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# Microscopic Derivation of Ginzburg–Landau Theory
and the BCS Critical Temperature Shift
in a Weak Homogeneous Magnetic Field

Andreas Deuchert, Christian Hainzl, Marcel Schaub

###### Abstract

Starting from the Bardeen–Cooper–Schrieffer (BCS) free energy functional, we
derive the Ginzburg–Landau functional for the case of a weak homogeneous
magnetic field. We also provide an asymptotic formula for the BCS critical
temperature as a function of the magnetic field. This extends the previous
works [17, 18] of Frank, Hainzl, Seiringer and Solovej to the case of external
magnetic fields with non-vanishing magnetic flux through the unit cell.

###### Contents

  1. 1 Introduction and Main Results
    1. 1.1 Introduction
    2. 1.2 Gauge-periodic samples
    3. 1.3 The BCS functional
    4. 1.4 The translation-invariant BCS functional
    5. 1.5 The Ginzburg–Landau functional
    6. 1.6 Main results
    7. 1.7 Organization of the paper and strategy of proof
  2. 2 Preliminaries
    1. 2.1 Schatten classes
    2. 2.2 Gauge-periodic Sobolev spaces
  3. 3 Trial States and their BCS Energy
    1. 3.1 The Gibbs states $\Gamma_{\Delta}$
    2. 3.2 The BCS energy of the states $\Gamma_{\Delta}$
    3. 3.3 The upper bound on (1.21) and proof of Theorem 2 (a)
  4. 4 Proofs of the Results in Section 3
    1. 4.1 Schatten norm estimates for operators given by product kernels
    2. 4.2 Proof of Proposition 3.4
    3. 4.3 Proof of Theorem 3.5
    4. 4.4 Proof of Lemma 3.1 and Proposition 3.2
    5. 4.5 Proof of Proposition 3.6
  5. 5 The Structure of Low-Energy States
    1. 5.1 A lower bound for the BCS functional
    2. 5.2 The first decomposition result
    3. 5.3 Uniform estimate on $\|\Psi\|_{2}$
    4. 5.4 Proof of Theorem 5.1
  6. 6 The Lower Bound on (1.21) and Proof of Theorem 2 (b)
    1. 6.1 The BCS energy of low-energy states
    2. 6.2 Estimate on the relative entropy
    3. 6.3 Conclusion
    4. 6.4 Proof of Lemma 6.2
  7. A Estimates on Eigenvalues and Eigenfunctions of $K_{T_{\mathrm{c}},B}-V$
    1. A.1 Phase approximation
    2. A.2 Asymptotics

##  1 Introduction and Main Results

###  1.1 Introduction

In 1950 Ginzburg and Landau (GL) introduced a phenomenological theory of
superconductivity that is based on a system of nonlinear partial differential
equations for a complex-valued wave function (the order parameter) and an
effective magnetic field [25]. Their theory is macroscopic in nature and
contains no reference to a microscopic mechanism behind the phenomenon of
superconductivity. The GL equations show a rich mathematical structure, which
has been investigated in great detail, see, e.g., [7, 8, 45, 44, 10, 11] and
references therein. They also inspired interesting new concepts beyond the
realm of their original application.

The first generally accepted microscopic theory of superconductivity was
discovered seven years later by Bardeen, Cooper and Schrieffer (BCS) in [2].
In a major breakthrough they realized that a pairing mechanism between the
conduction electrons (formation of Cooper pairs) causes the resistance in
certain materials to drop down to absolute zero if their temperature is
sufficiently low. This pairing phenomenon at low temperatures is induced by an
effective attraction between the electrons mediated by phonons, that is, by
the quantized lattice vibrations of the crystal formed by the ion cores. In
recognition of this contribution BCS were awarded the Nobel prize in physics
in 1972.

In the physics literature BCS theory is often formulated in terms of the gap
equation, which, in the absence of external fields, is a nonlinear integral
equation for a complex-valued function called the gap function (the order
parameter of BCS theory). The name of the equation is related to the fact that
its solution allows to determine the spectral gap of an effective quadratic
Hamiltonian that is open only in the superconducting phase. BCS theory also
has a variational interpretation, where the gap equation arises as the
Euler–Langrange equation of the BCS free energy functional. This free energy
functional can be obtained from a full quantum mechanical description of the
system by restricting attention to quasi-free states, a point of view that was
emphasized by Leggett in [37], see also [24]. In this formulation, the system
is described in terms of a one-particle density matrix and a Cooper pair wave
function.

Although it was originally introduced to describe the phase transition from
the normal to the superconducting state in metals and alloys, BCS theory can
also be applied to describe the phase transition to the superfluid state in
cold fermionic gases. In this case, the usual non-local phonon-induced
interaction in the gap equation needs to be replaced by a local pair
potential. From a mathematical point of view, the gap equation has been
studied for interaction kernels suitable to describe the physics of conduction
electrons in solids in [42, 3, 48, 50, 40, 51]. We refer to [28, 16, 31, 32,
22, 4, 19, 13] for works that investigate the translation-invariant BCS
functional with a local pair interaction. BCS theory in the presence of
external fields has been studied in [33, 5, 20, 12, 6].

A relation between the macroscopic GL theory and the microscopic BCS theory
was established by Gor’kov in 1959 [26]. He showed that, close to the critical
temperature, where the order parameters of both models are expected to be
small, GL theory arises from BCS theory when the free energy is expanded in
powers of the gap function. The first mathematically rigorous proof of this
relation was given by Frank, Hainzl, Seiringer and Solovej in 2012 [17]. They
showed that in the presence of weak and macroscopic external fields, the
macroscopic variations of the Cooper pair wave function of the system are
correctly described by GL theory if the temperature is close to the critical
temperature of the sample in an appropriate sense. The precise parameter
regime is as follows: The external electric field $W$ and the vector potential
$A$ of the external magnetic field are given by $h^{2}W(x)$ and $hA(hx)$,
respectively. Here $0<h\ll 1$ denotes the ratio between the microscopic and
the macroscopic length scale of the system. Such external fields change the
energy by an amount of the order $h^{2}$ and it is therefore natural to
consider temperatures $T=T_{\mathrm{c}}(1-Dh^{2})$ with $D>0$, where
$T_{\mathrm{c}}$ denotes the critical temperature of the sample in the absence
of external fields. Within this setup it has been shown in [17] that the
correction to the BCS free energy on the order $h^{4}$ is correctly described
by GL theory. Moreover, the Cooper pair wave function of the system is, to
leading order in $h$, given by

$\alpha(x,y)=h\,\alpha_{*}(x-y)\,\psi\left(\frac{h(x+y)}{2}\right).$ (1.1)

Here, $\psi$ denotes the order parameter of GL theory and $\alpha_{*}(x-y)$ is
related to the Cooper pair wave function in the absence of external fields.

External electric and magnetic fields may change the critical temperature of a
superconductor and this shift is expected to be described by GL theory. A
justification of this claim has been provided in [18]. More precisely, it has
been shown that, within the setup of [17] described above, the critical
temperature of the sample obeys the asymptotic expansion

$T_{\mathrm{c}}(h)=T_{\mathrm{c}}(1-D_{\mathrm{c}}h^{2})+o(h^{2}),$ (1.2)

where the constant $D_{\mathrm{c}}$ can be computed using linearized GL
theory.

One crucial assumption in [17] and [18] is that the vector potential related
to the external magnetic field is periodic. In this case the magnetic flux
through the unit cell equals zero. An important step towards an extension of
the results in [18] to the case of a magnetic field with non-vanishing
magnetic flux through the unit cell has been provided by Frank, Hainzl and
Langmann in [15]. In this article the authors consider the problem of
computing the BCS critical temperature shift in the presence of a weak
homogeneous magnetic field within linearized BCS theory. Heuristically, this
approximation is justified by the fact that linearized GL theory is sufficient
to predict the critical temperature shift, see the discussion in the previous
paragraph. In the physics literature this approximation appears in [34, 49,
36, 35], for instance.

The aim of the present article is to extend the results in [17] and [18] to a
setting with an external magnetic field having non-zero flux through the unit
cell. More precisely, we consider a large periodic sample of fermionic
particles subject to a weak homogeneous magnetic field
$\mathbf{B}\in\mathbb{R}^{3}$. The temperatures $T$ is chosen such that
$(T_{\mathrm{c}}-T)/T_{\mathrm{c}}=D|\mathbf{B}|$ with $D\in\mathbb{R}$. We
show that the correction of the BCS free energy of the sample at the order
$|\mathbf{B}|^{2}$ is given by GL theory. Moreover, to leading order in
$|\mathbf{B}|$ the Cooper pair wave function of the system is given by (1.1)
with $h$ replaced by $|\mathbf{B}|^{\nicefrac{{1}}{{2}}}$. We also show that
the BCS critical temperature shift caused by the external magnetic field is
given by (1.2) with $D_{\mathrm{c}}$ determined by linearized GL theory. Our
analysis yields the same formula that was computed within the framework of
linearized BCS theory in [15]. This can be interpreted as a justification of
the approximation to use linearized BCS theory to compute the BCS critical
temperature shift. The main new ingredient of our proof are a priori bounds
for certain low-energy states of the BCS functional that include the magnetic
field.

###  1.2 Gauge-periodic samples

We consider a $3$-dimensional sample of fermionic particles described by BCS
theory that is subject to an external magnetic field $\mathbf{B}:=Be_{3}$ with
strength $B>0$, pointing in the $e_{3}$-direction. We choose the magnetic
vector potential $\mathbf{A}(x):=\frac{1}{2}\mathbf{B}\wedge x$ so that
$\operatorname{curl}\mathbf{A}=\mathbf{B}$, where $\mathbf{B}\wedge
x\in\mathbb{R}^{3}$ denotes the cross product of two vectors. The
corresponding magnetic momentum operator $\pi:=-\mathrm{i}\nabla+\mathbf{A}$
commutes with the magnetic translations $T(v)$, defined by

$\displaystyle T(v)f(x)$
$\displaystyle:=\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}\cdot(v\wedge
x)}f(x+v),$ $\displaystyle v$ $\displaystyle\in\mathbb{R}^{3}.$ (1.3)

The family $\\{T(v)\\}_{v\in\mathbb{R}^{3}}$ obeys the relation
$T(v+w)=\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}\cdot(v\wedge
w)}\;T(v)T(w)$, that is, it is a unitary representation of the Heisenberg
group. We assume that our system is periodic with respect to the Bravais
lattice $\Lambda_{B}=\sqrt{2\pi B^{-1}}\,\mathbb{Z}^{3}$ with fundamental cell

$\displaystyle Q_{B}$ $\displaystyle:=\bigl{[}0,\sqrt{2\pi
B^{-1}}\bigr{]}^{3}\subseteq\mathbb{R}^{3}.$ (1.4)

The magnetic flux through the unit cell $Q_{B}$ equals
$\mathbf{B}\cdot(b_{1}\wedge b_{2})=2\pi$, where $b_{i}=\sqrt{2\pi
B^{-1}}\,e_{i}$ are the basis vectors spanning $\Lambda_{B}$. This assures
that the abelian subgroup $\\{T(\lambda)\\}_{\lambda\in\Lambda_{B}}$ is a
unitary representation of the lattice group.

###  1.3 The BCS functional

In BCS theory a state is described by a generalized fermionic one-particle
density matrix, that is, by a self-adjoint operator $\Gamma$ on
$L^{2}(\mathbb{R}^{3})\oplus L^{2}(\mathbb{R}^{3})$ which satisfies
$0\leqslant\Gamma\leqslant 1$ and is of the form

$\displaystyle\Gamma=\begin{pmatrix}\gamma&\alpha\\\
\overline{\alpha}&1-\overline{\gamma}\end{pmatrix}.$ (1.5)

Here, $\overline{\alpha}=J\alpha J$ with the Riesz identification operator
$J\colon L^{2}(\mathbb{R}^{3})\rightarrow L^{2}(\mathbb{R}^{3})$,
$f\mapsto\overline{f}$, realized by complex conjugation. The condition
$\Gamma=\Gamma^{*}$ implies that the one-particle density matrix $\gamma$ is a
self-adjoint operator. It also implies that the Cooper pair wave function
$\alpha(x,y)$, the kernel of $\alpha$, is symmetric under the exchange of its
coordinates. The symmetry of $\alpha$ is due to the fact that we exclude spin
variables from our description and assume that Cooper pairs are in a spin
singlet state. The condition $0\leqslant\Gamma\leqslant 1$ implies
$0\leqslant\gamma\leqslant 1$ as well as that $\gamma$ and $\alpha$ are
related through the operator inequality

$\displaystyle\alpha\alpha^{*}\leqslant\gamma(1-\gamma).$ (1.6)

A BCS state $\Gamma$ is called _gauge-periodic_ if
$\mathbf{T}(\lambda)\,\Gamma\,\mathbf{T}(\lambda)^{*}=\Gamma$ holds for every
$\lambda\in\Lambda_{B}$, with the magnetic translations $\mathbf{T}(\lambda)$
on $L^{2}(\mathbb{R}^{3})\oplus L^{2}(\mathbb{R}^{3})$ defined by

$\displaystyle\mathbf{T}(v)$ $\displaystyle:=\begin{pmatrix}T(v)&0\\\
0&\overline{T(v)}\end{pmatrix},$ $\displaystyle v$
$\displaystyle\in\mathbb{R}^{3}.$

For $\gamma$ and $\alpha$, this implies $T(\lambda)\gamma
T(\lambda)^{*}=\gamma$ and
$T(\lambda)\,\alpha\,\overline{T(\lambda)}^{*}=\alpha$ or, in terms of their
kernels,

$\displaystyle\gamma(x,y)$
$\displaystyle=\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}\cdot(\lambda\wedge(x-y))}\;\gamma(x+\lambda,y+\lambda),$
$\displaystyle\alpha(x,y)$
$\displaystyle=\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}\cdot(\lambda\wedge(x+y))}\;\alpha(x+\lambda,y+\lambda),$
$\displaystyle\lambda\in\Lambda_{B}.$ (1.7)

######  Remark 1.1.

Since we are interested in the situation of a constant magnetic field it seems
natural to consider magnetically translation-invariant BCS states, that is,
states obeying $\mathbf{T}(v)\,\Gamma\,\mathbf{T}(v)^{*}=\Gamma$ for every
$v\in\mathbb{R}^{3}$. However, in this case one obtains a trivial model
because the Cooper pair wave function $\alpha$ of a magnetically translation-
invariant state necessarily vanishes. To see this, we note that $\alpha$
satisfies $T(v)\,\alpha\,\overline{T(v)}^{*}=\alpha$ for all
$v\in\mathbb{R}^{3}$. Using this and the relation
$T(v+w)\,\alpha\,\overline{T(v+w)}^{*}=\mathrm{e}^{\mathrm{i}\mathbf{B}\cdot(v\wedge
w)}\,T(v)T(w)\,\alpha\,\overline{T(w)}^{*}\,\overline{T(v)}^{*}$, we conclude
that $\alpha=0$.

A gauge-periodic BCS state $\Gamma$ is said to be _admissible_ if

$\displaystyle\operatorname{Tr}\bigl{[}\gamma+(-\mathrm{i}\nabla+\mathbf{A})^{2}\gamma\bigr{]}<\infty$
(1.8)

holds. Here $\operatorname{Tr}[A]$ denotes the trace per unit volume of $A$,
i.e.,

$\displaystyle\operatorname{Tr}[A]$
$\displaystyle:=\frac{1}{|Q_{B}|}\operatorname{Tr}_{L^{2}(Q_{B})}[\chi
A\chi],$ (1.9)

with the characteristic function $\chi$ of the cube $Q_{B}$ in (1.4). By
$\operatorname{Tr}_{L^{2}(Q_{B})}[A]$ we denote the usual trace of an operator
$A$ on $L^{2}(Q_{B})$. The condition in (1.8) is meant to say that $\gamma$
and $(-\mathrm{i}\nabla+\mathbf{A})^{2}\gamma$ are locally trace class, that
is, they are trace class with respect to the trace in (1.9). Eq. (1.8), the
same inequality with $\gamma$ replaced by $\overline{\gamma}$, and the
inequality in (1.6) imply that $\alpha$,
$(-\mathrm{i}\nabla+\mathbf{A})\alpha$, and
$(-\mathrm{i}\nabla+\mathbf{A})\overline{\alpha}$ are locally Hilbert–Schmidt.
In Section 2 below we will express this property in terms of
$H^{1}$-regularity of the kernel of $\alpha$.

For any admissible BCS state $\Gamma$, we define the Bardeen–Cooper–Schrieffer
free energy functional (in the following: BCS functional) at temperature
$T\geqslant 0$ by

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma)$
$\displaystyle:=\operatorname{Tr}\bigl{[}\bigl{(}(-\mathrm{i}\nabla+\mathbf{A})^{2}-\mu\bigr{)}\gamma\bigr{]}-T\,S(\Gamma)-\frac{1}{|Q_{B}|}\int_{Q_{B}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;V(r)\,|\alpha(X,r)|^{2},$
(1.10)

with the von Neumann entropy per unit volume
$S(\Gamma)=-\operatorname{Tr}[\Gamma\ln(\Gamma)]$ and the chemical potential
$\mu\in\mathbb{R}$. The particles interact via a two-body potential $V\in
L^{\nicefrac{{3}}{{2}}}(\mathbb{R}^{3})+L_{\varepsilon}^{\infty}(\mathbb{R}^{3})$.
Furthermore, we introduced center-of-mass and relative coordinates
$X=\frac{x+y}{2}$ and $r=x-y$. Here and in the following, we abuse notation
slightly by writing $\alpha(X,r)\equiv\alpha(x,y)$.

######  Remark 1.2.

We opt for the above set-up because the solution of the problem for the
constant magnetic field already contains the main difficulties of the case of
a general magnetic field. This is related to the fact that the vector
potential of any magnetic field with non-zero flux through the unit cell can
be written as a sum of a vector potential of a homogeneous magnetic field and
a periodic vector potential, see e.g. [47, Proposition 4.1]. The latter can be
treated in some sense as a perturbation, see [17, 18]. However, this is not
true for the constant magnetic field, see Remark 1.6 (a) below. To solve the
general case it is therefore crucial to understand the case of a homogeneous
magnetic field. To keep the presentation to a reasonable length and to be able
to convey the main ideas more clearly, we therefore decided to present this
case first. We plan to extend our treatment to the case of a general magnetic
field in a second paper. One motivation to treat general periodic magnetic
fields with non-zero flux through the unit cell stems from the fact that it is
an interesting and highly relevant problem to consider magnetic fields that
are chosen self-consistently.

The BCS functional is bounded from below and coercive on the set of admissible
states. More precisely, it can be shown that the kinetic energy dominates the
entropy and the interaction energy, i.e., there is a constant $C>0$ such that
for all admissible $\Gamma$, we have

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma)$
$\displaystyle\geqslant\frac{1}{2}\operatorname{Tr}\bigl{[}\gamma+(-\mathrm{i}\nabla+\mathbf{A})^{2}\gamma\bigr{]}-C.$
(1.11)

The unique minimizer of the BCS functional among admissible states with
$\alpha=0$ is given by

$\displaystyle\Gamma_{0}$ $\displaystyle:=\begin{pmatrix}\gamma_{0}&0\\\
0&1-\overline{\gamma}_{0}\end{pmatrix},$ $\displaystyle\gamma_{0}$
$\displaystyle:=\frac{1}{1+\mathrm{e}^{((-\mathrm{i}\nabla+\mathbf{A})^{2}-\mu)/T}}.$
(1.12)

Since $\Gamma_{0}$ is also the unique minimizer of the BCS functional for
sufficiently large temperatures $T$, it is called the normal state. We define
the BCS free energy by

$\displaystyle
F^{\mathrm{BCS}}(B,T):=\inf\bigl{\\{}\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma_{0}):\Gamma\text{
admissible}\bigr{\\}}$ (1.13)

and say that our system is superconducting if $F^{\mathrm{BCS}}(B,T)<0$, that
is, if the minimal energy is strictly smaller than that of the normal state.
In this work we are interested in the regime of weak magnetic fields $0<B\ll
1$. Our goal is to obtain an asymptotic expansion of $F^{\mathrm{BCS}}(B,T)$
in powers of $B$ that allows us to derive Ginzburg–Landau theory, and to show
how the BCS critical temperature depends on the magnetic field $B$. For our
main results to hold, we need the following assumptions concerning the
regularity of the interaction potential $V$.

######  Assumption 1.3.

We assume that the interaction potential $V$ is a nonnegative, radial function
such that $(1+|\cdot|^{2})V\in L^{\infty}(\mathbb{R}^{3})$.

######  Remark 1.4.

Our main results Theorem 1 and Theorem 2 still hold if the assumption
$V\geqslant 0$ is dropped. We only use it in Appendix A when we investigate
the spectral properties of a certain linear operator involving $V$. These
statements still hold in the case of potentials without a definite sign but
their proof is longer. A proof of these statements in the general setting can
be found in the PhD thesis of M. Schaub. We expect our results to be true also
if $V$ has moderate local singularities. Furthermore, it may be possible to
slightly weaken the decay assumptions of $V$. We choose to work with the
assumptions above to keep the presentation at a reasonable length.

###  1.4 The translation-invariant BCS functional

If no external fields are present, i.e. if $\mathbf{B}=0$, we describe the
system by translation-invariant states, that is, we assume that the kernels of
$\gamma$ and $\alpha$ are of the form $\gamma(x-y)$ and $\alpha(x-y)$. To
define the trace per unit volume we choose a cube of side length $1$. The
resulting translation-invariant BCS functional and its infimum minus the free
energy of the normal state are denoted by
$\mathcal{F}^{\mathrm{BCS}}_{\mathrm{ti},T}$ and
$F^{\mathrm{BCS}}_{\mathrm{ti}}(T)$, respectively. This functional has been
studied in detail in [28], see also [27] and the references therein, where it
has been shown that there is a unique critical temperature
$T_{\mathrm{c}}\geqslant 0$ such that
$\mathcal{F}^{\mathrm{BCS}}_{\mathrm{ti},T}$ has a minimizer with $\alpha\neq
0$ if $T<T_{\mathrm{c}}$. For $T\geqslant T_{\mathrm{c}}$ the normal state in
(1.12) with $B=0$ is the unique minimizer. In terms of the energy, we have
$F^{\mathrm{BCS}}_{\mathrm{ti}}(T)<0$ for $T<T_{\mathrm{c}}$, while
$F^{\mathrm{BCS}}_{\mathrm{ti}}(T)=0$ if $T\geqslant T_{\mathrm{c}}$.

It has also been shown in [28] that the critical temperature $T_{\mathrm{c}}$
can be characterized via a linear criterion. More precisely, the critical
temperature is determined by the unique value of $T$ such that the operator

$K_{T}-V$

acting on $L^{2}_{\mathrm{sym}}(\mathbb{R}^{3})$, the space of reflection-
symmetric square-integrable functions, has zero as its lowest eigenvalue.
Here, $K_{T}=K_{T}(-\mathrm{i}\nabla)$ with the symbol

$\displaystyle K_{T}(p):=\frac{p^{2}-\mu}{\tanh\frac{p^{2}-\mu}{2T}}.$ (1.14)

It should be noted that the function $T\mapsto K_{T}(p)$ is strictly monotone
increasing for fixed $p\in\mathbb{R}^{3}$, and that $K_{T}(p)\geqslant 2T$ if
$\mu\geqslant 0$ and $K_{T}(p)\geqslant|\mu|/\tanh(|\mu|/(2T))$ if $\mu<0$.
Our assumptions on $V$ guarantee that the essential spectrum of the operator
$K_{T}-V$ equals $[2T,\infty)$ if $\mu\geqslant 0$ and
$[|\mu|/\tanh(|\mu|/(2T)),\infty)$ if $\mu<0$. Accordingly, an eigenvalue at
zero is necessarily isolated and of finite multiplicity.

The results in [28] have been obtained in the case where the Cooper pair wave
function $\alpha(x)$ is not necessarily an even function (as opposed to our
setup), which means that $K_{T_{\mathrm{c}}}(-\mathrm{i}\nabla)-V$ has to be
understood to act on $L^{2}(\mathbb{R}^{3})$. The results in [28], however,
equally hold if the symmetry of $\alpha$ is enforced.

We are interested in the situation where (a) $T_{\mathrm{c}}>0$ and (b) the
translation-invariant BCS functional has a unique minimizer with a radial
Cooper pair wave function ($s$-wave Cooper pairs) for $T$ close to
$T_{\mathrm{c}}$. This is implied by the following assumption. Part (b) should
be compared to [13, Theorem 2.8].

######  Assumption 1.5.

  1. (a)

We assume that $T_{\mathrm{c}}>0$. If $V\geqslant 0$ and it does not vanish
identically this is automatically implied, see [28, Theorem 3]. In the case of
an interaction potential without a definite sign it is a separate assumption.

  2. (b)

We assume that the lowest eigenvalue of $K_{T_{\mathrm{c}}}-V$ is simple.

In the following we denote by $\alpha_{*}$ the unique ground state of the
operator $K_{T_{\mathrm{c}}}-V$, i.e.,

$\displaystyle K_{T_{\mathrm{c}}}\alpha_{*}=V\alpha_{*}.$ (1.15)

We choose the normalization of $\alpha_{*}$ such that it is real-valued and
$\|\alpha_{*}\|_{L^{2}(\mathbb{R}^{3})}=1$. Since $V$ is a radial function and
$\alpha_{*}$ is the unique solution of (1.15) it follows that $\alpha_{*}$ is
radial, too.

###  1.5 The Ginzburg–Landau functional

We call a function $\Psi$ on $Q_{B}$ gauge-periodic if it is left invariant by
the magnetic translations of the form

$\displaystyle T_{B}(\lambda)\Psi(X)$
$\displaystyle:=\mathrm{e}^{\mathrm{i}\mathbf{B}\cdot(\lambda\wedge
X)}\;\Psi(X+\lambda),$ $\displaystyle\lambda$ $\displaystyle\in\Lambda_{B}.$
(1.16)

The operator $T(\lambda)$ in (1.3) coincides with $T_{B}(\lambda)$ when
$\mathbf{B}$ is replaced by $2\mathbf{B}$.

Let $\Lambda_{0},\Lambda_{2},\Lambda_{3}>0$ and $D\in\mathbb{R}$ be given. For
$B>0$ and a gauge periodic function $\Psi$, the Ginzburg–Landau functional is
defined by

$\displaystyle\mathcal{E}^{\mathrm{GL}}_{D,B}(\Psi)$
$\displaystyle:=\frac{1}{B^{2}}\frac{1}{|Q_{B}|}\int_{Q_{B}}\mathrm{d}X\;\bigl{\\{}\Lambda_{0}\;|(-\mathrm{i}\nabla+2\mathbf{A})\Psi(X)|^{2}-DB\,\Lambda_{2}\,|\Psi(X)|^{2}+\Lambda_{3}\,|\Psi(X)|^{4}\bigr{\\}}.$
(1.17)

We highlight the factor of $2$ in front of the magnetic potential in (1.17)
and that the definition of the magnetic translation in (1.16) differs from
that in (1.3) by a factor $2$. These two factors reflect the fact that $\Psi$
describes Cooper pairs, which carry twice the charge of a single particle. The
Ginzburg–Landau energy

$\displaystyle
E^{\mathrm{GL}}(D):=\inf\bigl{\\{}\mathcal{E}^{\mathrm{GL}}_{D,B}(\Psi):\Psi\in
H_{\mathrm{mag}}^{1}(Q_{B})\bigr{\\}}$

is independent of $B$ by scaling. More precisely, for given $\psi$ the
function

$\displaystyle\Psi(X)$
$\displaystyle:=\sqrt{B}\;\psi\bigl{(}\sqrt{B}\,X\bigr{)},$ $\displaystyle
X\in\mathbb{R}^{3},$ (1.18)

satisfies

$\displaystyle\mathcal{E}^{\mathrm{GL}}_{D,B}(\Psi)=\mathcal{E}_{D,1}^{\mathrm{GL}}(\psi).$
(1.19)

We also define the critical parameter

$\displaystyle D_{\mathrm{c}}$
$\displaystyle:=\frac{\Lambda_{0}}{\Lambda_{2}}\inf\operatorname{spec}_{L_{\mathrm{mag}}^{2}(Q_{1})}\bigl{(}(-\mathrm{i}\nabla+e_{3}\wedge
X)^{2}\bigr{)},$ (1.20)

where the infimum is taken over gauge-periodic square-integrable functions.
Its definition is motivated by the fact that $E^{\mathrm{GL}}(D)<0$ if
$D>D_{\mathrm{c}}$ and $E^{\mathrm{GL}}(D)=0$ if $D\leqslant D_{\mathrm{c}}$.
This should be compared to [18, Lemma 2.5]. In our situation with a constant
magnetic field the lowest eigenvalue of the Hamiltonian in (1.20) equals $2$,
see [47, Eq. (6.2)], and $D_{\mathrm{c}}$ is explicit. In the situation of
[18], where general external fields excluding the constant magnetic field are
present, the parameter $D_{\mathrm{c}}$ is not explicit.

###  1.6 Main results

Our first main result concerns the asymptotics of the BCS free energy in
(1.13) in the regime $B\ll 1$. It also contains a statement about the
asymptotics of the Cooper pair wave function of states $\Gamma$, whose energy
$\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma)$ has the same asymptotic
behavior as the BCS free energy (approximate minimizers). The precise
statement is captured in the following theorem.

######  Theorem 1.

Let Assumptions 1.3 and 1.5 hold, let $D\in\mathbb{R}$, and let the
coefficients $\Lambda_{0},\Lambda_{2},\Lambda_{3}>0$ be given by (3.22)-(3.24)
below. Then there are constants $C>0$ and $B_{0}>0$ such that for all
$0<B\leqslant B_{0}$, we have

$\displaystyle
F^{\mathrm{BCS}}(B,\,T_{\mathrm{c}}(1-DB))=B^{2}\;\bigl{(}E^{\mathrm{GL}}(D)+R\bigr{)},$
(1.21)

with $R$ satisfying the estimate

$\displaystyle CB\geqslant
R\geqslant-\mathcal{R}:=-CB^{\nicefrac{{1}}{{12}}}.$ (1.22)

Moreover, for any approximate minimizer $\Gamma$ of
$\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}$ at $T=T_{\mathrm{c}}(1-DB)$ in the
sense that

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma_{0})\leqslant
B^{2}\bigl{(}E^{\mathrm{GL}}(D)+\rho\bigr{)}$ (1.23)

holds for some $\rho\geqslant 0$, we have the decomposition

$\displaystyle\alpha(X,r)=\Psi(X)\,\alpha_{*}(r)+\sigma(X,r)$ (1.24)

for the Cooper pair wave function $\alpha=\Gamma_{12}$. Here, $\sigma$
satisfies

$\displaystyle\frac{1}{|Q_{B}|}\int_{Q_{B}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;|\sigma(X,r)|^{2}$
$\displaystyle\leqslant CB^{\nicefrac{{11}}{{6}}},$ (1.25)

$\alpha_{*}$ is the normalized zero energy eigenstate of
$K_{T_{\mathrm{c}}}-V$, and the function $\Psi$ obeys

$\displaystyle\mathcal{E}^{\mathrm{GL}}_{D,B}(\Psi)\leqslant
E^{\mathrm{GL}}(D)+\rho+\mathcal{R}.$ (1.26)

Our second main result concerns the shift of the BCS critical temperature that
is caused by the external magnetic field.

######  Theorem 2.

Let Assumptions 1.3 and 1.5 hold. Then there are constants $C>0$ and $B_{0}>0$
such that for all $0<B\leqslant B_{0}$ the following holds:

  1. (a)

Let $0<T_{0}<T_{\mathrm{c}}$. If the temperature $T$ satisfies

$T_{0}\leqslant T\leqslant
T_{\mathrm{c}}\,(1-B\,(D_{\mathrm{c}}+C\,B^{\nicefrac{{1}}{{2}}}))$ (1.27)

with $D_{\mathrm{c}}$ in (1.20), then we have

$F^{\mathrm{BCS}}(B,T)<0.$

  2. (b)

If the temperature $T$ satisfies

$T\geqslant T_{\mathrm{c}}\,(1-B\,(D_{\mathrm{c}}-\mathcal{R}))$ (1.28)

with $D_{\mathrm{c}}$ in (1.20) and $\mathcal{R}$ in (1.22), then we have

$\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma_{0})>0$

unless $\Gamma=\Gamma_{0}$.

######  Remarks 1.6.

  1. (a)

Theorem 1 and Theorem 2 extend similar results in [17, Theorem 1] and [18,
Theorem 2.4] to the case of a homogeneous magnetic field. Such a magnetic
field has a non-periodic vector potential and a non-zero magnetic flux through
the unit cell $Q_{B}$. The main reason why the problem with a homogeneous
magnetic field is more complicated is that it cannot be treated as a
perturbation of the Laplacian. More precisely, it was possible in [17, 18] to
work with a priori bounds for low-energy states that only involve the
Laplacian and not the external fields. As noticed in [15], see the discussion
below Remark 6, this is not possible in the case of a homogeneous magnetic
field. In the proof of comparable a priori estimates involving the homogeneous
magnetic field, see Theorem 5.1 below, we have to deal with the fact that the
components of the magnetic momentum operator do not commute, which leads to
significant technical difficulties.

  2. (b)

If we compare Theorem 1 to [17, Theorem 1] or Theorem 2 to [18, Theorem 2.4]
we note the following technical differences: (1) The parameter $h$ in [17, 18]
equals $B^{\nicefrac{{1}}{{2}}}$ in our work. (2) We use microscopic
coordinates while macroscopic coordinates are used in [17, 18]. (3) Our free
energy is normalized by a volume factor, see (1.9) and (1.10). This is not the
case in [17, 18]. (4) The leading order of the Cooper pair wave function in
[17, Theorem 1] is of the form

$\frac{1}{2}\alpha_{*}(x-y)(\Psi(x)+\Psi(y)).$ (1.29)

This should be compared to (1.24), where relative and center-of-mass
coordinates are used. Using the a priori bound for the $L^{2}$-norm of
$\nabla\Psi$ below (5.61) in [17], one can see that (1.29) equals the first
term in (1.24) to leading order in $h$. The analogue in our setup does not
seem to be correct.

  3. (c)

The Ginzburg–Landau energy appears at the order $B^{2}$. This should be
compared to the free energy of the normal state, which is of order $1$.

  4. (d)

To appreciate the bound in (1.25), we note that the first term in the
decomposition of $\alpha$ in (1.24) obeys

$\displaystyle\frac{1}{|Q_{B}|}\int_{Q_{B}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;|\Psi(X)\alpha_{*}(r)|^{2}$
$\displaystyle=O(B).$

  5. (e)

We stated Theorem 1 with fixed $D\in\mathbb{R}$. Our explicit error bounds
show that $D$ is allowed to vary with $B$ as long as there is a
$B$-independent constant $D_{0}>0$ such that $|D|\leqslant D_{0}$ holds.

  6. (f)

Theorem 2 gives bounds on the range of temperatures where superconductivity is
present, see (1.27), or absent, see (1.28). The interpretation of this theorem
is that for small magnetic fields $B$ the critical temperature obeys the
asymptotic expansion

$T_{\mathrm{c}}(B)=T_{\mathrm{c}}(1-D_{\mathrm{c}}B)+o(B).$ (1.30)

We highlight that $T_{\mathrm{c}}$ is determined by the translation-invariant
problem, and that $D_{\mathrm{c}}$ is given by the macroscopic (linearized) GL
theory. The same result has been obtained in [15, Theorem 4] in the case of
linearized BCS theory. Theorem 2 can therefore be interpreted as a
justification of this approximation. Eq. (1.30) allows us to compute the upper
critical field $B_{c2}$. That is, the magnetic field, above which, for a given
temperature $T$, superconductivity is absent. In particular, it allows us to
compute the derivative of $B_{c2}$ with respect to $T$ at the critical
temperature from the BCS functional. For more details we refer to [15,
Appendix A].

  7. (g)

We expect that the assumption $0<T_{0}\leqslant T$ for some arbitrary but
$B$-independent constant $T_{0}$ in Theorem 2 (a) is of technical nature. We
need this assumption, which similarly appears in [15, Theorem 4], because our
trial state analysis in Section 3 breaks down when the temperature $T$
approaches zero. This is related to the fact that the Fermi distribution
function $f_{T}(x)=(\mathrm{e}^{x/T}+1)^{-1}$ cannot be represented by a
Cauchy-integral uniformly in the temperature. We note that there is no such
restriction in Theorem 2 (b). It is also not needed in [18, Theorem 2.4].

###  1.7 Organization of the paper and strategy of proof

In Section 2 we complete the introduction of our mathematical setup. We recall
several properties of the trace per unit volume and introduce the relevant
spaces of gauge-periodic functions.

Section 3 is dedicated to a trial state analysis. We start by introducing a
class of Gibbs states, whose Cooper pair wave function is given by a product
of the form $\alpha_{*}(r)\Psi(X)$ to leading order in $B$ with $\alpha_{*}$
in (1.15) and with a gauge-periodic function $\Psi$ on $Q_{B}$. We state and
motivate several results concerning these Gibbs states and their BCS free
energy, whose proofs are deferred to Section 4. Afterwards, these statements
are used to prove the upper bound on (1.21) as well as Theorem 2 (a). As will
be explained below, they are also relevant for the proofs of the lower bound
in (1.21) and of Theorem 2 (b) in Section 6.

Section 4 contains the proof of the results concerning the Gibbs states and
their BCS free energy that have been stated without proof in Section 3. Our
analysis is based on an extension of the phase approximation method, which has
been pioneered in the framework of linearized BCS theory in [15], to our
nonlinear setting. The phase approximation is a well-known tool in the physics
literature, see, e.g., [34], and has also been used in the mathematical
literature to study spectral properties of Schrödinger operators involving a
magnetic field, for instance in [9, 41]. Our approach should be compared to
the trial state analysis in [17, 18], where a semi-classical expansion is
used. The main novelty of our trial state analysis is Lemma 4.2, where we
provide an alternative way to compute a certain trace function involving the
trial state. It should be compared to the related part in the proof of [17,
Theorem 2]. While the analysis in [17] uses a Cauchy integral representation
of the function $z\mapsto\ln(1+\mathrm{e}^{-z})$, our approach is based on a
product expansion of the hyperbolic cosine in terms of Matsubara frequencies.
In this way we obtain better decay properties in the subsequent resolvent
expansion, which, in our opinion, simplifies the analysis considerably.

Section 5 contains the proof of a priori estimates for BCS states, whose BCS
free energy is smaller than or equal to that of the normal state $\Gamma_{0}$
in (1.12) plus a correction of the order $B^{2}$ (low-energy states). The
result is captured in Theorem 5.1, which is the main novelty of the present
article. It states that the Cooper pair wave function of any low-energy state
in the above sense has a Cooper pair wave function, which is, to leading order
in $B$, given by a product of the form $\alpha_{*}(r)\Psi(X)$ with
$\alpha_{*}(r)$ in (1.15) and with a gauge-periodic function $\Psi(X)$ on
$Q_{B}$. Furthermore, the function $\Psi(X)$ obeys certain bounds, which show
that it is slowly varying and small in an appropriate sense. As explained in
Remark 1.6 (a), the main difficulty to overcome is that our a priori bounds
involve the magnetic field. Therefore, we have to deal with the non-
commutativity of the components of the magnetic momentum operator. The step
where this problem appears most prominently is in the proof of Proposition
5.7.

The proof of the lower bound on (1.21) and of Theorem 2 (b) is provided in
Section 6, which mostly follows the strategy in [17, Section 6] and [18,
Section 4.2]. Two main ingredients for the analysis in this section are the
trial state analysis in Section 3 and Section 4, and the a priori bounds for
low-energy states in Section 5. From Theorem 5.1 we know that the Cooper pair
wave function of any low-energy state has a product structure to leading order
in $B$. The main idea of the proof of the lower bound in (1.21) is to
construct a Gibbs state, whose Cooper pair wave function has the same
asymptotics to leading order in $B$. The precise characterization of the
Cooper pair wave function of the Gibbs state in Section 3 and the a priori
bounds in Theorem 5.1 then allow us to bound the BCS free energy of the
original state from below in terms of that of the Gibbs state. The latter has
been computed with sufficient precision in Section 3 and Section 4.

Throughout the paper, $c$ and $C$ denote generic positive constants that
change from line to line. We allow them to depend on the various fixed
quantities like $B_{0}$, $\mu$, $T_{\mathrm{c}}$, $V$, $\alpha_{*}$, etc.
Further dependencies are indexed.

##  2 Preliminaries

###  2.1 Schatten classes

In our proofs we frequently use Schatten norms of periodic operators, which
are defined with respect to the trace per unit volume in (1.9). In this
section we recall some basic facts about these norms.

A gauge-periodic operator $A$ belongs to the $p$${}^{\textrm{th}}$ local von-
Neumann–Schatten class $\mathcal{S}^{p}$ with $1\leqslant p<\infty$ if it has
finite $p$-norm, that is, if
$\|A\|_{p}^{p}:=\operatorname{Tr}(|A|^{p})<\infty$. By $\mathcal{S}^{\infty}$
we denote the set of bounded gauge-periodic operators and $\|\cdot\|_{\infty}$
is the usual operator norm. For the above norms the triangle inequality

$\displaystyle\|A+B\|_{p}\leqslant\|A\|_{p}+\|B\|_{p}$

holds for $1\leqslant p\leqslant\infty$. Moreover, for $1\leqslant
p,q,r\leqslant\infty$ with $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$ we have the
general Hölder inequality

$\displaystyle\|AB\|_{r}\leqslant\|A\|_{p}\|B\|_{q}.$ (2.1)

It is important to note that the above norms are not monotone decreasing in
the index $p$. This should be compared to the usual Schatten norms, where such
a property holds. The familiar inequality

$\displaystyle|\operatorname{Tr}A|\leqslant\|A\|_{1}$

is true also in the case of local Schatten norms.

The above inequalities can be reduced to the case of the usual Schatten norms,
see, e.g., [46], using the magnetic Bloch–Floquet decomposition. We refer to
[43, Section XIII.16] for an introduction to the Bloch–Floquet transformation
and to [23] for a particular treatment of the magnetic case. More
specifically, for a gauge-periodic operator $A$ we use the unitary equivalence

$\displaystyle A\cong\int^{\oplus}_{[0,\sqrt{2\pi B}]^{3}}\mathrm{d}k\;A_{k}$

to write the trace per unit volume as

$\operatorname{Tr}A=\int_{[0,\sqrt{2\pi
B}]^{3}}\mathrm{d}k\;\operatorname{Tr}_{L^{2}(Q_{B})}A_{k},$ (2.2)

where $\operatorname{Tr}_{L^{2}(Q_{B})}$ denotes the usual trace over
$L^{2}(Q_{B})$. The inequalities for the trace per unit volume from above
follow from the usual ones when we use that $(AB)_{k}=A_{k}B_{k}$ holds for
two gauge-periodic operators $A$ and $B$.

###  2.2 Gauge-periodic Sobolev spaces

In this section we introduce several spaces of gauge-periodic functions, which
will be used to describe the center-of-mass part of Cooper pair wave
functions.

For $1\leqslant p<\infty$, the space $L_{\mathrm{mag}}^{p}(Q_{B})$ consists of
all $L_{\mathrm{loc}}^{p}(\mathbb{R}^{3})$-functions $\Psi$, which satisfy
$T_{B}(\lambda)\Psi=\Psi$ for all $\lambda\in\Lambda_{B}$ with
$T_{B}(\lambda)$ in (1.16). The space is equipped with the usual $p$-norm per
unit volume

$\displaystyle\|\Psi\|_{L_{\mathrm{mag}}^{p}(Q_{B})}^{p}$
$\displaystyle:=\fint_{Q_{B}}\mathrm{d}X\;|\Psi(X)|^{p}:=\frac{1}{|Q_{B}|}\int_{Q_{B}}\mathrm{d}X\;|\Psi(X)|^{p},$
(2.3)

and we use the conventional abbreviation $\|\Psi\|_{p}$ when this does not
lead to confusion.

For $m\in\mathbb{N}_{0}$, the corresponding gauge-periodic Sobolev space is
defined by

$\displaystyle H_{\mathrm{mag}}^{m}(Q_{B})$ $\displaystyle:=\bigl{\\{}\Psi\in
L_{\mathrm{mag}}^{2}(Q_{B}):(-\mathrm{i}\nabla+2\mathbf{A})^{\nu}\Psi\in
L_{\mathrm{mag}}^{2}(Q_{B})\quad\forall\nu\in\mathbb{N}_{0}^{3},|\nu|_{1}\leqslant
m\bigr{\\}},$ (2.4)

where $|\nu|_{1}:=\sum_{i=1}^{3}\nu_{i}$ for $\nu\in\mathbb{N}_{0}^{3}$.
Equipped with the scalar product

$\displaystyle\langle\Phi,\Psi\rangle_{H_{\mathrm{mag}}^{m}(Q_{B})}$
$\displaystyle:=\sum_{|\nu|_{1}\leqslant
m}B^{-1-|\nu|_{1}}\;\langle(-\mathrm{i}\nabla+2\mathbf{A})^{\nu}\Phi,(-\mathrm{i}\nabla+2\mathbf{A})^{\nu}\Psi\rangle_{L_{\mathrm{mag}}^{2}(Q_{B})},$
(2.5)

it is a Hilbert space. We note that $(-\mathrm{i}\nabla+\mathbf{A})^{\nu}\Psi$
is a gauge-periodic function if $\Psi$ is gauge-periodic because the magnetic
momentum operator

$\displaystyle\Pi:=-\mathrm{i}\nabla+2\mathbf{A}$

commutes with the magnetic translations $T_{B}(\lambda)$ in (1.16). We also
note that $\Pi$ is a self-adjoint operator on $H_{\mathrm{mag}}^{1}(Q_{B})$.

At this point, we shall briefly explain the scaling behavior in $B$ of the
norms introduced in (2.3) and (2.5) in terms of the Ginzburg–Landau scaling in
(1.18). First, we note that if $\psi\in L_{\mathrm{mag}}^{p}(Q_{1})$ and
$\Psi$ is as in (1.18), then

$\displaystyle\|\Psi\|_{L_{\mathrm{mag}}^{p}(Q_{B})}=B^{\nicefrac{{1}}{{2}}}\,\|\psi\|_{L_{\mathrm{mag}}^{p}(Q_{1})}$
(2.6)

for every $1\leqslant p\leqslant\infty$. In contrast, the scaling of the norm
in (2.5) is chosen such that

$\displaystyle\|\Psi\|_{H_{\mathrm{mag}}^{m}(Q_{B})}=\|\psi\|_{H_{\mathrm{mag}}^{m}(Q_{1})}.$

This follows from (2.6) and the fact that
$\|(-\mathrm{i}\nabla+\mathbf{A})^{\nu}\Psi\|_{2}^{2}$ scales as
$B^{1+|\nu|_{1}}$ for $\nu\in\mathbb{N}_{0}^{3}$.

We also mention the following magnetic Sobolev inequality because it will be
used frequently in the course of the paper. For any $B>0$ and any $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{B})$, we have

$\displaystyle\|\Psi\|_{L_{\mathrm{mag}}^{6}(Q_{B})}^{2}$
$\displaystyle\leqslant
C\,B^{-1}\,\|(-\mathrm{i}\nabla+2\mathbf{A})\Psi\|_{L_{\mathrm{mag}}^{2}(Q_{B})}^{2}.$
(2.7)

###### Proof of (2.7).

Since $Q_{1}$ satisfies the cone property, [38, Theorem 8.8] implies

$\displaystyle\|\psi\|_{L_{\mathrm{mag}}^{6}(Q_{1})}^{2}$
$\displaystyle\leqslant
C\,\bigl{(}\|\psi\|_{L_{\mathrm{mag}}^{2}(Q_{1})}^{2}+\|\nabla|\psi|\,\|_{L_{\mathrm{mag}}^{2}(Q_{1})}^{2}\bigr{)}.$

From [47, Eq. (6.2)] we know that the bottom of the spectrum of
$(-\mathrm{i}\nabla+e_{3}\wedge X)^{2}$ equals $2$. For the first term on the
right side, this implies
$2\|\psi\|_{2}^{2}\leqslant\|(-\mathrm{i}\nabla+e_{3}\wedge X)\psi\|_{2}^{2}$.
To bound the second term, we apply the diamagnetic inequality
$|\nabla|\psi(X)||\leqslant|(-\mathrm{i}\nabla+e_{3}\wedge X)\psi(X)|$, see
[38, Theorem 7.21]. This proves (2.7) for $B=1$ and the scaling in (1.18)
yields (2.7) for $B>0$. ∎

As indicated below (1.9), the Cooper pair wave function $\alpha$ related to an
admissible state $\Gamma$ belongs to $\mathcal{S}^{2}$, the Hilbert–Schmidt
class introduced in Section 2.1. In terms of the center-of-mass and relative
coordinates, the gauge-periodicity and the symmetry of the kernel of $\alpha$
in (1.7) read

$\displaystyle\alpha(X,r)$
$\displaystyle=\mathrm{e}^{\mathrm{i}\mathbf{B}\cdot(\lambda\wedge
X)}\;\alpha(X+\lambda,r),\quad\lambda\in\Lambda_{B};$
$\displaystyle\alpha(X,r)$ $\displaystyle=\alpha(X,-r).$ (2.8)

That is, $\alpha(X,r)$ is a gauge-periodic function of the center-of-mass
coordinate $X$ and a reflection-symmetric function of the relative coordinate
$r\in\mathbb{R}^{3}$. We make use of the isometric identification of
$\mathcal{S}^{2}$ with the space

$\displaystyle
L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3}):=L_{\mathrm{mag}}^{2}(Q_{B})\otimes
L_{\mathrm{sym}}^{2}(\mathbb{R}^{3}),$

the square-integrable functions obeying (2.8), for which the norm

$\displaystyle\|\alpha\|_{{L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}}^{2}:=\fint_{Q_{B}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;|\alpha(X,r)|^{2}=\frac{1}{|Q_{B}|}\int_{Q_{B}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;|\alpha(X,r)|^{2}$

is finite. By (2.8), the identity
$\|\alpha\|_{2}=\|\alpha\|_{{L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}}$
holds. Therefore, we do not distinguish between the scalar products
$\langle\cdot,\cdot\rangle$ on
${L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}$ and $\mathcal{S}^{2}$ and
identify operators in $\mathcal{S}^{2}$ with their kernels whenever this
appears convenient.

Finally, the Sobolev space $H^{1}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})$
consists of all functions $\alpha\in
L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})$ with finite $H^{1}$-norm given
by

$\displaystyle\|\alpha\|_{H^{1}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}^{2}$
$\displaystyle:=\|\alpha\|_{2}^{2}+\|\Pi_{X}\alpha\|_{2}^{2}+\|\widetilde{\pi}_{r}\alpha\|_{2}^{2}.$
(2.9)

Here, we used the magnetic momentum operators

$\displaystyle\Pi_{X}$ $\displaystyle:=-\mathrm{i}\nabla_{X}+2\mathbf{A}(X),$
$\displaystyle\widetilde{\pi}_{r}$
$\displaystyle:=-\mathrm{i}\nabla_{r}+\frac{1}{2}\mathbf{A}(r),$ (2.10)

where $\mathbf{A}(x)=\frac{1}{2}\mathbf{B}\wedge x$. We note that the norm in
(2.9) is equivalent to the norm given by
$\operatorname{Tr}[\alpha\alpha^{*}]+\operatorname{Tr}[(-\mathrm{i}\nabla+\mathbf{A})\alpha\alpha^{*}(-\mathrm{i}\nabla+\mathbf{A})]+\operatorname{Tr}[(-\mathrm{i}\nabla+\mathbf{A})\alpha^{*}\alpha(-\mathrm{i}\nabla+\mathbf{A})]$,
which, in turn, is given by
$\|\alpha\|_{2}^{2}+\|(-\mathrm{i}\nabla+\mathbf{A})\alpha\|_{2}^{2}+\|\alpha(-\mathrm{i}\nabla+\mathbf{A})\|_{2}^{2}$.
See also the discussion below (1.9).

##  3 Trial States and their BCS Energy

The goal of this section is to provide the upper bound on (1.21) and the proof
of Theorem 2 (a). Both bounds are proved with a trial state argument using
Gibbs states $\Gamma_{\Delta}$ that are defined via a gap function $\Delta$ in
the effective Hamiltonian. In Proposition 3.2 we show that the Cooper pair
wave function $\alpha_{\Delta}$ of $\Gamma_{\Delta}$ is a product function
with respect to relative and center-of-mass coordinates to leading order
provided $\Delta$ is a product function that is small in a suitable sense. A
representation formula for the BCS energy in terms of the energy of these
states is provided in Proposition 3.4. Finally, in Theorem 3.5, we show that
certain parts of the BCS energy of the trial states $\Gamma_{\Delta}$ equal
the terms in the Ginzburg–Landau functional in (1.5) with sufficient precision
provided $T=T_{\mathrm{c}}(1-DB)$ for some fixed $D\in\mathbb{R}$. These
results, whose proofs are deferred to Section 4, are combined in Section 3.3
to give the proof of the results mentioned in the beginning of this paragraph.

###  3.1 The Gibbs states $\Gamma_{\Delta}$

For any $\Psi\in L_{\mathrm{mag}}^{2}(Q_{B})$, let us introduce the gap
function $\Delta\in L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})$, given by

$\displaystyle\Delta(X,r):=\Delta_{\Psi}(X,r)$
$\displaystyle:=-2\;V\alpha_{*}(r)\;\Psi(X).$ (3.1)

In our trial state analysis, $\Psi$ is going to be a minimizer of the
Ginzburg–Landau functional in (1.17). It therefore obeys the scaling in
(1.18), which implies that the local Hilbert-Schmidt norm $\|\Delta\|_{2}^{2}$
is of the order $B$. We highlight that the $L^{2}(\mathbb{R}^{3})$-norm of
$V\alpha_{*}$ is of the order $1$, that is, the size of $\|\Delta\|_{2}^{2}$
is determined by $\Psi$. In the proof of the lower bound we have less
information on $\Psi$. The related difficulties are discussed in Remark 3.3
below. With

$\displaystyle\mathfrak{h}_{B}$
$\displaystyle:=(-\mathrm{i}\nabla+\mathbf{A})^{2}-\mu,$ (3.2)

we define the Hamiltonian

$\displaystyle H_{\Delta}$
$\displaystyle:=H_{0}+\delta:=\begin{pmatrix}\mathfrak{h}_{B}&0\\\
0&-\overline{\mathfrak{h}_{B}}\end{pmatrix}+\begin{pmatrix}0&\Delta\\\
\overline{\Delta}&0\end{pmatrix}=\begin{pmatrix}\mathfrak{h}_{B}&\Delta\\\
\overline{\Delta}&-\overline{\mathfrak{h}_{B}}\end{pmatrix}$ (3.3)

and the corresponding Gibbs state at inverse temperature $\beta=T^{-1}>0$ as

$\displaystyle\begin{pmatrix}\gamma_{\Delta}&\alpha_{\Delta}\\\
\overline{\alpha_{\Delta}}&1-\overline{\gamma_{\Delta}}\end{pmatrix}=\Gamma_{\Delta}:=\frac{1}{1+\mathrm{e}^{\beta
H_{\Delta}}}.$ (3.4)

We note that the normal state $\Gamma_{0}$ in (1.12) corresponds to setting
$\Delta=0$ in (3.4).

######  Lemma 3.1 (Admissibility of $\Gamma_{\Delta}$).

Let Assumptions 1.3 and 1.5 hold. Then, for any $B>0$, any $T>0$, and any
$\Psi\in H_{\mathrm{mag}}^{1}(Q_{B})$, the state $\Gamma_{\Delta}$ in (3.4) is
admissible, where $\Delta\equiv\Delta_{\Psi}$ as in (3.1).

The states $\Gamma_{\Delta}$ are inspired by the following observation. Via
variational arguments it is straightforward to see that any minimizer of
$\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}$ in (1.10) solves the nonlinear
Bogolubov–de Gennes equation

$\displaystyle\Gamma$
$\displaystyle=\frac{1}{1+\mathrm{e}^{\beta\,\mathbb{H}_{V\alpha}}},$
$\displaystyle\mathbb{H}_{V\alpha}=\begin{pmatrix}\mathfrak{h}_{B}&-2\,V\alpha\\\
-2\,\overline{V\alpha}&-\overline{\mathfrak{h}_{B}}\end{pmatrix}.$ (3.5)

Here, $V\alpha$ is the operator given by the kernel $V(r)\alpha(X,r)$. As we
look for approximate minimizers of
$\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}$, we choose $\Gamma_{\Delta}$ in
order to approximately solve (3.5). As far as the leading term of
$\alpha_{\Delta}$ is concerned this is indeed the case, as the following
result shows. It should be compared to (1.24).

######  Proposition 3.2 (Structure of $\alpha_{\Delta}$).

Let Assumption 1.3 and 1.5 (a) be satisfied and let $T_{0}>0$ be given. Then,
there is a constant $B_{0}>0$ such that for any $0<B\leqslant B_{0}$, any
$T\geqslant T_{0}$, and any $\Psi\in H_{\mathrm{mag}}^{2}(Q_{B})$ the function
$\alpha_{\Delta}$ in (3.4) with $\Delta\equiv\Delta_{\Psi}$ as in (3.1) has
the decomposition

$\displaystyle\alpha_{\Delta}(X,r)$
$\displaystyle=\Psi(X)\alpha_{*}(r)-\eta_{0}(\Delta)(X,r)-\eta_{\perp}(\Delta)(X,r).$
(3.6)

The remainder functions $\eta_{0}(\Delta)$ and $\eta_{\perp}(\Delta)$ have the
following properties:

  1. (a)

The function $\eta_{0}$ satisfies the bound

$\displaystyle\|\eta_{0}\|_{H^{1}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}^{2}$
$\displaystyle\leqslant
C\;\bigl{(}B^{3}+B\,|T-T_{\mathrm{c}}|^{2}\bigr{)}\;\bigl{(}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{6}+\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{2}\bigr{)}.$
(3.7)

  2. (b)

The function $\eta_{\perp}$ satisfies the bound

$\displaystyle\|\eta_{\perp}\|_{{H^{1}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}}^{2}+\||r|\eta_{\perp}\|_{{L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}}^{2}$
$\displaystyle\leqslant C\;B^{3}\;\|\Psi\|_{H_{\mathrm{mag}}^{2}(Q_{B})}^{2}.$
(3.8)

  3. (c)

The function $\eta_{\perp}$ has the explicit form

$\displaystyle\eta_{\perp}(X,r)$
$\displaystyle=\int_{\mathbb{R}^{3}}\mathrm{d}Z\int_{\mathbb{R}^{3}}\mathrm{d}s\;k_{T}(Z,r-s)\,V\alpha_{*}(s)\,\bigl{[}\cos(Z\cdot\Pi_{X})-1\bigr{]}\Psi(X)$

with $k_{T}(Z,r)$ defined in Section 4 below (4.70). For any radial $f,g\in
L^{2}(\mathbb{R}^{3})$ the operator

$\displaystyle\iiint_{\mathbb{R}^{9}}\mathrm{d}Z\mathrm{d}r\mathrm{d}s\;f(r)\,k_{T}(Z,r-s)\,g(s)\,\bigl{[}\cos(Z\cdot\Pi)-1\bigr{]}$

commutes with $\Pi^{2}$, and, in particular, if $P$ and $Q$ are two spectral
projections of $\Pi^{2}$ with $PQ=0$, then $\eta_{\perp}$ satisfies the
orthogonality property

$\displaystyle\bigl{\langle}f(r)\,(P\Psi)(X),\,\eta_{\perp}(\Delta_{Q\Psi})\bigr{\rangle}=0.$
(3.9)

######  Remark 3.3.

The statement of Proposition 3.2 should be read in two different ways,
depending on whether we are interested in proving the upper or the lower bound
for the BCS free energy. When we prove the upper bound using trial states
$\Gamma_{\Delta}$, part (c) is irrelevant. In this case the gap function
$\Delta\equiv\Delta_{\Psi}$ is defined with a minimizer $\Psi$ of the GL
functional, whose $H_{\mathrm{mag}}^{2}(Q_{B})$-norm is uniformly bounded, and
all remainder terms can be estimated using (3.7) and (3.8).

In the proof of the lower bound for the BCS free energy in Section 6 we are
forced to work with a trial state $\Gamma_{\Delta}$, whose gap function is
defined via a function $\Psi$ that is related to a low-energy state of the BCS
functional, see Theorem 5.1 below. For such functions we only have a bound on
the $H_{\mathrm{mag}}^{1}(Q_{B})$-norm at our disposal. To obtain a function
in $H_{\mathrm{mag}}^{2}(Q_{B})$, we introduce a regularized version of $\Psi$
as in [17, Section 6], [18, Section 6], and [15, Section 7] by
$\Psi_{\leqslant}:=\mathbbs 1_{[0,\varepsilon]}(\Pi^{2})\Psi$ for some
$B\ll\varepsilon\ll 1$, see Corollary 5.2. The
$H_{\mathrm{mag}}^{2}(Q_{B})$-norm of $\Psi_{\leqslant}$ is not uniformly
bounded in $B$, see (5.5) below. This causes a certain error term, namely the
left side of (6.12) below, to be large, a priori.

To overcome this problem we use part (c) of Proposition 5.1. It exploits the
fact that the first term on the right side of (6.12) has an explicit form that
satisfies the orthogonality property in (3.9), which implies that the left
side of (6.12) is indeed small. This is the reason why we need to distinguish
between $\eta_{0}$ and $\eta_{\perp}$.

###  3.2 The BCS energy of the states $\Gamma_{\Delta}$

This section pertains to the BCS energy of the states $\Gamma_{\Delta}$, which
is given by the Ginzburg–Landau functional to leading order. We will see in
Section 4.2 that the BCS energy of $\Gamma_{\Delta}$ can be calculated in
terms of

$\displaystyle\operatorname{Tr}_{0}\Bigl{[}\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)}-\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{0}\bigr{)}\bigr{)}\Bigr{]}.$
(3.10)

Here, $\operatorname{Tr}_{0}$ is a weaker form of trace which will be
introduced later in (4.5). The operator inside the trace is closely related to
the relative entropy of $H_{\Delta}$ and $H_{0}$ but also incorporates the
interaction energy of $\alpha_{\Delta}$. We refer to (4.6) for more details.
In the following, we explain how the terms of the Ginzburg–Landau functional,
which appear in the energy expansion in (1.21), are obtained from the operator
in (3.10).

As pointed out in Remark 1.6, we should think of $\Delta$ as being small. In
order to expand the term in (3.10) in powers of $\Delta$, we use the
fundamental theorem to formally write (3.10) as

$\displaystyle\frac{\beta}{2}\operatorname{Tr}_{0}\Bigl{[}\int_{0}^{1}\mathrm{d}t\;\tanh\bigl{(}\frac{\beta}{2}H_{t\Delta}\bigr{)}\Bigl{(}\begin{matrix}0&\Delta\\\
\overline{\Delta}&0\end{matrix}\Bigr{)}\Bigr{]}.$ (3.11)

This identity is not rigorous because it ignores the subtlety that
$H_{t\Delta}$, $t\in[0,1]$, are unbounded operators which do not commute for
distinct values of $t$. We present a rigorous version of (3.11) in Lemma 4.2
below. For the sake of the following discussion it is legitimate to assume
that equality between (3.10) and (3.11) holds.

We use the Mittag-Leffler series expansion, see e.g. [15, Eq. (7)], to write
the hyperbolic tangent in (3.11) as

$\displaystyle\tanh\bigl{(}\frac{\beta}{2}z\bigr{)}$
$\displaystyle=-\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\frac{1}{\mathrm{i}\omega_{n}-z}$
(3.12)

with the Matsubara frequencies

$\displaystyle\omega_{n}$ $\displaystyle:=\pi(2n+1)T,\qquad n\in\mathbb{Z}.$
(3.13)

The convergence of (3.12) becomes manifest by combining the $+n$ and $-n$
terms. Thus,

$\displaystyle\tanh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}$
$\displaystyle=-\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\frac{1}{\mathrm{i}\omega_{n}-H_{\Delta}}.$
(3.14)

We use this representation to expand the operator in (3.11) in powers of
$\Delta$ using the resolvent equation. The first term obtained in this way is
$\langle\Delta,L_{T,B}\Delta\rangle$ with the linear operator
$L_{T,B}\colon{L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}\rightarrow{L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}$,
given by

$\displaystyle L_{T,B}\Delta$
$\displaystyle:=-\frac{2}{\beta}\sum_{n\in\mathbb{Z}}(\mathrm{i}\omega_{n}-\mathfrak{h}_{B})^{-1}\,\Delta\,(\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{B}})^{-1}.$
(3.15)

In the temperature regime we are interested in, we will obtain the quadratic
terms in the Ginzburg–Landau functional from
$\langle\Delta,L_{T,B}\Delta\rangle$.

The next term in the expansion of (3.14) is the quartic term
$\langle\Delta,N_{T,B}(\Delta)\rangle$ with the nonlinear map
$N_{T,B}\colon{H^{1}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}\rightarrow{L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}$
defined as

$\displaystyle N_{T,B}(\Delta)$
$\displaystyle:=\frac{2}{\beta}\sum_{n\in\mathbb{Z}}(\mathrm{i}\omega_{n}-\mathfrak{h}_{B})^{-1}\,\Delta\,(\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{B}})^{-1}\,\overline{\Delta}\,(\mathrm{i}\omega_{n}-\mathfrak{h}_{B})^{-1}\,\Delta\,(\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{B}})^{-1}.$
(3.16)

The expression $\langle\Delta,N_{T,B}(\Delta)\rangle$ will determine the
quartic term in the Ginzburg–Landau functional. All higher order terms in the
expansion of (3.11) in $\Delta$ will be summarized in a trace-class operator
called $\mathcal{R}_{T,B}(\Delta)$, whose local trace norm is small.

With the operators $L_{T,B}$ and $N_{T,B}$ at hand, we are in position to
state a representation formula for the BCS functional. It serves as the
fundamental equation, on which the proofs of Theorems 1 and 2 are based. In
particular, it will be applied in the proofs of upper and lower bounds, and we
therefore formulate the statement for a general state $\Gamma$ and not only
for Gibbs states.

######  Proposition 3.4 (Representation formula for the BCS functional).

Let $\Gamma$ be an admissible state. For any $B>0$, let $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{B})$ and let $\Delta\equiv\Delta_{\Psi}$ be as in
(3.1). For $T>0$ and if $V\alpha_{*}\in
L^{\nicefrac{{6}}{{5}}}(\mathbb{R}^{3})\cap L^{2}(\mathbb{R}^{3})$, there is
an operator $\mathcal{R}_{T,B}(\Delta)\in\mathcal{S}^{1}$ such that

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma_{0})$
$\displaystyle=-\frac{1}{4}\langle\Delta,L_{T,B}\Delta\rangle+\frac{1}{8}\langle\Delta,N_{T,B}(\Delta)\rangle+\|\Psi\|_{L_{\mathrm{mag}}^{2}(Q_{B})}^{2}\;\langle\alpha_{*},V\alpha_{*}\rangle_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle+\operatorname{Tr}\bigl{[}\mathcal{R}_{T,B}(\Delta)\bigr{]}$
$\displaystyle+\frac{T}{2}\mathcal{H}_{0}(\Gamma,\Gamma_{\Delta})-\fint_{Q_{B}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;V(r)\,\bigl{|}\alpha(X,r)-\alpha_{*}(r)\Psi(X)\bigr{|}^{2},$
(3.17)

where

$\displaystyle\mathcal{H}_{0}(\Gamma,\Gamma_{\Delta}):=\operatorname{Tr}_{0}\bigl{[}\Gamma(\ln\Gamma-\ln\Gamma_{\Delta})+(1-\Gamma)(\ln(1-\Gamma)-\ln(1-\Gamma_{\Delta}))\bigr{]}$
(3.18)

denotes the relative entropy of $\Gamma$ with respect to $\Gamma_{\Delta}$.
Moreover, $\mathcal{R}_{T,B}(\Delta)$ obeys the estimate

$\displaystyle\|\mathcal{R}_{T,B}(\Delta)\|_{1}\leqslant
C\;T^{-5}\;B^{3}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{6}.$

The relative entropy defined in (3.18) is based on the weaker form of trace
$\operatorname{Tr}_{0}$, whose introduction we postpone until (4.5).

The right side of (3.17) should be read as follows. The first line yields the
Ginzburg–Landau functional, see Theorem 3.5 below. The second and third line
consist of remainder terms. The second line is small in absolute value whereas
the techniques used to bound the third line differ for upper and lower bounds.
This is responsible for the different qualities of the upper and lower bounds
in Theorems 1 and 2, see (1.22). For an upper bound, when choosing
$\Gamma:=\Gamma_{\Delta}$ as a trial state, the relative entropy term
$\mathcal{H}_{0}(\Gamma_{\Delta},\Gamma_{\Delta})=0$ drops out and the last
term in (3.17) can be estimated with the help of Proposition 3.2. The last
term in (3.17) is actually nonpositive by our assumptions on $V$ but we do not
use this. For a lower bound, the third line needs to be bounded from below
using a relative entropy estimate that we provide in Section 6.

It remains to show that the first line of the right side of (3.17) is indeed
given by the Ginzburg–Landau functional. In order to state the result, we need
the function

$\displaystyle\widehat{V\alpha_{*}}(p):=\int_{\mathbb{R}^{3}}\mathrm{d}x\;\mathrm{e}^{-\mathrm{i}p\cdot
x}\,V(x)\alpha_{*}(x),$ (3.19)

which fixes our convention on the Fourier transform in this paper.

######  Theorem 3.5 (Calculation of the GL energy).

Let Assumptions 1.3 and 1.5 (a) hold and let $D\in\mathbb{R}$ be given. Then,
there is a constant $B_{0}>0$ such that for any $0<B\leqslant B_{0}$, any
$\Psi\in H_{\mathrm{mag}}^{2}(Q_{B})$, $\Delta\equiv\Delta_{\Psi}$ as in
(3.1), and $T=T_{\mathrm{c}}(1-DB)$, we have

$\displaystyle-\frac{1}{4}\langle\Delta,L_{T,B}\Delta\rangle+\frac{1}{8}\langle\Delta,N_{T,B}(\Delta)\rangle+\|\Psi\|_{L_{\mathrm{mag}}^{2}(Q_{B})}^{2}\;\langle\alpha_{*},V\alpha_{*}\rangle_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle=B^{2}\;\mathcal{E}^{\mathrm{GL}}_{D,B}(\Psi)+R(B).$ (3.20)

Here,

$\displaystyle|R(B)|\leqslant
C\,B^{3}\;\|\Psi\|_{H_{\mathrm{mag}}^{2}(Q_{B})}^{2}\;\bigl{[}1+\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{2}\bigr{]}$

and with the functions

$\displaystyle g_{1}(x)$
$\displaystyle:=\frac{\tanh(x/2)}{x^{2}}-\frac{1}{2x}\frac{1}{\cosh^{2}(x/2)},$
$\displaystyle g_{2}(x)$
$\displaystyle:=\frac{1}{2x}\frac{\tanh(x/2)}{\cosh^{2}(x/2)},$ (3.21)

the coefficients $\Lambda_{0}$, $\Lambda_{2}$, and $\Lambda_{3}$ in
$\mathcal{E}^{\mathrm{GL}}_{D,B}$ are given by

$\displaystyle\Lambda_{0}$
$\displaystyle:=\frac{\beta_{c}^{2}}{16}\int_{\mathbb{R}^{3}}\frac{\mathrm{d}p}{(2\pi)^{3}}\;|(-2)\widehat{V\alpha_{*}}(p)|^{2}\;\bigl{(}g_{1}(\beta_{c}(p^{2}-\mu))+\frac{2}{3}\beta_{c}\,p^{2}\,g_{2}(\beta_{c}(p^{2}-\mu))\bigr{)},$
(3.22) $\displaystyle\Lambda_{2}$
$\displaystyle:=\frac{\beta_{c}}{8}\int_{\mathbb{R}^{3}}\frac{\mathrm{d}p}{(2\pi)^{3}}\;\frac{|(-2)\widehat{V\alpha_{*}}(p)|^{2}}{\cosh^{2}(\frac{\beta_{c}}{2}(p^{2}-\mu))},$
(3.23) $\displaystyle\Lambda_{3}$
$\displaystyle:=\frac{\beta_{c}^{2}}{16}\int_{\mathbb{R}^{3}}\frac{\mathrm{d}p}{(2\pi)^{3}}\;|(-2)\widehat{V\alpha_{*}}(p)|^{4}\;\frac{g_{1}(\beta_{c}(p^{2}-\mu))}{p^{2}-\mu}.$
(3.24)

Let us comment on the positivity of the coefficients (3.22)-(3.24). First,
$\Lambda_{2}$ is trivially positive. Since $g_{1}(x)/x>0$ for all
$x\in\mathbb{R}$, the coefficient $\Lambda_{3}$ is positive as well. It cannot
be immediately seen that $\Lambda_{0}$ is positive, however. In order to prove
this, we introduce the positive function

$\displaystyle g_{3}(x)$
$\displaystyle:=\frac{2}{x^{2}}\frac{1}{\cosh^{2}(x/2)}-\frac{1}{x}\frac{1}{\tanh(x/2)}\frac{1}{\cosh^{2}(x/2)}$

and compute

$\displaystyle
2\operatorname{\mathrm{Re}}\langle\alpha_{*},x_{i}(K_{T_{\mathrm{c}}}-V)x_{i}\alpha_{*}\rangle$
$\displaystyle=(2\pi)^{-3}\langle\widehat{V\alpha_{*}},K_{T_{\mathrm{c}}}(p)^{-1}[-\mathrm{i}\partial_{p_{i}},[K_{T_{\mathrm{c}}}(p),-\mathrm{i}\partial_{p_{i}}]]K_{T_{\mathrm{c}}}(p)^{-1}\widehat{V\alpha_{*}}\rangle$
$\displaystyle=8\,\Lambda_{0}-2\beta_{c}^{3}\int_{\mathbb{R}^{3}}\frac{\mathrm{d}p}{(2\pi)^{3}}\;|\widehat{V\alpha_{*}}(p)|^{2}\;p_{i}^{2}\,g_{3}(\beta_{c}(p^{2}-\mu)).$
(3.25)

Since the left side is nonnegative, this proves that $\Lambda_{0}>0$. The idea
for this proof is borrowed from [17, Eq. (1.22)].

Let us comment on the connection between (3.20) and [15]. The two-particle
Birman–Schwinger operator
$1-V^{\nicefrac{{1}}{{2}}}L_{T,B}V^{\nicefrac{{1}}{{2}}}$ has been intensively
studied in [15] to identify temperature regimes, where the bottom of its
spectrum is positive or negative. This operator also appears in (3.20) because

$\displaystyle-\frac{1}{4}\langle\Delta,L_{T,B}\Delta\rangle+\|\Psi\|_{2}^{2}\,\langle\alpha_{*},V\alpha_{*}\rangle=\bigl{\langle}V^{\nicefrac{{1}}{{2}}}\alpha_{*}\Psi,\bigl{(}1-V^{\nicefrac{{1}}{{2}}}L_{T,B}V^{\nicefrac{{1}}{{2}}}\bigr{)}V^{\nicefrac{{1}}{{2}}}\alpha_{*}\Psi\bigr{\rangle}.$
(3.26)

That is, the question whether the bottom of the spectrum of
$1-V^{\nicefrac{{1}}{{2}}}L_{T,B}V^{\nicefrac{{1}}{{2}}}$ is positive or
negative is intimately related to the sign of (3.20), and thus of (3.17) and
(1.13). Accordingly, it is related to the question whether the systems
displays superconductivity or not. We highlight that the operator on the right
side of (3.26) acts on functions in $L^{2}(\mathbb{R}^{6})$ in [15], while it
acts on ${L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}$ in our case. Since
the lowest eigenvalue of the operator $(-\mathrm{i}\nabla+2\mathbf{A})^{2}$
equals $2B$ when understood to act on $L^{2}(\mathbb{R}^{3})$ or on
$L_{\mathrm{mag}}^{2}(Q_{B})$, we obtain the same asymptotic behavior of
$T_{\mathrm{c}}(B)$ as in [15, Theorem 4].

Theorem 3.5 is valid for the precise temperature scaling
$T=T_{\mathrm{c}}(1-DB)$. In order to prove Theorem 2 (a), we also need to
show that the system is superconducting for temperatures that are small
compared to $T_{\mathrm{c}}(1-DB)$. This is guaranteed by the following
proposition.

######  Proposition 3.6 (A priori bound on Theorem 2 (a)).

Let Assumptions 1.3 and 1.5 (a) hold. Then, for every $T_{0}>0$ there are
constants $B_{0}>0$ and $D_{0}>0$ such that for all $0<B\leqslant B_{0}$ and
all temperatures $T$ obeying

$\displaystyle T_{0}\leqslant T<T_{\mathrm{c}}(1-D_{0}B),$

there is an admissible BCS state $\Gamma$ with

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma_{0})<0.$
(3.27)

###  3.3 The upper bound on (1.21) and proof of Theorem 2 (a)

Using the results in the previous section, we provide the proofs of the upper
bound on (1.21) and of Theorem 2 (a). The statements in the previous section,
that is, Propositions 3.2 and 3.4, as well as Theorem 3.5 are proven in
Section 4.

###### Proof of the upper bound on (1.21).

Let $D\in\mathbb{R}$ be given, let $D_{0}:=1+|D|$, and let $\Psi$ be a
minimizer of the Ginzburg–Landau functional, i.e.,
$\mathcal{E}_{D,B}^{\mathrm{GL}}(\Psi)=E^{\mathrm{GL}}(D)$. We note that
$\Psi$ belongs to $H_{\mathrm{mag}}^{2}(Q_{B})$ and has uniformly bounded
$H_{\mathrm{mag}}^{2}(Q_{B})$-norm. Let $\Delta\equiv\Delta_{\Psi}$ be as in
(3.1) and let $T=T_{\mathrm{c}}(1-DB)$. We apply Proposition 3.4 with the
choice $\Gamma=\Gamma_{\Delta}$ and find

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma_{\Delta})-\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma_{0})$
$\displaystyle\leqslant-\frac{1}{4}\langle\Delta,L_{T,B}\Delta\rangle+\frac{1}{8}\langle\Delta,N_{T,B}(\Delta)\rangle+\|\Psi\|_{2}^{2}\,\langle\alpha_{*},V\alpha_{*}\rangle$
$\displaystyle-\int_{Q_{B}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;V(r)\,\bigl{|}\alpha_{\Delta}(X,r)-\alpha_{*}(r)\Psi(X)\bigr{|}^{2}+CB^{3}.$
(3.28)

The first term in the last line is bounded by $\|V\|_{\infty}\|\eta\|_{2}^{2}$
and a bound for the $L^{2}$-norm of $\eta:=\eta_{0}+\eta_{\perp}$ is provided
by (3.7) and (3.8). In fact, by Assumption 1.3, this term is nonpositive but
we do not need to use this here. By Theorem 3.5, this implies

$\displaystyle F^{\mathrm{BCS}}(T_{\mathrm{c}}(1-DB),B)$
$\displaystyle\leqslant B^{2}\,E^{\mathrm{GL}}(D)+CB^{3},$

which concludes the proof of the upper bound on (1.21). ∎

###### Proof of Theorem 2 (a).

Let $D_{0}>0$ be given and let us recall the definition of $D_{\mathrm{c}}$ in
(1.20). We show that there is a constant $D_{1}>0$ and appropriate trial
states such that (3.27) holds for all temperatures $T$ obeying

$\displaystyle T_{\mathrm{c}}(1-D_{0}B)\leqslant
T<T_{\mathrm{c}}(1-D_{\mathrm{c}}\,B-D_{1}\,B^{\nicefrac{{3}}{{2}}}),$ (3.29)

provided $B>0$ is small enough. Since Proposition 3.6 covers the remaining
range of $T$, this proves Theorem 2 (a).

We define $D:=\frac{T_{\mathrm{c}}-T}{BT_{\mathrm{c}}}$ and note that (3.29)
yields $D-D_{\mathrm{c}}>D_{1}B^{\nicefrac{{1}}{{2}}}$. Let $\psi\in
H_{\mathrm{mag}}^{2}(Q_{1})$ be a ground state of the linear operator in
(1.20) and let $\Psi$ be as in (1.18). Accordingly, we have
$(\Lambda_{0}/\Lambda_{2})\Pi^{2}\Psi=BD_{\mathrm{c}}\Psi$ and

$\displaystyle\inf_{\theta\in\mathbb{R}}\mathcal{E}^{\mathrm{GL}}_{D,B}(\theta\Psi)=-\frac{\Lambda_{2}^{2}(D-D_{\mathrm{c}})^{2}\|\psi\|_{2}^{4}}{4\Lambda_{3}\|\psi\|_{4}^{4}},$

where the optimal $\mathrm{\theta_{c}}$ satisfies
$\Lambda_{2}(D-D_{\mathrm{c}})\|\psi\|_{2}^{2}=2\Lambda_{3}\|\psi\|_{4}^{4}\,\mathrm{\theta_{c}}^{2}$.
We combine Proposition 3.4 and Theorem 3.5 applied to $\Gamma=\Gamma_{\Delta}$
with $\Delta=\Delta_{\mathrm{\theta_{c}}\Psi}$, to see that (3.28) holds in
this case as well. Let us note that (3.29) implies
$|T-T_{\mathrm{c}}|\leqslant CB$. Proposition 3.2 and (3.28) therefore allow
us to conclude that

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma_{\Delta})-\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma_{0})$
$\displaystyle\leqslant-\frac{\Lambda_{2}^{2}\|\psi\|_{2}^{4}}{4\Lambda_{3}\|\psi\|_{4}^{4}}\;(D-D_{\mathrm{c}})^{2}\;B^{2}+CB^{3}.$
(3.30)

The right side is negative provided $D_{1}>0$ is chosen large enough since
$D-D_{\mathrm{c}}>D_{1}B^{\nicefrac{{1}}{{2}}}$. This shows (3.27) for
temperatures $T$ satisfying (3.29) and completes the proof of Theorem 2 (a). ∎

##  4 Proofs of the Results in Section 3

###  4.1 Schatten norm estimates for operators given by product kernels

In this subsection we provide estimates for several norms of gauge-periodic
operators with integral kernels given by product functions of the form
$\tau(x-y)\Psi((x+y)/2)$, which will be used frequently in our proofs.

######  Lemma 4.1.

Let $B>0$, let $\Psi$ be a gauge-periodic function on $Q_{B}$ and let $\tau$
be an even and real-valued function on $\mathbb{R}^{3}$. Moreover, let the
operator $\alpha$ be defined via its integral kernel
$\alpha(X,r):=\tau(r)\Psi(X)$, i.e., $\alpha$ acts as

$\displaystyle\alpha f(x)$
$\displaystyle=\int_{\mathbb{R}^{3}}\mathrm{d}y\;\tau(x-y)\Psi\bigl{(}\frac{x+y}{2}\bigr{)}f(y),$
$\displaystyle f$ $\displaystyle\in L^{2}(\mathbb{R}^{3}).$

  1. (a)

Let $p\in\\{2,4,6\\}$. If $\Psi\in L_{\mathrm{mag}}^{p}(Q_{B})$ and $\tau\in
L^{\frac{p}{p-1}}(\mathbb{R}^{3})$, then $\alpha\in\mathcal{S}^{p}$ and

$\displaystyle\|\alpha\|_{p}\leqslant
C\;\|\tau\|_{\frac{p}{p-1}}\;\|\Psi\|_{p}.$

  2. (b)

For any $\nu>3$, there is a $C_{\nu}>0$, independent of $B$, such that if
$(1+|\cdot|)^{\nu}\tau\in L^{\nicefrac{{6}}{{5}}}(\mathbb{R}^{3})$ and
$\Psi\in L_{\mathrm{mag}}^{6}(Q_{B})$, then $\alpha\in\mathcal{S}^{\infty}$
and

$\displaystyle\|\alpha\|_{\infty}$ $\displaystyle\leqslant
C_{\nu}\,B^{-\nicefrac{{1}}{{4}}}\;\max\\{1,B^{\nicefrac{{\nu}}{{2}}}\\}\;\|(1+|\cdot|)^{\nu}\tau\|_{\nicefrac{{6}}{{5}}}\;\|\Psi\|_{6}.$

###### Proof.

The case $p=2$ of part (a) holds trivially with equality and $C=1$. Since
$\tau$ is even and real-valued, the kernel of $\alpha^{*}\alpha$ is given by

$\displaystyle\alpha^{*}\alpha(x,y)$
$\displaystyle=\int_{\mathbb{R}^{3}}\mathrm{d}z\;\tau(x-z)\overline{\Psi\bigl{(}\frac{x+z}{2}\bigr{)}}\tau(z-y)\Psi\bigl{(}\frac{z+y}{2}\bigr{)}.$

Using $\|\alpha\|_{4}^{4}=\|\alpha^{*}\alpha\|_{2}^{2}$ and the change of
variables $z\mapsto x-z$ and $y\mapsto x-y$, we see that

$\displaystyle\|\alpha\|_{4}^{4}$
$\displaystyle=\frac{1}{|Q_{B}|}\iint_{Q_{B}\times\mathbb{R}^{3}}\mathrm{d}x\mathrm{d}y\;\Bigl{|}\int_{\mathbb{R}^{3}}\mathrm{d}z\;\tau(z)\,\tau(y-z)\,\overline{\Psi\bigl{(}x-\frac{z}{2}\bigr{)}}\,\Psi\bigl{(}x-\frac{y+z}{2}\bigr{)}\Bigr{|}^{2}.$

By Hölder’s inequality, Young’s inequality and the fact that $|\Psi|$ is
periodic, we see that

$\displaystyle\|\alpha\|_{4}^{4}$
$\displaystyle\leqslant\|\Psi\|_{4}^{4}\;\int_{\mathbb{R}^{3}}\mathrm{d}y\,\Bigl{|}\int_{\mathbb{R}^{3}}\mathrm{d}z\;|\tau(z)\tau(y-z)|\Bigr{|}^{2}\leqslant
C\;\|\tau\|_{\nicefrac{{4}}{{3}}}^{4}\;\|\Psi\|_{4}^{4}$

holds. This proves part (a) for $p=4$. If $p=6$ we use
$\|\alpha\|_{6}^{6}=\|\alpha\,\alpha^{*}\alpha\|_{2}^{2}$ and a similar change
of variables to write

$\displaystyle\|\alpha\alpha^{*}\alpha\|_{2}^{2}$
$\displaystyle=\frac{1}{|Q_{B}|}\iint_{Q_{B}\times\mathbb{R}^{3}}\mathrm{d}x\mathrm{d}y\;\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}z_{1}\mathrm{d}z_{2}\;\tau(z_{1})\,\tau(z_{2}-z_{1})\,\tau(y-z_{2})\mathop{\times}$
$\displaystyle\hskip
120.0pt\times\Psi\bigl{(}x-\frac{z_{1}}{2}\bigr{)}\overline{\Psi\bigl{(}x-\frac{x_{1}+z_{2}}{2}\bigr{)}}\Psi\bigl{(}x-\frac{z_{2}+y}{2}\bigr{)}\Bigr{|}^{2}.$

We thus obtain
$\|\alpha\|_{6}^{6}\leqslant\|\Psi\|_{6}^{6}\;\|\tau*\tau*\tau\|_{2}^{2}$,
which, in combination with Young’s inequality, proves the claimed bound.

In case of part (b), we follow closely the strategy of the proof of [17, Eq.
(5.51)]. Let $f,g\in L^{2}(\mathbb{R}^{3})$ and let $\chi_{j}$ denote the
characteristic function of the cube with side length $\sqrt{2\pi B^{-1}}$
centered at $j\in\Lambda_{B}$. We estimate

$\displaystyle|\langle f,\alpha g\rangle|$
$\displaystyle\leqslant\sum_{j,k\in\Lambda_{B}}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}x\mathrm{d}y\;\bigl{|}\chi_{j}(x)f(x)\Psi\bigl{(}\frac{x+y}{2}\bigr{)}\tau(x-y)\chi_{k}(y)g(y)\bigr{|}.$
(4.1)

Let $|\cdot|_{\infty}$ and $|\cdot|$ denote the maximum norm and the euclidean
norm on $\mathbb{R}^{3}$, respectively. We observe that the estimates
$|x-j|_{\infty}\leqslant\frac{1}{2}\sqrt{2\pi B^{-1}}$ and
$|y-k|_{\infty}\leqslant\frac{1}{2}\sqrt{2\pi B^{-1}}$ imply
$|\frac{x+y}{2}-\frac{j+k}{2}|_{\infty}\leqslant\frac{1}{2}\sqrt{2\pi
B^{-1}}$. Accordingly, if $\chi_{j}(x)\chi_{k}(y)$ equals $1$, so does
$\chi_{\frac{j+k}{2}}(\frac{x+y}{2})$ and we may replace $\Psi$ on the right
side of (4.1) by $\chi_{\frac{j+k}{2}}\Psi$ without changing the term. The
above bounds for $|x-j|_{\infty}$ and $|y-k|_{\infty}$ also imply
$|j-k|\leqslant|x-y|+\sqrt{6\pi B^{-1}}$, which yields the lower bound

$\displaystyle|x-y|\geqslant\bigl{[}|j-k|-\sqrt{6\pi B^{-1}}\bigr{]}_{+}.$
(4.2)

We choose $\nu>3$, insert the factor $(\sqrt{2\pi B^{-1}}+|x-y|)^{\nu}$ and
its inverse in (4.1), use (4.2) to estimate the inverse, apply Cauchy–Schwarz
in the $x$-coordinate, and obtain

$\displaystyle|\langle f,\alpha g\rangle|$
$\displaystyle\leqslant\sum_{j,k\in\Lambda_{B}}\bigl{(}\sqrt{2\pi
B^{-1}}+\bigl{[}|j-k|-\sqrt{6\pi
B^{-1}}\bigr{]}_{+}\bigr{)}^{-\nu}\;\|\chi_{j}f\|_{2}\mathop{\times}$
$\displaystyle\times\Bigl{(}\int_{\mathbb{R}^{3}}\mathrm{d}x\,\Bigl{|}\int_{\mathbb{R}^{3}}\mathrm{d}y\;\bigl{|}(\chi_{\frac{j+k}{2}}\Psi)\bigl{(}\frac{x+y}{2}\bigr{)}\bigl{(}\sqrt{2\pi
B^{-1}}+|x-y|\bigr{)}^{\nu}\tau(x-y)\chi_{k}(y)g(y)\bigr{|}\;\Bigr{|}^{2}\Bigr{)}^{\nicefrac{{1}}{{2}}}.$

An application of Hölder’s inequality in the $y$-coordinate shows that the
second line is bounded by

$\displaystyle\Bigl{\|}\bigl{|}(\sqrt{2\pi
B^{-1}}+|\cdot|)^{\nu}\tau\bigr{|}^{\nicefrac{{6}}{{5}}}*|\chi_{k}g|^{\nicefrac{{6}}{{5}}}\Bigr{\|}_{\nicefrac{{5}}{{3}}}^{\nicefrac{{5}}{{6}}}$
$\displaystyle\leqslant\bigl{\|}\bigl{(}\sqrt{2\pi
B^{-1}}+|\cdot|\bigr{)}^{\nu}\tau\bigr{\|}_{\nicefrac{{6}}{{5}}}\,\|\chi_{k}g\|_{2}$

times $|Q_{B}|^{\nicefrac{{1}}{{6}}}\|\Psi\|_{L_{\mathrm{mag}}^{6}(Q_{B})}$.
We highlight that the $L_{\mathrm{mag}}^{6}(Q_{B})$-norm is defined via a
normalized integral, whence we needed to insert the factor of
$|Q_{B}|^{-\nicefrac{{1}}{{6}}}$. Hence,

$\displaystyle|\langle f,\alpha g\rangle|$ $\displaystyle\leqslant
CB^{-\nicefrac{{1}}{{4}}}\,\|\Psi\|_{6}\,\bigl{\|}\bigl{(}\sqrt{2\pi
B^{-1}}+|\cdot|\bigr{)}^{\nu}\tau\bigr{\|}_{\nicefrac{{6}}{{5}}}\mathop{\times}$
$\displaystyle\hskip 50.0pt\times\sum_{j,k\in\Lambda_{B}}\bigl{(}\sqrt{2\pi
B^{-1}}+\bigl{[}|j-k|-\sqrt{6\pi
B^{-1}}\bigr{]}_{+}\bigr{)}^{-\nu}\;\|\chi_{j}f\|_{2}\;\|\chi_{k}g\|_{2}.$

For $\lambda>0$ we estimate
$\|\chi_{j}f\|_{2}\|\chi_{k}g\|_{2}\leqslant\frac{\lambda}{2}\|\chi_{j}f\|_{2}^{2}+\frac{1}{2\lambda}\|\chi_{k}g\|_{2}^{2}$.
In each term, we carry out one of the sums and optimize the resulting
expression over $\lambda$. We find $\lambda=\|g\|_{2}\;\|f\|_{2}^{-1}$ as well
as

$\displaystyle|\langle f,\alpha g\rangle|$ $\displaystyle\leqslant
CB^{-\nicefrac{{1}}{{4}}}\|f\|_{2}\;\|g\|_{2}\;\|\Psi\|_{6}\;\frac{\|(\sqrt{2\pi
B^{-1}}+|\cdot|)^{\nu}\tau\|_{\nicefrac{{6}}{{5}}}}{(2\pi
B^{-1})^{\nicefrac{{\nu}}{{2}}}}\;\sum_{j\in\mathbb{Z}^{3}}\bigl{(}1+[|j|-\sqrt{3}]_{+}\bigr{)}^{-\nu}.$

The fraction involving $\tau$ is bounded by
$C_{\nu}\max\\{1,B^{\nicefrac{{\nu}}{{2}}}\\}\|(1+|\cdot|)^{\nu}\tau\|_{\nicefrac{{6}}{{5}}}$.
This proves the claim. ∎

###  4.2 Proof of Proposition 3.4

We recall the definitions of $\Delta(X,r)=-2\,V\alpha_{*}(r)\,\Psi(X)$ in
(3.1), the Hamiltonian $H_{\Delta}$ in (3.3) and
$\Gamma_{\Delta}=(1+\mathrm{e}^{\beta H_{\Delta}})^{-1}$ in (3.4). Throughout
this section we assume that the function $\Psi$ in the definition of $\Delta$
is in $H_{\mathrm{mag}}^{1}(Q_{B})$. From Lemma 3.1, which is proved in
Section 4.4 below, we know that $\Gamma_{\Delta}$ is an admissible BCS state
in this case. We define the anti-unitary operator

$\displaystyle\mathcal{J}$ $\displaystyle:=\begin{pmatrix}0&J\\\
-J&0\end{pmatrix}$

with $J$ defined below (1.5). The operator $H_{\Delta}$ obeys the relation
$\mathcal{J}H_{\Delta}\mathcal{J}^{*}=-H_{\Delta}$, which implies
$\mathcal{J}\Gamma_{\Delta}\mathcal{J}^{*}=1-\Gamma_{\Delta}$. Using this and
the cyclicity of the trace, we write the entropy of $\Gamma_{\Delta}$ as

$\displaystyle
S(\Gamma_{\Delta})=\frac{1}{2}\operatorname{Tr}[\varphi(\Gamma_{\Delta})],$
(4.3)

where $\varphi(x):=-[x\ln(x)+(1-x)\ln(1-x)]$ for $0\leqslant x\leqslant 1$.

In order to rewrite the BCS functional, it is useful to introduce a weaker
notion of trace per unit volume. More precisely, we call a gauge-periodic
operator $A$ acting on $L^{2}(\mathbb{R}^{3})\oplus L^{2}(\mathbb{R}^{3})$
weakly locally trace class if $P_{0}AP_{0}$ and $Q_{0}AQ_{0}$ are locally
trace class, where

$\displaystyle P_{0}=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$ (4.4)

and $Q_{0}=1-P_{0}$, and we define its weak trace per unit volume by

$\displaystyle\operatorname{Tr}_{0}(A):=\operatorname{Tr}\bigl{(}P_{0}AP_{0}+Q_{0}AQ_{0}\bigr{)}.$
(4.5)

If an operator is locally trace class then it is also weakly locally trace
class but the converse need not be true. It is true, however, in case of
nonnegative operators. If an operator is locally trace class then its weak
trace per unit volume and its usual trace per unit volume coincide.

Before their appearance in the context of BCS theory in [17, 18], weak traces
of the above kind appeared in [30, 21]. In [30, Lemma 1] it has been shown
that if two weak traces $\operatorname{Tr}_{P}$ and
$\operatorname{Tr}_{P^{\prime}}$ are defined via projections $P$ and
$P^{\prime}$ then $\operatorname{Tr}_{P}(A)=\operatorname{Tr}_{P^{\prime}}(A)$
holds for appropriate $A$ if $P-P^{\prime}$ is a Hilbert–Schmidt operator.

Let $\Gamma$ be an admissible BCS state and recall the normal state
$\Gamma_{0}$ in (1.12). In terms of the weak trace per unit volume, the BCS
functional can be written as

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma_{0})$
$\displaystyle=\frac{1}{2}\operatorname{Tr}\bigl{[}(H_{0}\Gamma-
H_{0}\Gamma_{0})-T\varphi(\Gamma)+T\varphi(\Gamma_{0})\bigr{]}-\fint_{Q_{B}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;V(r)\,|\alpha(X,r)|^{2}$
$\displaystyle=\frac{1}{2}\operatorname{Tr}_{0}\bigl{[}(H_{\Delta}\Gamma_{\Delta}-H_{0}\Gamma_{0})-T\varphi(\Gamma_{\Delta})+T\varphi(\Gamma_{0})\bigr{]}$
(4.6) $\displaystyle\hskip
55.0pt+\frac{1}{2}\operatorname{Tr}_{0}\bigl{[}(H_{\Delta}\Gamma-
H_{\Delta}\Gamma_{\Delta})-T\varphi(\Gamma)+T\varphi(\Gamma_{\Delta})\bigr{]}$
(4.7) $\displaystyle\hskip
55.0pt-\frac{1}{2}\operatorname{Tr}_{0}\begin{pmatrix}0&\Delta\\\
\overline{\Delta}&0\end{pmatrix}\Gamma-\fint_{Q_{B}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;V(r)\,|\alpha(X,r)|^{2}.$
(4.8)

Note that we added and subtracted the first term in (4.6) and that we added
and subtracted the first term in (4.8) to replace the Hamiltonian $H_{0}$ in
(4.7) by $H_{\Delta}$. The operators inside the traces in (4.6) and (4.7) are
not necessarily locally trace class, which is the reason we introduce the weak
local trace. We also note that (4.7) equals $\frac{T}{2}$ times the relative
entropy $\mathcal{H}_{0}(\Gamma,\Gamma_{\Delta})$ of $\Gamma$ with respect to
$\Gamma_{\Delta}$, defined in (3.18).

The first term in (4.8) can be written as

$\displaystyle-\frac{1}{2}\operatorname{Tr}_{0}\begin{pmatrix}0&\Delta\\\
\overline{\Delta}&0\end{pmatrix}\Gamma$
$\displaystyle=2\operatorname{\mathrm{Re}}\fint_{Q_{B}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;(V\alpha_{*})(r)\Psi(X)\;\overline{\alpha}(X,r).$
(4.9)

The integrands in (4.8) and (4.9) are equal to

$-|\alpha(X,r)|^{2}+2\operatorname{\mathrm{Re}}\alpha_{*}(r)\Psi(X)\;\overline{\alpha}(X,r)=-\bigl{|}\alpha(X,r)-\alpha_{*}(r)\Psi(X)\bigr{|}^{2}+\bigl{|}\alpha_{*}(r)\Psi(X)\bigr{|}^{2}.$

To rewrite (4.6) we need the following identities, whose proofs are
straightforward computations:

$\displaystyle\Gamma_{\Delta}$
$\displaystyle=\frac{1}{2}-\frac{1}{2}\tanh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)},$
$\displaystyle\ln(\Gamma_{\Delta})$
$\displaystyle=-\frac{\beta}{2}H_{\Delta}-\ln\bigl{(}2\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)},$
$\displaystyle 1-\Gamma_{\Delta}$
$\displaystyle=\frac{1}{2}+\frac{1}{2}\tanh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)},$
$\displaystyle\ln(1-\Gamma_{\Delta})$
$\displaystyle=\frac{\beta}{2}H_{\Delta}-\ln\bigl{(}2\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)}.$
(4.10)

Eq. (4.10) implies

$\displaystyle\Gamma_{\Delta}\ln(\Gamma_{\Delta})+(1-\Gamma_{\Delta})\ln(1-\Gamma_{\Delta})=-\ln\bigl{(}2\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)}+\frac{\beta}{2}H_{\Delta}\tanh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)},$
(4.11)

as well as

$\displaystyle\beta H_{\Delta}\Gamma_{\Delta}-\varphi(\Gamma_{\Delta})$
$\displaystyle=\frac{\beta}{2}H_{\Delta}-\ln\bigl{(}2\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)}.$

This allows us to rewrite (4.6) as

$\displaystyle\frac{1}{2\beta}\,\operatorname{Tr}_{0}\bigl{[}(\beta
H_{\Delta}\Gamma_{\Delta}-\beta
H_{0}\Gamma_{0})-\varphi(\Gamma_{\Delta})+\varphi(\Gamma_{0})\bigr{]}$
$\displaystyle=\frac{1}{4}\operatorname{Tr}_{0}\bigl{[}H_{\Delta}-H_{0}\bigr{]}-\frac{1}{2\beta}\operatorname{Tr}_{0}\bigl{[}\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)}-\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{0}\bigr{)}\bigr{)}\bigr{]}.$
(4.12)

We note that $H_{\Delta}-H_{0}$ is weakly locally trace class and that its
weak trace equals $0$. This, in particular, implies that the second term on
the right side of (4.12) is weakly locally trace class. To summarize, our
intermediate result reads

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{\mathbf{B},T}(\Gamma_{0})$
$\displaystyle=-\frac{1}{2\beta}\operatorname{Tr}_{0}\bigl{[}\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)}-\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{0}\bigr{)}\bigr{)}\bigr{]}$
$\displaystyle\hskip
30.0pt+\|\Psi\|_{L_{\mathrm{mag}}^{2}(Q_{B})}^{2}\;\langle\alpha_{*},V\alpha_{*}\rangle_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\hskip
30.0pt+\frac{T}{2}\mathcal{H}_{0}(\Gamma,\Gamma_{\Delta})-\fint_{Q_{B}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;V(r)\,\bigl{|}\alpha(X,r)-\alpha_{*}(r)\Psi(X)\bigr{|}^{2}.$
(4.13)

In order to compute the first term on the right side of (4.13), we need Lemma
4.2 below. It is the main technical novelty of our trial state analysis and
should be compared to the related part in the proof of [17, Theorem 2]. The
main difference between our proof of Lemma 4.2 and the relevant parts of the
proof of [17, Theorem 2] is that we use the product representation of the
hyperbolic cosine in (4.15) below instead of a Cauchy integral representation
of the function $z\mapsto\ln(1+e^{-z})$. In this way we obtain better decay
properties in the subsequent resolvent expansion, which simplifies the
analysis considerably.

As already noted above, the admissibility of $\Gamma_{\Delta}$ implies that
the difference between the two operators in the first term on the right side
of (4.13) is weakly locally trace class. We highlight that this is a
nontrivial statement because each of the two operators separately does not
share this property. We also highlight that our proof of Lemma 4.2 does not
require this as an assumption, it implies the statement independently.

In combination with (4.13), Lemma 4.2 below proves Proposition 3.4. Before we
state the lemma, we recall the definitions of the operators $L_{T,B}$ and
$N_{T,B}$ in (3.15) and (3.16), respectively.

######  Lemma 4.2.

Let $V\alpha_{*}\in L^{\nicefrac{{6}}{{5}}}(\mathbb{R}^{3})\cap
L^{2}(\mathbb{R}^{3})$. For any $B>0$, any $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{B})$, and any $T>0$, the operator

$\displaystyle\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)}-\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{0}\bigr{)}\bigr{)}$

is weakly locally trace class and its weak local trace equals

$\displaystyle-\frac{1}{2\beta}\operatorname{Tr}_{0}\Bigl{[}\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)}-\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{0}\bigr{)}\bigr{)}\Bigr{]}$
$\displaystyle=-\frac{1}{4}\langle\Delta,L_{T,B}\Delta\rangle+\frac{1}{8}\langle\Delta,N_{T,B}(\Delta)\rangle+\operatorname{Tr}\mathcal{R}_{T,B}(\Delta).$
(4.14)

The operator $\mathcal{R}_{T,B}(\Delta)$ is locally trace class and its trace
norm satisfies the bound

$\displaystyle\|\mathcal{R}_{T,B}(\Delta)\|_{1}$ $\displaystyle\leqslant
C\;T^{-5}\;B^{3}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{6}.$

###### Proof of Lemma 4.2.

We recall the Matsubara frequencies in (3.13) and write the hyperbolic cosine
in terms of the following product expansion, see [1, Eq. (4.5.69)],

$\displaystyle\cosh\bigl{(}\frac{\beta}{2}x\bigr{)}$
$\displaystyle=\prod_{k=0}^{\infty}\Bigl{(}1+\frac{x^{2}}{\omega_{k}^{2}}\Bigr{)}.$
(4.15)

We have

$\displaystyle 0$
$\displaystyle\leqslant\sum_{k=0}^{\infty}\ln\bigl{(}1+\frac{x^{2}}{\omega_{k}^{2}}\bigr{)}=\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}x\bigr{)}\bigr{)}\leqslant\frac{\beta}{2}\;|x|,$
$\displaystyle x$ $\displaystyle\in\mathbb{R},$

and accordingly

$\displaystyle\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)}=\sum_{k=0}^{\infty}\ln\bigl{(}1+\frac{H_{\Delta}^{2}}{\omega_{k}^{2}}\bigr{)}$

holds in a strong sense on the domain of $|H_{\Delta}|$. Since $\Delta$ is a
bounded operator by Lemma 4.1, the domains of $|H_{\Delta}|$ and $|H_{0}|$
coincide. The identity

$\displaystyle\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)}-\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{0}\bigr{)}\bigr{)}=\sum_{k=0}^{\infty}\bigl{[}\ln\bigl{(}\omega_{k}^{2}+H_{\Delta}^{2}\bigr{)}-\ln\bigl{(}\omega_{k}^{2}+H_{0}^{2}\bigr{)}\bigr{]}$
(4.16)

therefore holds in a strong sense on the domain of $|H_{0}|$. Elementary
arguments show that

$\displaystyle\ln\bigl{(}\omega^{2}+H_{\Delta}^{2}\bigr{)}-\ln\bigl{(}\omega^{2}+H_{0}^{2}\bigr{)}=-\lim_{R\to\infty}\int_{\omega}^{R}\mathrm{d}u\;\Bigl{[}\frac{2u}{u^{2}+H_{\Delta}^{2}}-\frac{2u}{u^{2}+H_{0}^{2}}\Bigr{]}$
(4.17)

holds for $\omega>0$ in a strong sense on the domain of $\ln(1+|H_{0}|)$.
Therefore, by (4.16) and (4.17), we have

$\displaystyle\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{\Delta}\bigr{)}\bigr{)}-\ln\bigl{(}\cosh\bigl{(}\frac{\beta}{2}H_{0}\bigr{)}\bigr{)}$
$\displaystyle=-\mathrm{i}\sum_{k=0}^{\infty}\int_{\omega_{k}}^{\infty}\mathrm{d}u\;\Bigl{[}\frac{1}{\mathrm{i}u-H_{\Delta}}-\frac{1}{\mathrm{i}u-H_{0}}+\frac{1}{\mathrm{i}u+H_{\Delta}}-\frac{1}{\mathrm{i}u+H_{0}}\Bigr{]}$
(4.18)

in a strong sense on the domain of $|H_{0}|$. By a slight abuse of notation,
we have incorporated the limit in (4.17) into the integral.

In the next step we use the resolvent expansion

$\displaystyle(z-H_{\Delta})^{-1}=(z-H_{0})^{-1}+(z-H_{0})^{-1}\;(H_{\Delta}-H_{0})\;(z-H_{\Delta})^{-1}$
(4.19)

to see that the right side of (4.18) equals

$\displaystyle\mathcal{O}_{1}+\mathcal{D}_{2}+\mathcal{O}_{3}+\mathcal{D}_{4}+\mathcal{O}_{5}-2\beta\,\mathcal{R}_{T,B}(\Delta),$

with two diagonal operators $\mathcal{D}_{2}$ and $\mathcal{D}_{4}$, three
offdiagonal operators $\mathcal{O}_{1}$, $\mathcal{O}_{3}$ and
$\mathcal{O}_{5}$ and a remainder term $\mathcal{R}_{T,B}(\Delta)$. The index
of the operators reflects the number of $\delta$ matrices appearing in their
definition. The diagonal operators $\mathcal{D}_{2}$ and $\mathcal{D}_{4}$ are
given by

$\displaystyle\mathcal{D}_{2}$
$\displaystyle:=-\mathrm{i}\sum_{k=0}^{\infty}\int_{\omega_{k}}^{\infty}\mathrm{d}u\;\Bigl{[}\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}+\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\Bigr{]},$
$\displaystyle\mathcal{D}_{4}$
$\displaystyle:=-\mathrm{i}\sum_{k=0}^{\infty}\int_{\omega_{k}}^{\infty}\mathrm{d}u\;\Bigl{[}\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}$
$\displaystyle\hskip
120.0pt+\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\Bigr{]}$

and the offdiagonal operators read

$\displaystyle\mathcal{O}_{1}$
$\displaystyle:=-\mathrm{i}\sum_{k=0}^{\infty}\int_{\omega_{k}}^{\infty}\mathrm{d}u\;\Bigl{[}\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}+\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\Bigr{]},$
$\displaystyle\mathcal{O}_{3}$
$\displaystyle:=-\mathrm{i}\sum_{k=0}^{\infty}\int_{\omega_{k}}^{\infty}\mathrm{d}u\;\Bigl{[}\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}$
$\displaystyle\hskip
85.0pt+\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\Bigr{]},$
$\displaystyle\mathcal{O}_{5}$
$\displaystyle:=-\mathrm{i}\sum_{k=0}^{\infty}\int_{\omega_{k}}^{\infty}\mathrm{d}u\;\Bigl{[}\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}$
$\displaystyle\hskip
85.0pt+\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\Bigr{]}.$

Since the operators $\mathcal{O}_{1}$, $\mathcal{O}_{3}$, and
$\mathcal{O}_{5}$ are offdiagonal, they are weakly locally trace class and
their weak local trace equals $0$. We also note that the operator
$\mathcal{O}_{1}$ is not necessarily locally trace class, which is why we need
to work with the weak local trace. The operator $\mathcal{R}_{T,B}(\Delta)$ is
defined by

$\displaystyle\mathcal{R}_{T,B}(\Delta)$
$\displaystyle:=\frac{\mathrm{i}}{2\beta}\sum_{k=0}^{\infty}\int_{\omega_{k}}^{\infty}\mathrm{d}u\;\Bigl{[}\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{\Delta}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta\frac{1}{\mathrm{i}u-H_{0}}$
$\displaystyle\hskip
50.0pt+\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{\Delta}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\frac{1}{\mathrm{i}u+H_{0}}\Bigr{]}.$

It remains to compute the traces of $\mathcal{D}_{2}$ and $\mathcal{D}_{4}$,
and to estimate the trace norm of $\mathcal{R}_{T,B}(\Delta)$. We first
consider $\mathcal{D}_{2}$ and use Hölder’s inequality in (2.1) to estimate

$\displaystyle\Bigl{\|}\frac{1}{\mathrm{i}u\pm
H_{0}}\delta\frac{1}{\mathrm{i}u\pm H_{0}}\delta\frac{1}{\mathrm{i}u\pm
H_{0}}\Bigr{\|}_{1}$ $\displaystyle\leqslant\Bigl{\|}\frac{1}{\mathrm{i}u\pm
H_{0}}\Bigr{\|}_{\infty}^{3}\;\|\delta\|_{2}^{2}=\frac{2}{u^{3}}\;\|\Delta\|_{2}^{2}.$
(4.20)

Therefore, Lemma 4.1 shows that the combination of the series and the integral
defining $\mathcal{D}_{2}$ converges absolutely in local trace norm. In
particular, $\mathcal{D}_{2}$ is locally trace class and we may arbitrarily
interchange the trace, the sum, and the integral to compute its trace. We do
this, use the cyclicity of the trace, and obtain

$\operatorname{Tr}\mathcal{D}_{2}=-\mathrm{i}\sum_{k=0}^{\infty}\int_{\omega_{k}}^{\infty}\mathrm{d}u\;\operatorname{Tr}\Bigl{[}\Bigl{(}\frac{1}{\mathrm{i}u-H_{0}}\Bigr{)}^{2}\delta\frac{1}{\mathrm{i}u-H_{0}}\delta+\Bigl{(}\frac{1}{\mathrm{i}u+H_{0}}\Bigr{)}^{2}\delta\frac{1}{\mathrm{i}u+H_{0}}\delta\Bigr{]}.$
(4.21)

Integration by parts shows

$\displaystyle\int_{\omega_{k}}^{\infty}\mathrm{d}u\;\Bigl{(}\frac{1}{\mathrm{i}u\pm
H_{0}}\Bigr{)}^{2}\delta\frac{1}{\mathrm{i}u\pm H_{0}}\delta$
$\displaystyle=-\mathrm{i}\;\frac{1}{\mathrm{i}\omega_{k}\pm
H_{0}}\delta\frac{1}{\mathrm{i}\omega_{k}\pm H_{0}}\delta$
$\displaystyle\hskip
80.0pt-\int_{\omega_{k}}^{\infty}\mathrm{d}u\;\frac{1}{\mathrm{i}u\pm
H_{0}}\delta\Bigl{(}\frac{1}{\mathrm{i}u\pm H_{0}}\Bigr{)}^{2}\delta,$

and another application of the cyclicity of the trace yields

$\displaystyle\operatorname{Tr}\int_{\omega_{k}}^{\infty}\mathrm{d}u\;\Bigl{(}\frac{1}{\mathrm{i}u\pm
H_{0}}\Bigr{)}^{2}\delta\frac{1}{\mathrm{i}u\pm H_{0}}\delta$
$\displaystyle=-\frac{\mathrm{i}}{2}\;\operatorname{Tr}\frac{1}{\mathrm{i}\omega_{k}\pm
H_{0}}\delta\frac{1}{\mathrm{i}\omega_{k}\pm H_{0}}\delta.$ (4.22)

Note that

$\displaystyle\frac{1}{\mathrm{i}\omega_{k}\pm
H_{0}}\,\delta\,\frac{1}{\mathrm{i}\omega_{k}\pm H_{0}}\,\delta$
$\displaystyle=\begin{pmatrix}\frac{1}{\mathrm{i}\omega_{k}\pm\mathfrak{h}_{B}}\,\Delta\frac{1}{\mathrm{i}\omega_{k}\mp\overline{\mathfrak{h}_{B}}}\,\overline{\Delta}\\\
&\frac{1}{\mathrm{i}\omega_{k}\mp\overline{\mathfrak{h}_{B}}}\,\overline{\Delta}\frac{1}{\mathrm{i}\omega_{k}\pm\mathfrak{h}_{B}}\,\Delta\end{pmatrix}.$
(4.23)

We combine this with (4.21) and (4.22) and summarize the cases $\pm$ into a
single sum over $n\in\mathbb{Z}$. This yields

$\displaystyle-\frac{1}{2\beta}\operatorname{Tr}\mathcal{D}_{2}$
$\displaystyle=\frac{1}{2\beta}\sum_{n\in\mathbb{Z}}\Bigl{\langle}\Delta,\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{B}}\Delta\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{B}}}\Bigr{\rangle}=-\frac{1}{4}\langle\Delta,L_{T,B}\Delta\rangle,$

where $L_{T,B}$ is the operator defined in (3.15).

We argue as above to see that the integrand in the definition of
$\mathcal{D}_{4}$ is bounded by $C\|\Delta\|_{4}^{4}\,u^{-5}$. Moreover, we
have $\|\Delta\|_{4}^{4}\leqslant
CB^{2}\|V\alpha_{*}\|_{\nicefrac{{4}}{{3}}}^{4}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{4}$
by (2.7) and Lemma 4.1. Therefore, the integral and the sum in
$\mathcal{D}_{4}$ are absolutely convergent with respect to the local trace
norm. The trace of $\mathcal{D}_{4}$ is computed similar to that of
$\mathcal{D}_{2}$. With $N_{T,B}$ defined in (3.16), the result reads

$\displaystyle-\frac{1}{2\beta}\operatorname{Tr}\mathcal{D}_{4}$
$\displaystyle=\frac{1}{8}\langle\Delta,N_{T,B}(\Delta)\rangle.$ (4.24)

In case of $\mathcal{R}_{T,B}(\Delta)$, we bound the trace norm of the
operator inside the integral by $u^{-7}\|\Delta\|_{6}^{6}$. Using (2.7) and
Lemma 4.1, we estimate the second factor by a constant times
$\|V\alpha_{*}\|_{\nicefrac{{6}}{{5}}}^{6}B^{-3}\|\Pi\Psi\|_{2}^{6}\leqslant
CB^{3}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{6}$. Finally, integration over
$u$ yields the term $6\pi^{-6}T^{-6}(2k+1)^{-6}$, which is summable in $k$.
This proves the claimed bound for the trace norm of
$\mathcal{R}_{T,B}(\Delta)$. ∎

###  4.3 Proof of Theorem 3.5

####  4.3.1 Magnetic resolvent estimates

In this preparatory subsection, we provide estimates for the magnetic
resolvent kernel

$\displaystyle G^{z}_{B}(x,y)$
$\displaystyle:=\frac{1}{z-\mathfrak{h}_{B}}(x,y),$ $\displaystyle x,y$
$\displaystyle\in\mathbb{R}^{3}.$

We also introduce the function

$\displaystyle g_{B}^{z}(x)$ $\displaystyle:=G_{B}^{z}(x,0),$ $\displaystyle
x$ $\displaystyle\in\mathbb{R}^{3}.$ (4.25)

The proof of the following statement can be found in [15, Lemma 8].

######  Lemma 4.3.

For all $B\geqslant 0$, $z\in\mathbb{C}\setminus[B,\infty)$ and
$x,y\in\mathbb{R}^{3}$ we have

  1. (a)

$g_{B}^{z}(-x)=g_{B}^{z}(x)$,

  2. (b)

$G_{B}^{z}(x,y)=\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}\cdot(x\wedge
y)}\;g_{B}^{z}(x-y)$.

We start our analysis by providing a decay estimate for the $L^{1}$-norm of
the resolvent kernel $g^{z}_{0}$ in (4.25) and its gradient in the case $B=0$.
For $g_{0}^{z}$ such an estimate has been provided in [15, Lemma 9]. Since we
additionally need an estimate for $\nabla g_{0}^{z}$, we repeat some of the
arguments here.

######  Lemma 4.4.

Let $a>-2$. There is a constant $C_{a}>0$ such that for
$t,\omega\in\mathbb{R}$, we have

$\displaystyle\left\|\,|\cdot|^{a}g_{0}^{\mathrm{i}\omega+t}\right\|_{1}$
$\displaystyle\leqslant C_{a}\;f(t,\omega)^{1+\frac{a}{2}},$ (4.26)

where

$\displaystyle
f(t,\omega):=\frac{|\omega|+|t+\mu|}{(|\omega|+(t+\mu)_{-})^{2}}$ (4.27)

and $x_{-}:=-\min\\{x,0\\}$. Furthermore, for any $a>-1$, there is a constant
$C_{a}>0$ with

$\displaystyle\left\|\,|\cdot|^{a}\nabla
g_{0}^{\mathrm{i}\omega+t}\right\|_{1}\leqslant
C_{a}\;f(t,\omega)^{\frac{1}{2}+\frac{a}{2}}\;\Bigl{[}1+\frac{|\omega|+|t+\mu|}{|\omega|+(t+\mu)_{-}}\Bigr{]}.$
(4.28)

###### Proof.

The resolvent kernel $g_{0}^{z}$ is given by

$\displaystyle g_{0}^{z}(x)$
$\displaystyle=-\frac{1}{4\pi|x|}\;\mathrm{e}^{-\sqrt{-(z+\mu)}\;|x|},$ (4.29)

where $\sqrt{\cdot}$ denotes the standard branch of the square root. As long
as $a>-2$ we have

$\displaystyle\left\|\,|\cdot|^{a}g_{0}^{z}\right\|_{1}$
$\displaystyle=\int_{\mathbb{R}^{3}}\mathrm{d}x\;|x|^{a}\;\Bigl{|}\frac{1}{4\pi|x|}\mathrm{e}^{-|x|\sqrt{-(z+\mu)}}\Bigr{|}=\frac{\Gamma(a+2)}{(\operatorname{\mathrm{Re}}\sqrt{-(z+\mu)})^{a+2}}.$
(4.30)

Moreover,

$\displaystyle(\operatorname{\mathrm{Re}}\sqrt{-z})^{2}=\frac{1}{2}(|z|-\operatorname{\mathrm{Re}}z)\geqslant\begin{cases}\frac{1}{4}\frac{|\operatorname{\mathrm{Im}}z|^{2}}{\operatorname{\mathrm{Re}}z+|\operatorname{\mathrm{Im}}z|}&\operatorname{\mathrm{Re}}z\geqslant
0,\\\ \frac{1}{2}|z|&\operatorname{\mathrm{Re}}z<0,\end{cases}$

and hence

$\displaystyle(\operatorname{\mathrm{Re}}\sqrt{-(t+\mu+\mathrm{i}\omega)})^{2}\geqslant\frac{1}{4}\frac{(|\omega|+(t+\mu)_{-})^{2}}{|\omega|+|t+\mu|}.$

This proves (4.26). To prove (4.28), we use (4.29) and estimate

$\displaystyle|\nabla
g_{0}^{z}(x)|\leqslant|z+\mu|^{\nicefrac{{1}}{{2}}}\;|g_{0}^{z}(x)|+|x|^{-1}|g_{0}^{z}(x)|.$

This shows the second estimate for $a>-1$. ∎

In the next step we prove estimates for the $L^{1}$-norms of $g_{B}^{z}$ and
$g_{B}^{z}-g_{0}^{z}$ and the gradient of these functions if $B\neq 0$. Once
more, some of the arguments in [15, Lemma 10] reappear in our proof below,
ensuring self-consistency.

######  Lemma 4.5.

For any $a\geqslant 0$, there are constants $\delta_{a},C_{a}>0$ such that for
all $t,\omega\in\mathbb{R}$ and for all $B\geqslant 0$ with
$f(t,\omega)^{2}\,B^{2}\leqslant\delta_{a}$, we have

$\displaystyle\bigl{\|}|\cdot|^{a}g_{B}^{\mathrm{i}\omega+t}\bigr{\|}_{1}$
$\displaystyle\leqslant C_{a}\,f(t,\omega)^{1+\frac{a}{2}},$
$\displaystyle\bigl{\|}|\cdot|^{a}\nabla
g_{B}^{\mathrm{i}\omega+t}\bigr{\|}_{1}$ $\displaystyle\leqslant
C_{a}\,f(t,\omega)^{\frac{1}{2}+\frac{a}{2}}\Bigl{[}1+\frac{|\omega|+|t+\mu|}{|\omega|+(t+\mu)_{-}}\Bigr{]},$
(4.31)

and

$\displaystyle\bigl{\|}\,|\cdot|^{a}(g_{B}^{\mathrm{i}\omega+t}-g_{0}^{\mathrm{i}\omega+t})\bigr{\|}_{1}$
$\displaystyle\leqslant C_{a}\,B^{2}\,f(t,\omega)^{3+\frac{a}{2}},$
$\displaystyle\bigl{\|}\,|\cdot|^{a}(\nabla g_{B}^{\mathrm{i}\omega+t}-\nabla
g_{0}^{\mathrm{i}\omega+t})\bigr{\|}_{1}$ $\displaystyle\leqslant
C_{a}\,B^{2}\,f(t,\omega)^{\frac{5}{2}+\frac{a}{2}}\Bigl{[}1+\frac{|\omega|+|t+\mu|}{|\omega|+(t+\mu)_{-}}\Bigr{]}$
(4.32)

with the function $f(t,\omega)$ in (4.27).

###### Proof.

During the proof we use the notation $z=\mathrm{i}\omega+t$. We define the
function

$\displaystyle h^{z}(x):=\frac{1}{4}\,|e_{3}\wedge x|^{2}\,g_{0}^{z}(x)$
(4.33)

and choose $\delta_{a}$ such that $2\delta_{a}D_{a}C_{2}=1$. Here $C_{2}$
denotes the constant in (4.26) and $D_{a}:=1$ if $0\leqslant a\leqslant 1$ and
$D_{a}:=2^{a}$ if $a>1$. Lemma 4.4 and the bound
$\|h^{z}\|_{1}\leqslant\||\cdot|^{2}g_{0}^{z}\|_{1}$ imply

$\displaystyle B^{2}D_{a}\|h^{z}\|_{1}\leqslant\frac{1}{2}$ (4.34)

for all $\omega$, $t$, and $B$ that are allowed by our assumptions. We define
the operator $\widetilde{G}_{B}^{z}$ by the kernel

$\displaystyle\widetilde{G}_{B}^{z}(x,y)$
$\displaystyle:=\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}(x\wedge
y)}g_{0}^{z}(x-y)$

and note that

$\displaystyle(z-\mathfrak{h}_{B})\widetilde{G}_{B}^{z}=1-T_{B}^{z},$ (4.35)

where $T_{B}^{z}$ is the operator given by the kernel

$\displaystyle T_{B}^{z}(x,y)$
$\displaystyle:=\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}(x\wedge
y)}\bigl{[}\mathbf{B}\wedge(x-y)(-\mathrm{i}\nabla_{x})g_{0}^{z}(x-y)+B^{2}\;h^{z}(x-y)\bigr{]}.$

The first term in square brackets equals $0$ because $g_{0}^{z}$ is a radial
function, which implies that the vector $\nabla g_{0}^{z}(x-y)$ is
perpendicular to $\mathbf{B}\wedge(x-y)$. Multiplication of (4.35) with
$(z-\mathfrak{h}_{B})^{-1}$ from the left yields

$\displaystyle
G_{B}^{z}(x,y)-\widetilde{G}_{B}^{z}(x,y)=\int_{\mathbb{R}^{3}}\mathrm{d}v\;G_{B}^{z}(x,v)T_{B}^{z}(v,y).$

We set $y=0$, change variables $v\mapsto x-v$, and find

$\displaystyle
g_{B}^{z}(x)-g_{0}^{z}(x)=B^{2}\int_{\mathbb{R}^{3}}\mathrm{d}v\;\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}\cdot(v\wedge
x)}\;g_{B}^{z}(v)\;h^{z}(x-v).$ (4.36)

This implies

$\displaystyle\|g_{B}^{z}-g_{0}^{z}\|_{1}$ $\displaystyle\leqslant
B^{2}\||g_{B}^{z}|*|h^{z}|\|_{1}\leqslant
B^{2}\;\|g_{B}^{z}-g_{0}^{z}\|_{1}\;\|h^{z}\|_{1}+B^{2}\;\|g_{0}^{z}\|_{1}\;\|h^{z}\|_{1}.$
(4.37)

A straightforward calculation involving (4.35) and the Neumann series shows
that $g_{B}^{z}-g_{0}^{z}$ belongs to $L^{1}(\mathbb{R}^{3})$. Therefore,
(4.34) and (4.37) imply

$\displaystyle\|g_{B}^{z}-g_{0}^{z}\|_{1}$
$\displaystyle\leqslant\|g_{0}^{z}\|_{1}$ (4.38)

for all $t,\omega$, and $B$ that are allowed by our assumptions.

We use this estimate as a basis to prove the bounds claimed in the lemma and
start with the first bound in (4.32). By (4.36), we have

$\displaystyle\|\,|\cdot|^{a}(g_{B}^{z}-g_{0}^{z})\|_{1}$
$\displaystyle\leqslant
D_{a}B^{2}\bigl{[}\|\,|\cdot|^{a}g_{0}^{z}\|_{1}\,\|h^{z}\|_{1}+\|\,|\cdot|^{a}(g_{B}^{z}-g_{0}^{z})\|_{1}\,\|h^{z}\|_{1}$
$\displaystyle\hskip
80.0pt+\|g_{B}^{z}-g_{0}^{z}\|_{1}\,\||\cdot|^{a}h^{z}\|_{1}+\|g_{0}^{z}\|_{1}\,\|\,|\cdot|^{a}h^{z}\|_{1}\bigr{]}.$

A similar argument to the one above (4.38) shows that
$|\cdot|^{a}(g_{B}^{z}-g_{0}^{z})$ belongs to $L^{1}(\mathbb{R}^{3})$. In
combination with (4.38), we therefore obtain

$\displaystyle\|\,|\cdot|^{a}(g_{B}^{z}-g_{0}^{z})\|_{1}$
$\displaystyle\leqslant
2D_{a}B^{2}\;\bigl{[}\||\cdot|^{a}g_{0}^{z}\|_{1}\,\|h^{z}\|_{1}+2\,\|g_{0}^{z}\|_{1}\,\||\cdot|^{a}h^{z}\|_{1}\bigr{]}.$
(4.39)

With the help of Lemma 4.4 and (4.34), we read off the first bound in (4.32).
Moreover, the triangle inequality, Lemma 4.4, the first bound in (4.32) and
the bound $f(t,\omega)^{2}\,B^{2}\leqslant\delta_{a}$ imply the first bound in
(4.31).

Next, we consider the bounds in (4.31) and (4.32) involving the gradient. As a
preparation, the bound $|\nabla
h^{z}(x)|\leqslant|x|\,|g_{0}^{z}(x)|+|x|^{2}\,|\nabla g_{0}^{z}(x)|$ and
Lemma 4.4 show

$\displaystyle\|\,|\cdot|^{a}\nabla h^{z}\|_{1}$ $\displaystyle\leqslant
C_{a}\;f(t,\omega)^{\frac{3}{2}+\frac{a}{2}}\Bigl{[}1+\frac{|\omega|+|t+\mu|}{|\omega|+(t+\mu)_{-}}\Bigr{]}.$
(4.40)

We use $|\nabla\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}\cdot(v\wedge
x)}|\leqslant B|v|$ and (4.36) to see that

$\displaystyle|\nabla g_{B}^{z}(x)-\nabla g_{0}^{z}(x)|$
$\displaystyle\leqslant
B^{2}\int_{\mathbb{R}^{3}}\mathrm{d}v\;\bigl{[}B\,|v|\,|g_{B}^{z}(v)|\,|h^{z}(x-v)|+|g_{B}^{z}(v)|\,|\nabla
h^{z}(x-v)|\bigr{]}$

as well as

$\displaystyle\|\,|\cdot|^{a}(\nabla g_{B}^{z}-\nabla g_{0}^{z})\|_{1}$
$\displaystyle\leqslant
D_{a}B^{2}\bigl{[}B\,\|\,|\cdot|^{a+1}g_{B}^{z}\|_{1}\,\|h^{z}\|_{1}+B\,\|\,|\cdot|g_{B}^{z}\|_{1}\,\|\,|\cdot|^{a}h^{z}\|_{1}$
$\displaystyle\hskip 60.0pt+\|\,|\cdot|^{a}g_{B}^{z}\|_{1}\;\|\nabla
h^{z}\|_{1}+\|g_{B}^{z}\|_{1}\;\|\,|\cdot|^{a}\nabla h^{z}\|_{1}\bigr{]}.$
(4.41)

When we combine (4.41), the first estimates in (4.31) and (4.32), the bound in
(4.40) and Lemma 4.4, we see that

$\displaystyle\|\,|\cdot|^{a}(\nabla g_{B}^{z}-\nabla g_{0}^{z})\|_{1}$
$\displaystyle\leqslant
C_{a}\,B^{2}\,f(t,\omega)^{\frac{5}{2}+\frac{a}{2}}\bigl{[}1+Bf(t,\omega)\bigr{]}.$

An application of the assumption $B^{2}f(t,\omega)^{2}\leqslant\delta_{a}$
proves the second bound in (4.32). Finally, the triangle inequality, the
second bound in (4.32), and Lemma 4.4 show

$\displaystyle\|\,|\cdot|^{a}\nabla g_{B}^{z}\|_{1}$ $\displaystyle\leqslant
C_{a}\,f(t,\omega)^{\frac{1}{2}+\frac{a}{2}}\Bigl{[}1+\frac{|\omega|+|t+\mu|}{|\omega|+(t+\mu)_{-}}\Bigr{]}\bigl{[}1+B^{2}f(t,\omega)^{2}\bigr{]}.$

Another application of $B^{2}f(t,\omega)^{2}\leqslant\delta_{a}$ on the right
side proves the second bound in (4.31). ∎

####  4.3.2 A representation formula for $L_{T,B}$ and an outlook on the
quadratic terms

In this subsection we compute the terms in (3.20) involving the linear
operator $L_{T,B}$ defined in (3.15). Our starting point is the representation
formula for $L_{T,B}$ in [15, Lemma 11], which expresses the operator
explicitly in terms of the relative and the center-of-mass coordinate.

######  Lemma 4.6.

The operator
$L_{T,B}\colon{L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}\rightarrow{L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}$
in (3.15) acts as

$\displaystyle(L_{T,B}\alpha)(X,r)$
$\displaystyle=\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T,B}(Z,r,s)\;(\cos(Z\cdot\Pi_{X})\alpha)(X,s)$

with

$\displaystyle k_{T,B}(Z,r,s)$
$\displaystyle:=\frac{2}{\beta}\sum_{n\in\mathbb{Z}}k_{T,B}^{n}(Z,r-s)\;\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}\cdot(r\wedge
s)}$ (4.42)

and

$\displaystyle k_{T,B}^{n}(Z,r)$
$\displaystyle:=g_{B}^{\mathrm{i}\omega_{n}}\bigl{(}Z-\frac{r}{2}\bigr{)}\;g_{B}^{-\mathrm{i}\omega_{n}}\bigl{(}Z+\frac{r}{2}\bigr{)}.$
(4.43)

We analyze the operator $L_{T,B}$ in three steps. In the first two steps we
introduce two operators of increasing simplicity in their dependence on $B$:

$\displaystyle
L_{T,B}=(L_{T,B}-\widetilde{L}_{T,B})+(\widetilde{L}_{T,B}-M_{T,B})+M_{T,B},$
(4.44)

where $\widetilde{L}_{T,B}$ and $M_{T,B}$ are defined below in (4.45) and
(4.70), respectively. To obtain $\widetilde{L}_{T,B}$ we replace the functions
$g_{B}^{z}$ in the definition of $L_{T,B}$ by $g_{0}^{z}$. Moreover, $M_{T,B}$
is obtained from $\widetilde{L}_{T,B}$ when we replace $k_{T,B}$ by $k_{T,0}$,
i.e., when we additionally replace the magnetic phase
$\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}\cdot(r\wedge s)}$ by $1$. In
Section 4.3.3 we prove that the terms in the brackets in (4.44) are small in a
suitable sense. The third step consists of a careful analysis of the operator
$M_{T,B}$, which takes place in Section 4.3.4. There, we expand the operator
$\cos(Z\cdot\Pi_{X})$ in powers of $Z\cdot\Pi_{X}$ up to second order and
extract the quadratic terms of the Ginzburg–Landau functional in (1.17) as
well as a term that cancels the last term on the left side of (3.20). In
Section 4.3.5 we summarize our findings.

We remark that the operator $\widetilde{L}_{T,B}$ is called $M_{T,B}$ in [15]
and that $M_{T,B}$ is called $N_{T,B}$. The reason why we did not follow the
notation in [15] is that $N_{T,B}$ is reserved for the nonlinear term in the
present paper. We note that our decomposition of $L_{T,B}$ in (4.44) already
appeared in [15]. Parts of our analysis follow the analysis of $L_{T,B}$ in
Section 4 and Section 5 in that reference. However, we additionally need
${H^{1}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}$-norm bounds that are not
provided in [15]. It should also be noted that $L_{T,B}$ acts on
$L^{2}(\mathbb{R}^{6})$ in [15], while it acts on
${L^{2}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}$ in our case.

####  4.3.3 Approximation of $L_{T,B}$

#####  4.3.3.1 The operator $\widetilde{L}_{T,B}$.

The operator $\widetilde{L}_{T,B}$ is defined by

$\displaystyle\widetilde{L}_{T,B}\alpha(X,r)$
$\displaystyle:=\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;\widetilde{k}_{T,B}(Z,r,s)\;(\cos(Z\cdot\Pi_{X})\alpha)(X,s)$
(4.45)

with

$\displaystyle\widetilde{k}_{T,B}(Z,r,s):=\frac{2}{\beta}\sum_{n\in\mathbb{Z}}k_{T,0}^{n}(Z,r-s)\;\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}\cdot(r\wedge
s)}$ (4.46)

and $k_{T,0}^{n}$ in (4.43). In the following proposition we provide an
estimate that allows us to replace $L_{T,B}$ by $\widetilde{L}_{T,B}$ in our
computations.

######  Proposition 4.7.

For any $T_{0}>0$ there is $B_{0}>0$ such that for any $0<B\leqslant B_{0}$,
any $T\geqslant T_{0}$ and whenever $|\cdot|^{k}V\alpha_{*}\in
L^{2}(\mathbb{R}^{3})$ for $k\in\\{0,1\\}$, $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{B})$, and $\Delta\equiv\Delta_{\Psi}$ as in (3.1), we
have

$\displaystyle\|L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta\|_{H^{1}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}^{2}\leqslant
C\;B^{5}\;\bigl{(}\|V\alpha_{*}\|_{2}^{2}+\|\,|\cdot|V\alpha_{*}\|_{2}^{2}\bigr{)}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{2}.$

######  Remark 4.8.

For the proof of Theorem 3.5 we only need a bound for
$\langle\Delta,(L_{T,B}-\widetilde{L}_{T,B})\Delta\rangle$, which is easier to
obtain. This bound follows directly from Proposition 4.7, Lemma 4.1 and an
application of the Cauchy–Schwarz inequality. The more general bound in
Proposition 4.7 is needed in the proof of Proposition 3.2.

Before we start with the proof of Proposition 4.7 let $a\geqslant 0$ and
introduce the functions

$\displaystyle F_{T,B}^{a}$
$\displaystyle:=\smash{\frac{2}{\beta}\sum_{n\in\mathbb{Z}}}\;\bigl{(}|\cdot|^{a}\,|g_{B}^{\mathrm{i}\omega_{n}}-g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*|g_{B}^{-\mathrm{i}\omega_{n}}|+|g_{B}^{\mathrm{i}\omega_{n}}-g_{0}^{\mathrm{i}\omega_{n}}|*\bigl{(}|\cdot|^{a}\,|g_{B}^{-\mathrm{i}\omega_{n}}|\bigr{)}$
$\displaystyle\hskip
50.0pt+\bigl{(}|\cdot|^{a}\,|g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*|g_{B}^{-\mathrm{i}\omega_{n}}-g_{0}^{-\mathrm{i}\omega_{n}}|+|g_{0}^{\mathrm{i}\omega_{n}}|*\bigl{(}|\cdot|^{a}\,|g_{B}^{-\mathrm{i}\omega_{n}}-g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}$
(4.47)

and

$\displaystyle G_{T,B}$
$\displaystyle:=\smash{\frac{2}{\beta}\sum_{n\in\mathbb{Z}}}\;|\nabla
g_{B}^{\mathrm{i}\omega_{n}}-\nabla
g_{0}^{\mathrm{i}\omega_{n}}|*|g_{B}^{-\mathrm{i}\omega_{n}}|+|g_{B}^{\mathrm{i}\omega_{n}}-g_{0}^{\mathrm{i}\omega_{n}}|*|\nabla
g_{B}^{-\mathrm{i}\omega_{n}}|$ $\displaystyle\hskip 80.0pt+|\nabla
g_{0}^{\mathrm{i}\omega_{n}}|*|g_{B}^{-\mathrm{i}\omega_{n}}-g_{0}^{-\mathrm{i}\omega_{n}}|+|g_{0}^{\mathrm{i}\omega_{n}}|*|\nabla
g_{B}^{-\mathrm{i}\omega_{n}}-\nabla g_{0}^{-\mathrm{i}\omega_{n}}|$ (4.48)

with the Matsubara frequencies $\omega_{n}$ in (3.13) and the resolvent kernel
$g_{B}^{z}$ in (4.25). We claim that for any $a\geqslant 0$ there is a
constant $B_{0}>0$ such that for $0\leqslant B\leqslant B_{0}$ we have

$\displaystyle\|F_{T,B}^{a}\|_{1}+\|G_{T,B}\|_{1}\leqslant C_{a}B^{2}.$ (4.49)

To prove (4.49) we note that the function $f(t,\omega)$ in (4.27) satisfies

$\displaystyle f(0,\omega_{n})$ $\displaystyle\leqslant
C\;(T^{-1}+T^{-2})\;|2n+1|^{-1}$ (4.50)

and that

$\displaystyle\frac{|\omega_{n}|+|\mu|}{|\omega_{n}|+\mu_{-}}\leqslant
C\;(1+T^{-1}).$ (4.51)

Since $T\geqslant T_{0}>0$, Lemmas 4.4 and 4.5 prove (4.49).

###### Proof of Proposition 4.7.

We write

$\displaystyle\|L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta\|_{H^{1}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}^{2}$
$\displaystyle=\|L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta\|_{2}^{2}+\|\Pi_{X}(L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta)\|_{2}^{2}+\|\widetilde{\pi}_{r}(L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta)\|_{2}^{2}$
(4.52)

and claim that

$\displaystyle\|L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta\|_{2}^{2}$
$\displaystyle\leqslant
4\;\|\Psi\|_{2}^{2}\;\|F_{T,B}^{0}*|V\alpha_{*}|\,\|_{2}^{2}.$ (4.53)

If this is true, Young’s inequality, (2.5), and (4.49) prove

$\|L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta\|_{2}^{2}\leqslant
CB^{5}\;\|V\alpha_{*}\|_{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{2}.$
(4.54)

To see that (4.53) holds, we expand the squared modulus in the Hilbert–Schmidt
norm and obtain

$\displaystyle\|L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta\|_{2}^{2}$
$\displaystyle\leqslant
4\int_{\mathbb{R}^{3}}\mathrm{d}r\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}Z^{\prime}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}s\mathrm{d}s^{\prime}\;|V\alpha_{*}(s)|\;|V\alpha_{*}(s^{\prime})|\mathop{\times}$
$\displaystyle\hskip
120.0pt\times|k_{T,B}(Z,r,s)-\widetilde{k}_{T,B}(Z,r,s)|\mathop{\times}$
$\displaystyle\hskip
120.0pt\times|k_{T,B}(Z^{\prime},r,s^{\prime})-\widetilde{k}_{T,B}(Z^{\prime},r,s^{\prime})|\mathop{\times}$
$\displaystyle\hskip
30.0pt\times\fint_{Q_{B}}\mathrm{d}X\;|\cos(Z\cdot\Pi_{X})\Psi(X)|\;|\cos(Z^{\prime}\cdot\Pi_{X})\Psi(X)|.$
(4.55)

The operator $\cos(Z\cdot\Pi_{X})$ is bounded by $1$ and we have

$\displaystyle\fint_{Q_{B}}\mathrm{d}X\;|\cos(Z\cdot\Pi)\Psi(X)|\;|\cos(Z^{\prime}\cdot\Pi)\Psi(X)|$
$\displaystyle\leqslant\|\Psi\|_{2}^{2}.$ (4.56)

By (4.55), this implies

$\displaystyle\|L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta\|_{2}^{2}$
$\displaystyle\leqslant
4\;\|\Psi\|_{2}^{2}\;\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;|k_{T,B}(Z,r,s)-\widetilde{k}_{T,B}(Z,r,s)|\;|V\alpha_{*}(s)|\Bigr{|}^{2},$
(4.57)

where the integrand is bounded by

$\displaystyle|k_{T,B}(Z,r,s)-\widetilde{k}_{T,B}(Z,r,s)|$
$\displaystyle\leqslant\frac{2}{\beta}\,\smash{\sum_{n\in\mathbb{Z}}}\,\bigl{[}|g_{B}^{\mathrm{i}\omega_{n}}-g_{0}^{\mathrm{i}\omega_{n}}|\bigl{(}Z-\frac{r}{2}\bigr{)}\;|g_{B}^{-\mathrm{i}\omega_{n}}|\bigl{(}Z+\frac{r}{2}\bigr{)}$
$\displaystyle\hskip
40.0pt+|g_{0}^{\mathrm{i}\omega_{n}}|\bigl{(}Z-\frac{r}{2}\bigr{)}\;|g_{B}^{-\mathrm{i}\omega_{n}}-g_{0}^{-\mathrm{i}\omega_{n}}|\bigl{(}Z+\frac{r}{2}\bigr{)}\bigr{]}.$
(4.58)

For $a\geqslant 0$, we have the estimate

$\displaystyle|Z|^{a}$
$\displaystyle\leqslant\bigl{|}Z+\frac{r}{2}\bigr{|}^{a}+\bigl{|}Z-\frac{r}{2}\bigr{|}^{a}.$
(4.59)

This, (4.58), and the fact that $g_{B}^{\pm\mathrm{i}\omega_{n}}$ is an even
function imply

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}Z\;|Z|^{a}\;|k_{T,B}(Z,r,s)-\widetilde{k}_{T,B}(Z,r,s)|\leqslant
F_{T,B}^{a}(r-s),$ (4.60)

where $F_{T,B}^{a}$ is the function in (4.47). We apply the case $a=0$ to
(4.57) and read off (4.53).

We claim that the second term on the right side of (4.52) is bounded by

$\displaystyle\|\Pi_{X}(L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta)\|_{2}^{2}$
$\displaystyle\leqslant
CB^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{2}\;\|(F_{T,B}^{0}+F_{T,B}^{1})*|V\alpha_{*}|\,\|_{2}^{2}.$
(4.61)

If this is true, Young’s inequality and (4.49) show the claimed bound for this
term. To prove (4.61), we use (4.55) with $\cos(Z\cdot\Pi_{X})$ replaced by
$\Pi_{X}\cos(Z\cdot\Pi_{X})$, that is, we need to replace (4.56) by

$\displaystyle\fint_{Q_{B}}\mathrm{d}X\;|\Pi\cos(Z\cdot\Pi)\Psi(X)|\;|\Pi\cos(Z^{\prime}\cdot\Pi)\Psi(X)|$
$\displaystyle\leqslant\|\Pi\cos(Z\cdot\Pi)\Psi\|_{2}\;\|\Pi\cos(Z^{\prime}\cdot\Pi)\Psi\|_{2}.$

In Lemma 5.12 in Section 5 we prove intertwining relations for
$\cos(Z\cdot\Pi)$ with various magnetic momenta. The intertwining relation
(5.40) therein and (2.5) show

$\displaystyle\|\Pi\cos(Z\cdot\Pi)\Psi\|_{2}$
$\displaystyle\leqslant\|\Pi\Psi\|_{2}+2B|Z|\;\|\Psi\|_{2}\leqslant
C\,B\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}\;(1+|Z|),$ (4.62)

which yields

$\displaystyle\|\Pi_{X}(L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta)\|_{2}^{2}$
$\displaystyle\leqslant
C\,B^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{2}\mathop{\times}$
$\displaystyle\times\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;(1+|Z|)\;|k_{T,B}(Z,r,s)-\widetilde{k}_{T,B}(Z,r,s)|\;|V\alpha_{*}(s)|\Bigr{|}^{2}.$
(4.63)

We apply the cases $a=0$ and $a=1$ of (4.60) to this and obtain (4.61).

Concerning the third term on the right side of (4.52) we claim that

$\displaystyle\|\widetilde{\pi}_{r}(L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta)\|_{2}^{2}$
$\displaystyle\leqslant
C\;\|\Psi\|_{2}^{2}\;\bigl{\|}\bigl{(}G_{T,B}+F_{T,B}^{1}\bigr{)}*|V\alpha_{*}|+F_{T,B}^{0}*|\cdot|\,|V\alpha_{*}|\,\bigr{\|}_{2}^{2}.$
(4.64)

If this is true, Young’s inequality, (4.49), and (2.5) show the relevant bound
for this term. To prove (4.64), we estimate

$\displaystyle\|\widetilde{\pi}_{r}(L_{T,B}\Delta-\widetilde{L}_{T,B}\Delta)\|_{2}^{2}$
$\displaystyle\leqslant
4\;\|\Psi\|_{2}^{2}\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;|\widetilde{\pi}_{r}k_{T,B}(Z,r,s)-\widetilde{\pi}_{r}\widetilde{k}_{T,B}(Z,r,s)|\;|V\alpha_{*}(s)|\Bigr{|}^{2}.$
(4.65)

Using $\frac{1}{4}|\mathbf{B}\wedge r|\leqslant B(|r-s|+|s|)$ we see that the
integrand on the right side is bounded by

$\displaystyle|\widetilde{\pi}_{r}k_{T,B}(Z,r,s)-\widetilde{\pi}_{r}\widetilde{k}_{T,B}(Z,r,s)|$
$\displaystyle\leqslant\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\Big{[}|\nabla_{r}k_{T,B}^{n}(Z,r-s)-\nabla_{r}k_{T,0}^{n}(Z,r-s)|$
$\displaystyle\hskip 30.0pt+B|r-s|\;|k_{T,B}^{n}(Z,r-s)-k_{T,0}^{n}(Z,r-s)|$
$\displaystyle\hskip
30.0pt+B|s|\;|k_{T,B}^{n}(Z,r-s)-k_{T,0}^{n}(Z,r-s)|\Big{]}.$ (4.66)

We also have

$\displaystyle|\nabla_{r}k_{T,B}^{n}(Z,r)-\nabla_{r}k_{T,0}^{n}(Z,r)|$
$\displaystyle\leqslant|\nabla g_{B}^{\mathrm{i}\omega_{n}}-\nabla
g_{0}^{\mathrm{i}\omega_{n}}|\bigl{(}Z+\frac{r}{2}\bigr{)}\;|g_{B}^{-\mathrm{i}\omega_{n}}|\bigl{(}Z-\frac{r}{2}\bigr{)}$
$\displaystyle\hskip
30.0pt+|g_{B}^{\mathrm{i}\omega_{n}}-g_{0}^{\mathrm{i}\omega_{n}}|\bigl{(}Z+\frac{r}{2}\bigr{)}\;|\nabla
g_{B}^{-\mathrm{i}\omega_{n}}|\bigl{(}Z-\frac{r}{2}\bigr{)}$
$\displaystyle\hskip 30.0pt+|\nabla
g_{0}^{\mathrm{i}\omega_{n}}|\bigl{(}Z+\frac{r}{2}\bigr{)}\;|g_{B}^{-\mathrm{i}\omega_{n}}-g_{0}^{-\mathrm{i}\omega_{n}}|\bigl{(}Z-\frac{r}{2}\bigr{)}$
$\displaystyle\hskip
30.0pt+|g_{0}^{\mathrm{i}\omega_{n}}|\bigl{(}Z+\frac{r}{2}\bigr{)}|\nabla
g_{B}^{-\mathrm{i}\omega_{n}}-\nabla
g_{0}^{-\mathrm{i}\omega_{n}}|\bigl{(}Z-\frac{r}{2}\bigr{)}.$

Since $g_{B}^{\pm\mathrm{i}\omega_{n}}$ is an even function this implies

$\displaystyle\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}^{3}}\mathrm{d}Z\;|\nabla
k_{T,B}^{n}(Z,r)-\nabla k_{T,0}^{n}(Z,r)|\leqslant G_{T,B}(r).$ (4.67)

Moreover, the estimate

$\displaystyle|r-s|$
$\displaystyle=\bigl{|}\frac{r-s}{2}+Z+\frac{r-s}{2}-Z\bigr{|}\leqslant\bigl{|}Z-\frac{r-s}{2}\bigr{|}+\bigl{|}Z+\frac{r-s}{2}\bigr{|}$
(4.68)

shows that $|r-s|\,F_{T,B}^{0}(r-s)\leqslant F_{T,B}^{1}(r-s)$. We conclude
the estimate

$\displaystyle\smash{\int_{\mathbb{R}^{3}}}\mathrm{d}Z\;|\widetilde{\pi}_{r}k_{T,B}(Z,r,s)-\widetilde{\pi}_{r}\widetilde{k}_{T,B}(Z,r,s)|$
$\displaystyle\leqslant
G_{T,B}(r-s)+B\;F_{T,B}^{1}(r-s)+B\;F_{T,B}^{0}(r-s)\;|s|.$ (4.69)

From (4.69) we deduce (4.64), which proves the claim. ∎

#####  4.3.3.2 The operator $M_{T,B}$.

The operator $M_{T,B}$ is defined by

$\displaystyle
M_{T,B}\alpha(X,r):=\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T}(Z,r-s)\;(\cos(Z\cdot\Pi_{X})\alpha)(X,s),$
(4.70)

where $k_{T}(Z,r):=k_{T,0}(Z,r,0)$ with $k_{T,0}$ in (4.42). The following
proposition allows us to replace $\widetilde{L}_{T,B}$ by $M_{T,B}$ in our
computations.

######  Proposition 4.9.

For any $T_{0}>0$ there is $B_{0}>0$ such that for any $0<B\leqslant B_{0}$,
any $T\geqslant T_{0}$, and whenever $|\cdot|^{k}V\alpha_{*}\in
L^{2}(\mathbb{R}^{3})$ for $k\in\\{0,1\\}$, $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{B})$, and $\Delta\equiv\Delta_{\Psi}$ as in (3.1), we
have

$\displaystyle\|\widetilde{L}_{T,B}\Delta-
M_{T,B}\Delta\|_{H^{1}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}^{2}$
$\displaystyle\leqslant
C\;B^{3}\;\bigl{(}\|V\alpha_{*}\|_{2}^{2}+\|\,|\cdot|V\alpha_{*}\|_{2}^{2}\bigr{)}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{2}.$
(4.71)

If instead $|\cdot|^{2}V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$ then

$\displaystyle|\langle\Delta,\widetilde{L}_{T,B}\Delta-M_{T,B}\Delta\rangle|$
$\displaystyle\leqslant
C\;B^{3}\;\|\,|\cdot|^{2}V\alpha_{*}\|_{2}^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{2}.$
(4.72)

######  Remark 4.10.

The ${H^{1}(Q_{B}\times\mathbb{R}_{\mathrm{s}}^{3})}$-norm bound in (4.71) is
needed for the proof of Proposition 3.2 and the quadratic form bound in (4.72)
is needed for the proof of Theorem 3.5. We highlight that the bound in (4.71)
is insufficient as far as the proof of Theorem 3.5 is concerned. More
precisely, if we apply Cauchy–Schwarz to the left side of (4.72), and use
(4.71) as well as the Lemma 4.1 to estimate $\|\Delta\|_{2}$ we obtain a bound
of the order $B^{2}$ only. This is not good enough because $B^{2}$ is the
order of the Ginzburg–Landau energy.

To obtain the desired quality for the quadratic form bound (4.72), we exploit
the fact that $V\alpha_{*}$ is real-valued, which allows us to replace the
magnetic phase factor $\exp(\frac{\mathrm{i}}{4}\mathbf{B}(r\wedge s))$ in
$\widetilde{k}_{T,B}$ in (4.46) by $\cos(\frac{1}{4}\mathbf{B}(r\wedge s))$.
This improves the error estimate by an additional factor of $B$.

Before we start with the proof of Proposition 4.9, let $a\in\mathbb{N}_{0}$
and define the functions

$\displaystyle
F_{T}^{a}:=\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\sum_{b=0}^{a}\binom{a}{b}\;\bigl{(}|\cdot|^{b}\,|g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*\bigl{(}|\cdot|^{a-b}\,|g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}$
(4.73)

and

$\displaystyle G_{T}$
$\displaystyle:=\smash{\frac{2}{\beta}\sum_{n\in\mathbb{Z}}}\bigl{(}|\cdot|\,|\nabla
g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*|g_{0}^{-\mathrm{i}\omega_{n}}|+|\nabla
g_{0}^{\mathrm{i}\omega_{n}}|*\bigl{(}|\cdot|\,|g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}$
$\displaystyle\hskip
110.0pt+\bigl{(}|\cdot|\,|g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*|\nabla
g_{0}^{-\mathrm{i}\omega_{n}}|+|g_{0}^{\mathrm{i}\omega_{n}}|*\bigl{(}|\cdot|\,|\nabla
g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}.$ (4.74)

For $T\geqslant T_{0}>0$ and $a\in\mathbb{N}_{0}$, by Lemma 4.4, (4.50), and
(4.51), we have

$\displaystyle\|F_{T}^{a}\|_{1}+\|G_{T}\|_{1}$ $\displaystyle\leqslant C_{a}.$
(4.75)

###### Proof of Proposition 4.9.

We start with the proof of (4.71), which is similar to the proof of
Proposition 4.7. We claim that

$\displaystyle\|\widetilde{L}_{T,B}\Delta-M_{T,B}\Delta\|_{2}^{2}$
$\displaystyle\leqslant
4\;B^{2}\;\|\Psi\|_{2}^{2}\;\|F_{T}^{1}*|\cdot|\,|V\alpha_{*}|\,\|_{2}^{2}.$
(4.76)

If this is true Young’s inequality, (2.5), and (4.75) prove the claimed bound
for this term. To see that (4.76) holds, we argue as in (4.55)-(4.57) and find

$\displaystyle\|\widetilde{L}_{T,B}\Delta-M_{T,B}\Delta\|_{2}^{2}$
$\displaystyle\leqslant
4\|\Psi\|_{2}^{2}\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Bigl{|}\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;\bigl{|}k_{T,0}^{n}(Z,r-s)\bigl{[}\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}\cdot(r\wedge
s)}-1\bigr{]}\bigr{|}\;|V\alpha_{*}(s)|\Bigr{|}^{2}.$

Since $|r\wedge s|\leqslant|r-s|\,|s|$, we have
$|\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}\cdot(r\wedge s)}-1|\leqslant
B\,|r-s|\,|s|$ as well as

$\displaystyle|k_{T,0}^{n}(Z,r-s)|\bigl{|}\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}\cdot(r\wedge
s)}-1\bigr{|}$ $\displaystyle\leqslant
B\;|g_{0}^{\mathrm{i}\omega_{n}}|\bigl{(}Z-\frac{r-s}{2}\bigr{)}\;|g_{0}^{-\mathrm{i}\omega_{n}}|\bigl{(}Z+\frac{r-s}{2}\bigr{)}\;|r-s|\;|s|.$

In combination with the estimate for $|r-s|$ in (4.68) and the bound for
$|Z|^{a}$ in (4.59), this proves

$\displaystyle\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}^{3}}\mathrm{d}Z\;|Z|^{a}\,|k_{T,0}^{n}(Z,r-s)|\,\bigl{|}\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}\cdot(r\wedge
s)}-1\bigr{|}$ $\displaystyle\leqslant B\;F_{T}^{a+1}(r-s)\;|s|$ (4.77)

for $a\in\mathbb{N}_{0}$. The case $a=0$ implies (4.76). A computation similar
to the one leading to (4.63) shows

$\displaystyle\|\Pi_{X}(\widetilde{L}_{T,B}\Delta-M_{T,B}\Delta)\|_{2}^{2}$
$\displaystyle\leqslant
C\,B^{4}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{B})}^{2}\;\|(F_{T}^{1}+F_{T}^{2})*|\cdot|\,|V\alpha_{*}|\,\|_{2}^{2}.$

To obtain the result we also used (4.62) and (4.77). We apply Young’s
inequality and use (4.75) to prove the claimed bound for this term. Finally, a
computation similar to the one that leads to (4.65) shows

$\displaystyle\|\widetilde{\pi}_{r}(\widetilde{L}_{T,B}\Delta-
M_{T,B}\Delta)\|_{2}^{2}$ $\displaystyle\leqslant
4\,\|\Psi\|_{2}^{2}\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\bigl{|}\widetilde{\pi}_{r}k_{T,0}^{n}(Z,r-s)\bigl{[}\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}(r\wedge
s)}-1\bigr{]}\bigr{|}\;|V\alpha_{*}(s)|\Bigr{|}^{2}.$

We argue as in the proof of (4.69) to see that

$\displaystyle\smash{\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}^{3}}\mathrm{d}Z}\;\bigl{|}\widetilde{\pi}_{r}k_{T,0}^{n}(Z,r-s)\bigl{[}\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}(r\wedge
s)}-1\bigr{]}\bigr{|}$ $\displaystyle\leqslant
C\,B\;\bigl{(}G_{T}(r-s)\;|s|+F_{T}^{1}(r-s)+F_{T}^{0}(r-s)\;|s|\bigr{)}.$

With the help of Young’s inequality and (4.75), these considerations prove
(4.71).

It remains to prove (4.72). The term we need to estimate reads

$\displaystyle\langle\Delta,\widetilde{L}_{T,B}\Delta-M_{T,B}\Delta\rangle$
$\displaystyle=4\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}r\mathrm{d}s\;\bigl{(}\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}\cdot(r\wedge
s)}-1\bigr{)}V\alpha_{*}(r)V\alpha_{*}(s)\mathop{\times}$
$\displaystyle\times\int_{\mathbb{R}^{3}}\mathrm{d}Z\;\frac{2}{\beta}\sum_{n\in\mathbb{Z}}k_{T,0}^{n}(Z,r-s)\fint_{Q_{B}}\mathrm{d}X\;\overline{\Psi(X)}\cos(Z\cdot\Pi_{X})\Psi(X).$
(4.78)

Except for the factor $\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}(r\wedge
s)}$, the right side is symmetric under the exchange of the coordinates $r$
and $s$. The exponential factor acquires a minus sign in the exponent when
this transformation is applied. When we add the right side of (4.78) and the
same term with the roles of $r$ and $s$ interchanged, we get

$\displaystyle\langle\Delta,\widetilde{L}_{T,B}\Delta-M_{T,B}\Delta\rangle$
$\displaystyle=-8\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}r\mathrm{d}s\;\sin^{2}\bigl{(}\frac{1}{8}\,\mathbf{B}\cdot(r\wedge
s)\bigr{)}\;V\alpha_{*}(r)V\alpha_{*}(s)\mathop{\times}$
$\displaystyle\times\int_{\mathbb{R}^{3}}\mathrm{d}Z\;\frac{2}{\beta}\sum_{n\in\mathbb{Z}}k_{T,0}^{n}(Z,r-s)\fint_{Q_{B}}\mathrm{d}X\;\overline{\Psi(X)}\cos(Z\cdot\Pi_{X})\Psi(X).$
(4.79)