{1}_{0}(0,T)}\mathcal{I}(w).$ ∎

The _adjoint equation_ to the linear magnetic Schrödinger equation is defined
as

$\begin{split}i\partial_{t}\zeta&=(-i\nabla-A)^{2}\zeta+V\zeta+V_{\operatorname{TD}}(t)\zeta,\
(x,t)\in\mathbb{R}^{3}\times(0,T)\\\ \zeta(T)&=S\psi(T).\end{split}$ (5.2)

It is easy to see that the vanishing of the Gateaux derivative of the cost
functional $(D_{h}\mathcal{I})(u)=0$ for all admissible functions $h$ is
equivalent to a differential equation which is often referred to as the
_optimality equation_. For the linear Schrödinger equation this equation reads

$-2\kappa
u^{\prime\prime}(t)=\left\langle\zeta(t),V_{\operatorname{con}}\psi(t)\right\rangle_{L^{2}(\mathbb{R}^{3})},\
u\in H^{1}_{0}(0,T),$

with parameter $\kappa$ as in the control functional (5.1), and $\zeta$ the
solution to the adjoint problem. Various algorithms [IK07, IK09, WG07] to
numerically solve the OCP (5.1) rely on solving the triplet consisting of
Schrödinger equation (1.1), adjoint equation (5.2), and optimality
differential equation. The following example shows that the optimal control
can in general not be computed by that approach.

Our findings of the previous sections showed that it is possible to compute
the solution to the Schrödinger equation by restricting it to an auxiliary
BVP. The following example shows that this is not possible for optimal control
problems.

######  Example 2.

Consider an odd control potential $V$ and an even initial state $\varphi_{0}$
to the Schrödinger equation

$\begin{split}i\psi_{u}^{\prime}(t,x)=\left(-\partial_{x}^{2}+u(t)V\right)\psi_{u}(t,x),\quad\psi(0)=\varphi_{0}.\end{split}$

We also take $S$ to be an even function in the window that the algorithm
samples such that there exists a non-zero optimal control with $L^{2}$
distance to any other optimal control of at least $2\varepsilon.$ Our
assumptions guarantee that if $u$ is an optimal control, then so is $-u$ as
the solution satisfies $\psi_{u}(x)=\psi_{u}(-x)$ and $S$ is assumed to be an
even function in the window the algorithm samples.

It suffices now to continue $S$ in such a way that the control functional
(5.1) favors $u$ over $-u$. The algorithm necessarily has to include both $u$
and $-u$ and will therefore be off by a Hausdorff distance $2\varepsilon$ at
least.

Thus, it is impossible to numerically approximate optimal controls numerically
by sampling only a finite amount of data. However, it is possible to
approximate the value of the control functional for a positive Schrödinger
operator $S=(-i\nabla-A)^{2}+U$ with magnetic potential $A$ and control
potential $U$ (possibly different from the potentials governing the evolution
of the equation) satisfying the conditions of the static potential in
Assumption 1.

###### Proof.

The result follows straight from the result we have already established in
previous sections: We first observe that by Theorems 2 and 6, we can compute
up to any precision the value of the control functional (5.1) for any fixed
control $u$ in the domain of the computational problem. Thus, by computing the
value of the control functional for any fixed control, we obtain an upper
bound on the minimal value of (5.1) and thus also on the $H^{1}_{0}$ bound of
any optimal control. This yields a bound on the absolute value of Fourier
coefficients on the optimal control. This implies an error that does not
exceed $\varepsilon$ in the optimal control functional, as the solution is
easily seen to be Lipschitz continuous with respect to controls in
$L^{1}(0,T)$.∎

##  6\. Numerical examples

(a)  (b)  (c)

Figure 1. Propagation of Gaussian initial state (6.1) under the linear
Schrödinger equation. The colormap depicts the probability amplitude of the
state in the respective square (a) initial state at time $t=0$; (b) evolution
after time $t=2s$ with magnetic field where the Lorentz force shifts the
direction of propagation; (c) evolution after time $t=2s$ without magnetic
field.

We consider a Gaussian initial state

$\psi_{0}(x)=\frac{1}{\pi}\operatorname{exp}\left(-\frac{(x_{1}+5/2)^{2}}{2}-\frac{(x_{2}-5/2)^{2}}{2}\right)$
(6.1)

and a potential that is quadratically confining in $x_{1}$ direction and
quadratically spreading (exponentially fast in the dynamics) in $x_{2}$
direction

$V_{\operatorname{reg}}(x)=x_{1}^{2}-x_{2}^{2}.$

In addition, we consider a constant magnetic field of unit strength $B_{0}=1.$
We then simulate the dynamics for two seconds with $100$ time steps with
Crank-Nicholson method and Strang splitting scheme in Figure 1. The wavepacket
then propagates exponentially fast away from the origin, which is consistent
with the Gronwall estimates from our theoretical part, but the entire dynamics
can still be captured in a bounded domain with Dirichlet boundary conditions.

## References

  * [B05] Baudouin, L. (2005). _Existence and Regularity of the Solution of a Time Dependent Hartree-Fock Equation Coupled with a Classical Nuclear Dynamics_ , Rev. Mat. Complut. 2005, 18; Num. 2, 285-314.
  * [BD11] Bao, W. and Dong, X. (2011). _Numerical methods for computing ground states and dynamics of nonlinear relativistic Hartree equation for boson stars_. Journal of Computational Physics. Elsevier.
  * [BH20] Becker, S. and Hansen, A. (2020). _Computing solutions of Schrödinger equations
\- On the brink of numerical algorithms_, arXiv:2010.16347.

  * [BKP05] Baudouin, L., Kavian, O., and Puel, J. (2005). _Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control_ , Journal of Differential Equations, Volume 216, Issue 1, Pages 188-222.
  * [BM91] Brezzi, F. and Markowich, P. _The three-dimensional Wigner-Poisson problem: Existence, uniqueness and approximation_ , Math. Methods Appl. Sci. 14 (1991), 35-61.
  * [DM94] Hans De Raedt, and Kristel Michielsen. (1994) _Algorithm to solve the time-dependent Schrödinger equation for a charged particle in an inhomogeneous magnetic field: Application to the Aharonov–Bohm effect_. Computers in Physics 8, 600
  * [F04] Fröhlich, M. (2004). _Exponentielle Integrationsverfahren für die Schrödinger-Poisson-Gleichung_ , Doctoral Thesis, Universität Tübingen.
  * [IK07] Ito, K. and Kunisch, K. (2007). _Optimal Bilinear Control of an Abstract Schrödinger Equation._ SIAM Journal on Control and Optimization, Vol. 46, No. 1 : pp. 274-287.
  * [IK09] Ito, K. and Kunisch, K. (2009). _Asymptotic properties of feedback solutions for a class of quantum control problems._
  * [IS63] Segal, I. (1963) _Non-Linear Semi-Groups_. Annals of Mathematics, Second Series, Vol. 78, No. 2, pp. 339-364.
  * [IZL94] R. Illner, P.F. Zweifel, H. Lange, _Global existence, uniqueness and asymptotic behaviour of solutions of the Wigner-Poisson and Schrödinger-Poisson systems_ , Math. Methods Appl. Sci. 17 (1994), 349–376.
  * [JG07] Jiang, S. and Greengard, L. (2007). _Efficient representation of nonreflecting boundary conditions for the time‐dependent Schrödinger equation in two dimensions_. Communications on Pure and Applied MathematicsVolume 61, Issue 2.
  * [KG18] Kaye, J. and Greengard, L. (2020). _Transparent Boundary Conditions for the Time-Dependent Schrödinger Equation with a Vector Potential_. arXiv:1812.04200.
  * [L08] Lubich, C. (2008). _On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations_ , Mathematics of Computation, Volume 77, Number 264, Pages 2141-2153
  * [RS75] Reed, M. and Simon, B. (1975) _Methods of Modern Mathematical Physics_.
  * [SS07] Soffer, A., and Stucchio, C. _Open boundaries for the nonlinear Schrödinger equation._ Journal of Computational Physics 225.2 (2007): 1218-1232.
  * [SS09] Soffer, A., and Stucchio, C. _Multiscale resolution of shortwave‐longwave interaction._ Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences 62.1 (2009): 82-124.
  * [WG07] Werschnik, J. and Gross, E. (2007). _Quantum optimal control theory_. Journal of Physics B: Atomic, Molecular and Optical Physics, Volume 40, Number 18.
  * [WL07] Wu, X. and Li, X. _Absorbing boundary conditions for the time-dependent Schrödinger-type equations in $\mathbb{R}^{3}.$_ Phys. Rev. E 101, 013304.
  * [ZWW19] Zhai, S., Wang, D., Weng, Z. et al. Error Analysis and Numerical Simulations of Strang Splitting Method for Space Fractional Nonlinear Schrödinger Equation. J Sci Comput 81, 965–989 (2019).