# Inequalities between Dirichlet and Neumann eigenvalues of the magnetic
Laplacian

Vladimir Lotoreichik  Department of Theoretical Physics, Nuclear Physics
Institute, Czech Academy of Sciences, Hlavní 130, 25068 Řež, Czech Republic
<EMAIL_ADDRESS>
###### Abstract.

We consider the magnetic Laplacian with the homogeneous magnetic field in two
and three dimensions. We prove that the $(k+1)$-th magnetic Neumann eigenvalue
of a bounded convex planar domain is not larger than its $k$-th magnetic
Dirichlet eigenvalue. In three dimensions, we restrict our attention to convex
domains, which are invariant under rotation by an angle of $\pi$ around an
axis parallel to the magnetic field. For such domains, we prove that the
$(k+2)$-th magnetic Neumann eigenvalue is not larger than the $k$-th magnetic
Dirichlet eigenvalue provided that this Dirichlet eigenvalue is simple. The
proofs rely on a modification of the strategy due to Levine and Weinberger.

###### Key words and phrases:

magnetic Laplacian, homogeneous magnetic field, inequalities between Dirichlet
and Neumann eigenvalues, min-max principle

###### 1991 Mathematics Subject Classification:

35P15, 81Q10

##  1\. Introduction

The aim of the present paper is to obtain inequalities between Dirichlet and
Neumann eigenvalues of the magnetic Laplacian on a bounded domain in the
spirit of the Levine-Weinberger and Friedlander-Filonov inequalities for the
usual (non-magnetic) Laplacian. The Levine-Weinberger inequality [LW86] states
that on a $d$-dimensional ($d\geq 2$) bounded convex domain the $(k+d)$-th
Neumann eigenvalue of the Laplacian does not exceed its $k$-th Dirichlet
eigenvalue. Without the convexity assumption, it was proved by Friedlander
[F91] that the $(k+1)$-th Neumann eigenvalue of the Laplacian does not exceed
its $k$-th Dirichlet eigenvalue. This inequality was later shown by Filonov
[F05] to be strict and under weaker regularity assumptions on the domain. A
discussion of these results can be found in the recent monograph [LMP].

Our analysis is motivated by the question to what extent these eigenvalue
inequalities survive the presence of the homogeneous magnetic field. We focus
on physically motivated two- and three-dimensional settings. Inequalities
between Dirichlet and Neumann eigenvalues of the magnetic Laplacian are
already considered by Frank and Laptev in [FL10]. The present paper is
partially inspired by their considerations and contains improvements of their
results under additional assumptions.

First, it is shown in [FL10] in two dimensions that, below any Landau level,
the number of magnetic Neumann eigenvalues exceeds at least by one the number
of magnetic Dirichlet eigenvalues. This result suggests that an inequality
between individual eigenvalues with an index shift can hold. In the present
paper, we prove that on any bounded planar convex domain the $(k+1)$-th
Neumann eigenvalue of the magnetic Laplacian does not exceed its $k$-th
Dirichlet eigenvalue. It is also proved in [FL10] that in three dimensions the
$(k+1)$-th magnetic Neumann eigenvalue is smaller than the $k$-th magnetic
Dirichlet eigenvalue. A natural question is whether one can improve the index
shift in this inequality. We restrict our attention to convex domains, which
coincide with itself upon rotation by an angle of $\pi$ around an axis
parallel to the vector of the magnetic field. For such symmetric domains, we
prove that the $(k+2)$-th Neumann eigenvalue of the magnetic Laplacian does
not exceed its $k$-th Dirichlet eigenvalue provided that the latter Dirichlet
eigenvalue is simple.

Recently, Rohleder [R23] established, using a new variational principle, that
in two dimensions the Levine-Weinberger inequality is valid for all
sufficiently smooth bounded simply-connected domains (without the convexity
assumption). The validity of the Levine-Weinberger inequality for general
bounded domains in all space dimensions is a prominent open problem [LMP,
Conjecture 3.2.41]. In this perspective, one may ask whether the inequalities
obtained in the present paper are also valid without the convexity assumption.

###  1.1. Main results

Let us describe our results in more detail. To this aim we need first to
define the operators we consider. Let $\Omega\subset{\mathbb{R}}^{d}$ with
$d\in\\{2,3\\}$ be a bounded Lipschitz domain. Note that any bounded convex
domain has Lipschitz boundary (see [Gr, Corollary 1.2.2.3]). We choose the
vector potential ${\bf A}\colon\Omega\rightarrow{\mathbb{R}}^{d}$ of the
homogeneous magnetic field of intensity $b>0$ in the Landau gauge

$\displaystyle{\bf A}(x)=\begin{cases}(0,bx_{1})^{\top},&d=2,\\\
(0,bx_{1},0)^{\top},&d=3,\end{cases}$

where we use the convention $x=(x_{1},x_{2})$ for $d=2$ and
$x=(x_{1},x_{2},x_{3})$ for $d=3$. In two dimensions, the respective magnetic
field is given by $\mathrm{curl}\,{\bf
A}=\partial_{1}A_{2}-\partial_{2}A_{1}=b$ and it points in the direction
orthogonal to the plane. The particular choice of the gauge of the homogeneous
magnetic field in two dimensions is not restrictive for further
considerations, because our results concern convex domains, for which there is
gauge invariance of the spectra for the magnetic Laplacians; cf. [FH, Appendix
D]. In three dimensions, we get after a simple computation that the magnetic
field is given by $\mathrm{curl}\,{\bf A}=(0,0,b)^{\top}$ and thus points
along the $x_{3}$-axis. The direction of the vector of the magnetic field in
three dimensions is essential in our analysis.

Let $H^{1}(\Omega)$ be the standard $L^{2}$-based first-order Sobolev space
and let $H^{1}_{0}(\Omega)$ be the closure of $C^{\infty}_{0}(\Omega)$ in
$H^{1}(\Omega)$. Consider the following closed, densely defined, non-negative
quadratic forms

$\displaystyle\mathfrak{h}_{b,{\rm D}}^{\Omega}[u]$
$\displaystyle:=\|(\nabla-{\mathsf{i}}{\bf
A})u\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{d})},\qquad\mathrm{dom}\,\mathfrak{h}_{b,{\rm
D}}^{\Omega}:=H^{1}_{0}(\Omega),$ (1.1) $\displaystyle\mathfrak{h}_{b,{\rm
N}}^{\Omega}[u]$ $\displaystyle:=\|(\nabla-{\mathsf{i}}{\bf
A})u\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{d})},\qquad\mathrm{dom}\,\mathfrak{h}_{b,{\rm
N}}^{\Omega}:=H^{1}(\Omega),$

in the Hilbert space $L^{2}(\Omega)$; cf. [FH, Section 1.2]. Let
${\mathsf{H}}_{b,{\rm D}}^{\Omega}$ and ${\mathsf{H}}_{b,{\rm N}}^{\Omega}$ be
the self-adjoint operators in $L^{2}(\Omega)$ associated with the forms
$\mathfrak{h}_{b,{\rm D}}^{\Omega}$ and $\mathfrak{h}_{b,{\rm N}}^{\Omega}$,
respectively. Compactness of the embedding of $H^{1}(\Omega)$ into
$L^{2}(\Omega)$ yields that the spectra of both ${\mathsf{H}}_{b,\rm
D}^{\Omega}$ and ${\mathsf{H}}_{b,\rm N}^{\Omega}$ are purely discrete. Denote
by $\\{\lambda_{k}^{b}(\Omega)\\}_{k\in{\mathbb{N}}}$ and
$\\{\mu_{k}^{b}(\Omega)\\}_{k\in{\mathbb{N}}}$ the eigenvalues of,
respectively, ${\mathsf{H}}_{b,\rm D}^{\Omega}$ and ${\mathsf{H}}_{b,\rm
N}^{\Omega}$, enumerated in the non-decreasing order and repeated with
multiplicities taken into account.

It was proved in two dimensions by Frank and Laptev in [FL10, Theorem 3.1 (1)]
that

$\\#\big{\\{}k\in{\mathbb{N}}\colon\lambda_{k}^{b}(\Omega)\leq
b(2q+1)\big{\\}}+1\leq\\#\big{\\{}k\in{\mathbb{N}}\colon\mu_{k}^{b}(\Omega)<b(2q+1)\big{\\}},\qquad\text{for
all}\,\,q\in{\mathbb{N}}_{0}.$

Namely, for any $q\in{\mathbb{N}}_{0}$, if the operator ${\mathsf{H}}_{b,\rm
D}^{\Omega}$ has $j\in{\mathbb{N}}_{0}$ eigenvalues less or equal to
$b(2q+1)$, then the operator ${\mathsf{H}}_{b,\rm N}^{\Omega}$ has at least
$j+1$ eigenvalues less than $b(2q+1)$. As we have already mentioned, this
result suggests that an inequality with an index shift between the individual
eigenvalues of ${\mathsf{H}}_{b,\rm D}^{\Omega}$ and ${\mathsf{H}}_{b,\rm
N}^{\Omega}$ may hold. This inequality under an additional convexity
assumption is the first main result of the present paper.

######  Theorem 1.1.

Let $\Omega\subset{\mathbb{R}}^{2}$ be a bounded, convex domain. Then, for any
$b>0$,

$\mu_{k+1}^{b}(\Omega)\leq\lambda_{k}^{b}(\Omega),\qquad\text{for
all}\,\,k\in{\mathbb{N}}.$

In the same paper, Frank and Laptev established in [FL10, Theorem 3.1 (2)]
that, in three dimensions the inequality
$\mu_{k+1}^{b}(\Omega)<\lambda_{k}^{b}(\Omega)$ holds for any $b>0$ and all
$k\in{\mathbb{N}}$. In our second main result, we prove a (non-strict)
eigenvalue inequality with a higher index shift under the convexity assumption
in the class of domains invariant under rotation by angle of $\pi$ around the
$x_{3}$-axis (the axis along which the magnetic field is oriented). In order
to formulate this result we introduce the mapping

${\mathsf{J}}\colon{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}^{3},\qquad{\mathsf{J}}x=(-x_{1},-x_{2},x_{3})^{\top}.$
(1.2)

######  Theorem 1.2.

Let $\Omega\subset{\mathbb{R}}^{3}$ be a bounded, convex domain such that
${\mathsf{J}}(\Omega)=\Omega$. Let $b>0$ be fixed. Let $k\in{\mathbb{N}}$ be
such that $\lambda_{k}^{b}(\Omega)$ is a simple eigenvalue of
${\mathsf{H}}_{b,\rm D}^{\Omega}$. Then it holds that

$\mu_{k+2}^{b}(\Omega)\leq\lambda_{k}^{b}(\Omega).$

Theorem 1.2 applies to all domains rotationally invariant around the
$x_{3}$-axis including the ball or properly oriented ellipsoid with two semi-
axes of equal length. It also applies to some symmetric polyhedral domains.
For example, it can be applied to the octahedron oriented so that one of its
diagonals lies on the $x_{3}$-axis.

The Levine-Weinberger inequality is sharp in the sense that, in general, the
index $k+d$ can not be replaced by $k+d+1$. Indeed, as is well-known, for the
disk the fourth Neumann eigenvalue of the Laplacian is larger than its first
Dirichlet eigenvalue; cf. [R23, Example 4.6]. We provide a numerical evidence
in Section 5 below, that in the magnetic case the third Neumann eigenvalue of
the disk is not larger than its first Dirichlet eigenvalue. Thus, an example
showing sharpness of Theorem 1.1 (if it exists) should either be not the disk
or should be based on comparison of higher eigenvalues for the disk. In view
of this observation, it remains an open question whether the index $k+1$ in
Theorem 1.1 can be replaced by $k+2$. The assumption of simplicity of the
$k$-th Dirichlet eigenvalue in Theorem 1.2 is technical and we expect that it
can be removed. However, the symmetry of the domain is essential for the
argument and it is not a priori clear whether the inequality in Theorem 1.2
holds for general convex domains.

Both theorems are first proved for convex polyhedral domains. The results for
general convex domains are deduced via approximation. The eigenvalue
inequalities for polyhedral domains are proved via the min-max principle,
where as the trial subspace we select the span of the first $k$ magnetic
Dirichlet eigenfunctions and of all but one first-order partial derivatives of
the $k$-th Dirichlet eigenfunction. In the original proof [LW86] of the
Levine-Weinberger inequality for the Laplacian it was possible to use all
first-order partial derivatives of the $k$-th Dirichlet eigenfunction in the
construction of the trial subspace. In the magnetic case, we need to exclude
the partial derivative with respect to the variable $x_{1}$, on which the
vector potential ${\bf A}$ depends. Similar idea was used in [LR17] for the
Laplacian with mixed boundary conditions and in [R21] for the Schrödinger
operator on a bounded domain with (electric) potential dependent on the number
of directions strictly less than the space dimension. The main novel
difficulties of the magnetic problem is that the perturbation is non-additive
and that the eigenfunctions are genuinely complex-valued. The fact that the
eigenfunctions of the magnetic Dirichlet Laplacian are complex-valued leads to
additional difficulties. It is the main reason for the symmetry assumption
imposed in the formulation of Theorem 1.2.

###  1.2. Overview of the literature

The results of Levine, Weinberger, Friedlander, Filonov, and Rohleder
mentioned in the beginning of the introduction were preceded by eigenvalue
inequalities in more special settings or under additional assumptions. Pólya
proved in [P52] that the second Neumann eigenvalue of the Laplacian on a
bounded smooth planar domain is smaller than its first Dirichlet eigenvalue.
For convex, two-dimensional domains with $C^{2}$-boundary Payne proved in
[P55] that the $(k+2)$-th Neumann eigenvalue of the Laplacian is smaller than
its $k$-th Dirichlet eigenvalue, the inequality which was later generalized by
Levine and Weinberger to higher dimensions. The Friedlander-Filonov inequality
was conjectured by Payne in [P91] and preceded by the paper of Aviles [A86],
in which this inequality was proved under an additional assumption on the mean
curvature of the boundary.

Similar type inequalities are also obtained for other types of differential
operators. Besides the already mentioned papers: [FL10] on the sub-Laplacian
on the Heisenberg group and the magnetic Laplacian, [LR17] on the Laplacian
with mixed boundary conditions, [R21] on the Schrödinger operator, we also
refer in this context to recent results [DT24] on the Stokes operator and to
[L24, Pr19] on the biharmonic and more general polyharmonic operators. The
literature on this subject is quite extensive and it is difficult to mention
all related results.

Recently, geometric bounds [CLPS23, EKP16, KL23, LS15] and isoperimetric
inequalities [CLPS24, FFG+23] for the eigenvalues of the magnetic Neumann
Laplacian attracted considerable attention. The eigenvalues inequalities
obtained in the present paper can also be viewed as estimates of magnetic
Neumann eigenvalues but in terms of magnetic Dirichlet eigenvalues.

### Structure of the paper

In Section 2, we collect properties of the magnetic Laplacian on a bounded
domain and recall a formula for integration by parts on polyhedral domains.
Theorem 1.1 is proved in Section 3. In turn, a proof of Theorem 1.2 is
provided in Section 4. Section 5 contains a numerical analysis of the case of
the disk, which is motivated by the intention to check sharpness of the index
shift in Theorem 1.1. The paper is complemented by Appendix A, in which we
show lower semicontinuity of magnetic Neumann eigenvalues on varying convex
domains satisfying an inclusion property. This auxiliary statement is used in
extending our results from polyhedral domains to general convex domains.

##  2\. Preliminaries

###  2.1. Magnetic Laplacian on a bounded domain

Let $\Omega\subset{\mathbb{R}}^{d}$, $d\in\\{2,3\\}$, be a bounded Lipschitz
domain. We denote by $\partial\Omega$ the boundary of $\Omega$ and by $\nu$
the outer unit normal vector to $\partial\Omega$. This normal vector is well
defined almost everywhere on $\partial\Omega$ thanks to Rademacher’s theorem.
By the magnetic Laplacian we understand the following second-order
differential expression

$-(\nabla-{\mathsf{i}}{\bf
A})^{2}=-\Delta+2{\mathsf{i}}bx_{1}\partial_{2}+b^{2}x_{1}^{2}.$

It is a special case of the most general second-order differential expression
considered in [McL, Chapter 4] (see also [BR12, BR20]).

The magnetic Dirichlet Laplacian ${\mathsf{H}}_{b,\rm D}^{\Omega}$ associated
with the quadratic form $\mathfrak{h}_{b,\rm D}^{\Omega}$ in (1.1) can be
characterised as (see [FH, Section 1.2])

${\mathsf{H}}_{b,\rm D}^{\Omega}u=-(\nabla-{\mathsf{i}}{\bf
A})^{2}u,\qquad\mathrm{dom}\,{\mathsf{H}}_{b,\rm D}^{\Omega}=\big{\\{}u\in
H^{1}_{0}(\Omega)\colon\Delta u\in L^{2}(\Omega)\big{\\}}.$ (2.1)

It follows directly from [Gr, Theorem 3.2.1.2] that

$\Omega\,\,\text{is
convex}\qquad\Longrightarrow\qquad\mathrm{dom}\,{\mathsf{H}}_{b,{\rm
D}}^{\Omega}=H^{2}(\Omega)\cap H^{1}_{0}(\Omega).$ (2.2)

Recall that for any $u\in H^{1}(\Omega)$ its trace on the boundary
$u|_{\partial\Omega}$ is well defined as a function in
$H^{1/2}(\partial\Omega)$ and for $u\in H^{1}(\Omega)$ with $\Delta u\in
L^{2}(\Omega)$ its trace of the normal derivative
$\partial_{\nu}u|_{\partial\Omega}$ is well defined as a distribution in
$H^{-1/2}(\partial\Omega)$; see [BGM22] for details.

The magnetic Neumann Laplacian ${\mathsf{H}}_{b,\rm N}^{\Omega}$ associated
with the quadratic form $\mathfrak{h}_{b,\rm N}^{\Omega}$ in (1.1) can be
characterised as (see [FH, Section 1.2] and [BR20, Section 5])

$\displaystyle{\mathsf{H}}_{b,\rm N}^{\Omega}u$
$\displaystyle=-(\nabla-{\mathsf{i}}{\bf A})^{2}u,$ (2.3)
$\displaystyle\mathrm{dom}\,{\mathsf{H}}_{b,\rm N}^{\Omega}$
$\displaystyle=\big{\\{}u\in H^{1}(\Omega)\colon\Delta u\in
L^{2}(\Omega),\,\nu\cdot(\nabla-{\mathsf{i}}{\bf
A})u=0~{}~{}\text{on}~{}\partial\Omega\big{\\}},$

where $\nu\cdot(\nabla-{\mathsf{i}}{\bf A})u$ on the boundary $\partial\Omega$
should be identified with
$\partial_{\nu}u|_{\partial\Omega}-{\mathsf{i}}\nu\cdot{\bf
A}|_{\partial\Omega}u|_{\partial\Omega}$ being, by the above discussion, well
defined as a distribution in $H^{-1/2}(\partial\Omega)$. Recall also that for
any $u\in H^{1}(\Omega)$ and any $v\in H^{1}(\Omega)$ with $\Delta v\in
L^{2}(\Omega)$ we have by [BGM22, Eq. (5.112)] the following formula for
integration by parts

$((\nabla-{\mathsf{i}}{\bf A})u,(\nabla-{\mathsf{i}}{\bf
A})v)_{L^{2}(\Omega;{\mathbb{C}}^{d})}\\!=\\!(u,-(\nabla-{\mathsf{i}}{\bf
A})^{2}v)_{L^{2}(\Omega)}\\!+\\!\langle
u|_{\partial\Omega},\partial_{\nu}v|_{\partial\Omega}-{\mathsf{i}}\nu\cdot{\bf
A}|_{\partial\Omega}v|_{\partial\Omega}\rangle_{\frac{1}{2},-\frac{1}{2}},$
(2.4)

where $\langle\cdot,\cdot\rangle_{1/2,-1/2}$ stands for the duality product
between $H^{1/2}(\partial\Omega)$ and $H^{-1/2}(\partial\Omega)$, which is
compatible with the inner product in $L^{2}(\partial\Omega)$ (linear in the
first entry).

The eigenvalues of the magnetic Laplacians ${\mathsf{H}}_{b,\rm D}^{\Omega}$
and ${\mathsf{H}}_{b,\rm N}^{\Omega}$ can be characterised by the min-max
principle (see, e.g. , [FLW23, Theorem 1.28])

$\lambda_{k}^{b}(\Omega)=\min_{\begin{smallmatrix}{\mathcal{L}}\subset
H^{1}_{0}(\Omega)\\\
\dim{\mathcal{L}}=k\end{smallmatrix}}\max_{u\in{\mathcal{L}}\setminus\\{0\\}}\frac{\|(\nabla-{\mathsf{i}}{\bf
A})u\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{d})}}{\|u\|^{2}_{L^{2}(\Omega)}},\qquad
k\in{\mathbb{N}},$ (2.5a)
$\mu_{k}^{b}(\Omega)=\min_{\begin{smallmatrix}{\mathcal{L}}\subset
H^{1}(\Omega)\\\
\dim{\mathcal{L}}=k\end{smallmatrix}}\max_{u\in{\mathcal{L}}\setminus\\{0\\}}\frac{\|(\nabla-{\mathsf{i}}{\bf
A})u\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{d})}}{\|u\|^{2}_{L^{2}(\Omega)}},\qquad
k\in{\mathbb{N}},$ (2.5b)

where the minima are taken with respect to $k$-dimensional linear subspaces
${\mathcal{L}}$ of respective Sobolev spaces and the maxima are taken with
respect to all non-zero functions in the subspace ${\mathcal{L}}$. The minima
are attained on the spans of $k$ orthonormal eigenfunctions of respective
operators corresponding to their first $k$ eigenvalues while the maximum is
subsequently attained on the eigenfunctions corresponding to the $k$-th
eigenvalue. As a direct consequence of the above variational characterisation
we get that for any $k\in{\mathbb{N}}$ there exists a subspace
${\mathcal{L}}\subset H^{1}_{0}(\Omega)$ with $\dim{\mathcal{L}}=k$ such that

$\mathfrak{h}_{b,{\rm
D}}^{\Omega}[u]\leq\lambda_{k}^{b}(\Omega)\|u\|^{2}_{L^{2}(\Omega)},\qquad\text{for
all}\,\,u\in{\mathcal{L}}.$ (2.6)

The next lemma is a special case of [BR12, Proposition 2.5].

######  Lemma 2.1.

Let $\Omega\subset{\mathbb{R}}^{d}$, $d\in\\{2,3\\}$, be a bounded connected
Lipschitz domain. Let an open subset $\omega\subset\partial\Omega$ be non-
empty. Let $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ be such that
$-(\nabla-{\mathsf{i}}{\bf A})^{2}u=\lambda u$ with some
$\lambda\in{\mathbb{R}}$ and that $\partial_{\nu}u|_{\omega}=0$. Then there
holds $u\equiv 0$.

######  Remark 2.2.

Under the assumption $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ the Neumann
trace $\partial_{\nu}u|_{\partial\Omega}$, is more regular and belongs to
$L^{2}(\partial\Omega)$ (see [BGM22, Eq. (1.18)]). Thus, its restriction
$\partial_{\nu}u|_{\omega}$ to $\omega\subset\partial\Omega$ is naturally
defined.

###  2.2. Integration by parts in convex polygons and polyhedra

In this subsection we recall a formula for integration by parts on polyhedral
domains, which is derived in [LR17, Appendix A] in all space dimensions. Its
two-dimensional version can be found in the classical monograph by Grisvard
(see [Gr, Lemma 4.3.1.2]). Before we proceed, we recall the definition of
polyhedral domains in ${\mathbb{R}}^{d}$ with $d=2,3$. The definition can be
extended to dimensions $d>3$, but it is not needed for our purposes.

######  Definition 2.3.

Let $\Omega\subset{\mathbb{R}}^{d}$, $d\in\\{2,3\\}$, be a bounded, connected
Lipschitz domain.

  * (i)

For $d=2$, $\Omega$ is a polyhedral (or polygonal) domain if its boundary
$\partial\Omega$ is the union of finitely many line segments.

  * (ii)

For $d=3$, $\Omega$ is a polyhedral domain if for each plane
$H\subset{\mathbb{R}}^{3}$ the intersection $H\cap\Omega$ is either a polygon
in ${\mathbb{R}}^{2}$ (where we identify $H$ with ${\mathbb{R}}^{2}$) or
empty.

We remark that any polyhedral domain is Lipschitz. Now we are ready to state a
formula for integration by parts on convex polyhedral domains. In the below
proposition and in the following, we will occasionally use the abbreviation
for the partial derivatives $\partial_{ij}u=\frac{\partial^{2}u}{\partial
x_{i}\partial x_{j}}$ for $1\leq i,j\leq d$. We will also use the abbreviation
$\partial_{i}^{2}u=\frac{\partial^{2}u}{\partial x_{i}^{2}}$ for $1\leq i\leq
d$.

######  Proposition 2.4 ([LR17, Lemma A.1]).

Let $\Omega\subset{\mathbb{R}}^{d}$, $d\in\\{2,3\\}$, be a bounded convex
polyhedral domain and let $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ be real-
valued. Then

$\int_{\Omega}(\partial_{km}u)(\partial_{kj}u){\,\mathrm{d}}x=\int_{\Omega}(\partial_{mj}u)(\partial_{kk}u){\,\mathrm{d}}x$

for all $j,k,m\in\\{1,2,3\\}$ if $d=3$ or for all $j,k,m\in\\{1,2\\}$ if
$d=2$.

A direct consequence of the above proposition will be useful in the following.

######  Corollary 2.5.

Let $\Omega\subset{\mathbb{R}}^{d}$, $d\in\\{2,3\\}$, be a bounded convex
polyhedral domain and let $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$. Then

${\rm
Re}\,\left\\{\int_{\Omega}(\partial_{km}u)(\overline{\partial_{kj}u}){\,\mathrm{d}}x\right\\}={\rm
Re}\,\left\\{\int_{\Omega}(\partial_{mj}u)(\overline{\partial_{kk}u}){\,\mathrm{d}}x\right\\}$

for all $j,k,m\in\\{1,2,3\\}$ if $d=3$ or for all $j,k,m\in\\{1,2\\}$ if
$d=2$.

###### Proof.

Clearly, we have ${\rm Re}\,u,{\rm Im}\,u\in H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$. By Proposition 2.4 we get

$\displaystyle\int_{\Omega}\partial_{km}({\rm Re}\,u)\partial_{kj}({\rm
Re}\,u){\,\mathrm{d}}x$ $\displaystyle=\int_{\Omega}\partial_{mj}({\rm
Re}\,u)\partial_{kk}({\rm Re}\,u){\,\mathrm{d}}x$
$\displaystyle\int_{\Omega}\partial_{km}({\rm Im}\,u)\partial_{kj}({\rm
Im}\,u){\,\mathrm{d}}x$ $\displaystyle=\int_{\Omega}\partial_{mj}({\rm
Im}\,u)\partial_{kk}({\rm Im}\,u){\,\mathrm{d}}x,$

from which the desired identity immediately follows. ∎

##  3\. Proof of Theorem 1.1

We will prove the theorem first for convex polygons. The eigenvalue inequality
for general convex domains in ${\mathbb{R}}^{2}$ will be then derived from the
eigenvalue inequality for convex polygons via approximation. Assume until the
last step of the proof that $\Omega\subset{\mathbb{R}}^{2}$ is a convex
polygon.

Step 1: construction of a trial subspace of $H^{1}(\Omega)$. For the sake of
brevity we use the notation $\lambda=\lambda_{k}^{b}(\Omega)$. By the
consequence (2.6) of the min-max principle for the magnetic Laplacian with
Dirichlet boundary conditions (2.5a) there exists a subspace
${\mathcal{L}}\subset H^{1}_{0}(\Omega)$ with $\dim{\mathcal{L}}=k$ such that

$\mathfrak{h}_{b,{\rm
D}}^{\Omega}[u]\leq\lambda\|u\|^{2}_{L^{2}(\Omega)},\qquad\text{for
all}\,\,u\in{\mathcal{L}}.$ (3.1)

Let $v\in H^{1}_{0}(\Omega)$ be an eigenfunction of ${\mathsf{H}}_{b,\rm
D}^{\Omega}$ corresponding to the eigenvalue $\lambda$. By (2.2) we infer that
$v\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ and, in particular, we get as a
consequence that $\partial_{2}v\in H^{1}(\Omega)$. Let us define the subspace
${\mathcal{M}}$ of $H^{1}(\Omega)$ by

${\mathcal{M}}:={\mathcal{L}}+{\rm span}\,\\{\partial_{2}v\\}.$ (3.2)

Our next aim is to verify that $\dim{\mathcal{M}}=k+1$. To this aim it
suffices to show that $\partial_{2}v\notin H^{1}_{0}(\Omega)$. We will prove
this statement by contradiction. Suppose for the moment that $\partial_{2}v\in
H^{1}_{0}(\Omega)$. Let us choose the side $\Gamma\subset\partial\Omega$ of
the polygon $\Omega$ such that the (outer) unit normal vector
$\nu_{\Gamma}=(\nu_{\Gamma,1},\nu_{\Gamma,2})^{\top}$ to this side is not
parallel to the $x_{1}$-axis. It is clearly possible to do, because in any
polygon there are sides not parallel to each other. Let us also denote by
$\tau_{\Gamma}=(-\nu_{\Gamma,2},\nu_{\Gamma,1})^{\top}$ a unit tangential
vector to $\Gamma$. It follows from $v|_{\partial\Omega}=0$ that the
tangential derivative of $v$ on $\Gamma$ vanishes

$\tau_{\Gamma}\cdot(\nabla
v)|_{\Gamma}=-\nu_{\Gamma,2}\partial_{1}v|_{\Gamma}+\nu_{\Gamma,1}\partial_{2}v|_{\Gamma}=-\nu_{\Gamma,2}\partial_{1}v|_{\Gamma}=0.$

As a consequence we get since $\nu_{\Gamma,2}\neq 0$ that
$\partial_{1}v|_{\Gamma}=0$. Altogether, the function $v\in H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$ satisfies the differential equation
$-(\nabla-{\mathsf{i}}{\bf A})^{2}v=\lambda v$ in $\Omega$ and its Neumann
trace on $\Gamma$ vanishes
$\partial_{\nu}v|_{\Gamma}=\nu_{\Gamma,1}\partial_{1}v|_{\Gamma}+\nu_{\Gamma,2}\partial_{2}v|_{\Gamma}=0$.
Hence, we get by Lemma 2.1 that $v$ is identically zero in $\Omega$ leading to
a contradiction.

Step 2: application of the min-max principle. Any $w\in{\mathcal{M}}\subset
H^{1}(\Omega)$ can be represented as

$w=u+c\partial_{2}v$ (3.3)

with some $c\in{\mathbb{C}}$ and $u\in{\mathcal{L}}$. Substituting $w$ into
the quadratic form for the magnetic Neumann Laplacian on the convex polygon
$\Omega$ we get

$\displaystyle\mathfrak{h}_{b,\rm N}^{\Omega}[w]$
$\displaystyle=\|(\nabla-{\mathsf{i}}{\bf
A})u\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{2})}$ (3.4) $\displaystyle\qquad+2{\rm
Re}\,\left\\{\left((\nabla-{\mathsf{i}}{\bf A})u,(\nabla-{\mathsf{i}}{\bf
A})c\partial_{2}v\right)_{L^{2}(\Omega;{\mathbb{C}}^{2})}\right\\}+\|(\nabla-{\mathsf{i}}{\bf
A})c\partial_{2}v\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{2})}.$

We will analyse the three terms appearing on the right-hand side in the above
equation separately. We derive from (3.1)

$\|(\nabla-{\mathsf{i}}{\bf
A})u\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{2})}\leq\lambda\|u\|^{2}_{L^{2}(\Omega)}.$
(3.5)

Note that

$-(\nabla-{\mathsf{i}}{\bf
A})^{2}\partial_{2}v=-\Delta\partial_{2}v+2{\mathsf{i}}bx_{1}\partial_{2}^{2}v+b^{2}x_{1}^{2}\partial_{2}v\\\
=-\partial_{2}(\nabla-{\mathsf{i}}{\bf A})^{2}v=\lambda\partial_{2}v\in
L^{2}(\Omega),$

where we first evaluated $\Delta\partial_{2}v$ in the distributional sense and
used that $v$ is an eigenfunction of ${\mathsf{H}}_{b,\rm D}^{\Omega}$
corresponding to the eigenvalue $\lambda$. In particular, we also get as a
consequence

$\Delta\partial_{2}v=2{\mathsf{i}}bx_{1}\partial_{2}^{2}v+b^{2}x_{1}^{2}\partial_{2}v-\lambda\partial_{2}v\in
L^{2}(\Omega).$

By (2.4) we can perform the integration by parts

$\left((\nabla-{\mathsf{i}}{\bf A})u,(\nabla-{\mathsf{i}}{\bf
A})c\partial_{2}v\right)_{L^{2}(\Omega;{\mathbb{C}}^{2})}=-\left(u,(\nabla-{\mathsf{i}}{\bf
A})^{2}c\partial_{2}v\right)_{L^{2}(\Omega)}=\lambda(u,c\partial_{2}v)_{L^{2}(\Omega)},$
(3.6)

where the boundary terms vanished since $u\in H^{1}_{0}(\Omega)$.

Next, we decompose $v=v_{1}+{\mathsf{i}}v_{2}$ with real-valued
$v_{1},v_{2}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$. Then we get

$\displaystyle\|(\nabla-{\mathsf{i}}{\bf
A})\partial_{2}v\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{2})}$
$\displaystyle=\|\nabla\partial_{2}v_{1}+{\bf
A}\partial_{2}v_{2}\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{2})}+\|\nabla\partial_{2}v_{2}-{\bf
A}\partial_{2}v_{1}\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{2})}$ (3.7)
$\displaystyle=\int_{\Omega}\left(|\partial_{2}^{2}v_{1}+bx_{1}\partial_{2}v_{2}|^{2}+|\partial_{1}\partial_{2}v_{1}|^{2}\right){\,\mathrm{d}}x$
$\displaystyle\qquad+\int_{\Omega}\left(|\partial_{2}^{2}v_{2}-bx_{1}\partial_{2}v_{1}|^{2}+|\partial_{1}\partial_{2}v_{2}|^{2}\right){\,\mathrm{d}}x.$

Since $\Omega$ is a convex polygon and $v_{1},v_{2}\in H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$ are real-valued, we can apply the identity in Proposition
2.4 to get

$\int_{\Omega}|\partial_{1}\partial_{2}v_{j}|^{2}{\,\mathrm{d}}x=\int_{\Omega}\partial_{1}^{2}v_{j}\partial_{2}^{2}v_{j}{\,\mathrm{d}}x,\qquad
j=1,2.$ (3.8)

Recall that $v$ is an eigenfunction of ${\mathsf{H}}_{b,\rm D}^{\Omega}$
corresponding to the eigenvalue $\lambda$. Thus, $v$ satisfies the partial
differential equation

$-\Delta v+2{\mathsf{i}}bx_{1}\partial_{2}v+b^{2}x_{1}^{2}v=\lambda v.$ (3.9)

Using the above decomposition $v=v_{1}+{\mathsf{i}}v_{2}$ with real-valued
$v_{1},v_{2}$ we derive from (3.9)

$\begin{cases}-\Delta
v_{1}-2bx_{1}\partial_{2}v_{2}+b^{2}x_{1}^{2}v_{1}=\lambda v_{1},\\\ -\Delta
v_{2}+2bx_{1}\partial_{2}v_{1}+b^{2}x_{1}^{2}v_{2}=\lambda v_{2}.\end{cases}$
(3.10)

From the above two equations we can easily express $\partial_{1}^{2}v_{1}$ and
$\partial_{1}^{2}v_{2}$ as follows

$\displaystyle\partial_{1}^{2}v_{1}$
$\displaystyle=-\partial_{2}^{2}v_{1}-2bx_{1}\partial_{2}v_{2}+b^{2}x_{1}^{2}v_{1}-\lambda
v_{1},$ (3.11) $\displaystyle\partial_{1}^{2}v_{2}$
$\displaystyle=-\partial_{2}^{2}v_{2}+2bx_{1}\partial_{2}v_{1}+b^{2}x_{1}^{2}v_{2}-\lambda
v_{2}.$

Combining the identity (3.8) for $j=1$ with the first equation in (3.11) we
obtain

$\displaystyle\int_{\Omega}|\partial_{1}\partial_{2}v_{1}|^{2}{\,\mathrm{d}}x$
$\displaystyle=\int_{\Omega}(-|\partial_{2}^{2}v_{1}|^{2}-2bx_{1}\partial_{2}v_{2}\partial_{2}^{2}v_{1}+b^{2}x_{1}^{2}v_{1}\partial_{2}^{2}v_{1}-\lambda
v_{1}\partial_{2}^{2}v_{1}){\,\mathrm{d}}x$ (3.12)
$\displaystyle=\int_{\Omega}(-|\partial_{2}^{2}v_{1}|^{2}-2bx_{1}\partial_{2}v_{2}\partial_{2}^{2}v_{1}-b^{2}x_{1}^{2}|\partial_{2}v_{1}|^{2}+\lambda|\partial_{2}v_{1}|^{2}){\,\mathrm{d}}x,$

where we performed twice the integration by parts in the second step.
Analogously we find from the identity (3.8) for $j=2$ combined with the second
equation in (3.11)

$\displaystyle\int_{\Omega}|\partial_{1}\partial_{2}v_{2}|^{2}{\,\mathrm{d}}x$
$\displaystyle=\int_{\Omega}\big{(}-|\partial_{2}^{2}v_{2}|^{2}+2bx_{1}\partial_{2}v_{1}\partial_{2}^{2}v_{2}+bx_{1}^{2}v_{2}\partial_{2}^{2}v_{2}-\lambda
v_{2}\partial_{2}^{2}v_{2}\big{)}{\,\mathrm{d}}x$ (3.13)
$\displaystyle=\int_{\Omega}\big{(}-|\partial_{2}^{2}v_{2}|^{2}+2bx_{1}\partial_{2}v_{1}\partial_{2}^{2}v_{2}-b^{2}x_{1}^{2}|\partial_{2}v_{2}|^{2}+\lambda|\partial_{2}v_{2}|^{2}\big{)}{\,\mathrm{d}}x.$

It follows from (3.12) and (3.13) that

$\int_{\Omega}\big{(}|\partial_{1}\partial_{2}v_{1}|^{2}+|\partial_{1}\partial_{2}v_{2}|^{2}\big{)}{\,\mathrm{d}}x=\lambda\|\partial_{2}v\|^{2}_{L^{2}(\Omega)}-\int_{\Omega}|\partial_{2}^{2}v_{1}+bx_{1}\partial_{2}v_{2}|^{2}{\,\mathrm{d}}x-\int_{\Omega}|\partial_{2}^{2}v_{2}-bx_{1}\partial_{2}v_{1}|^{2}{\,\mathrm{d}}x.$

Substituting the last identity into (3.7) we get

$\|(\nabla-{\mathsf{i}}{\bf
A})\partial_{2}v\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{2})}=\lambda\|\partial_{2}v\|^{2}_{L^{2}(\Omega)}.$
(3.14)

Finally, we get from (3.5), (3.6) and (3.14) that

$\displaystyle\mathfrak{h}_{b,\rm N}^{\Omega}[w]$
$\displaystyle\leq\lambda\|u\|^{2}_{L^{2}(\Omega)}+2\lambda{\rm
Re}\,\big{\\{}(u,c\partial_{2}v)_{L^{2}(\Omega)}\big{\\}}+|c|^{2}\lambda\|\partial_{2}v\|^{2}_{L^{2}(\Omega)}$
$\displaystyle=\lambda\|u+c\partial_{2}v\|^{2}_{L^{2}(\Omega)}=\lambda\|w\|^{2}_{L^{2}(\Omega)}.$

We have shown that for any $w\in{\mathcal{M}}\subset H^{1}(\Omega)$ it holds
that $\mathfrak{h}_{b,\rm N}^{\Omega}[w]\leq\lambda\|w\|^{2}_{L^{2}(\Omega)}$.
Since ${\mathcal{M}}$ is a $(k+1$)-dimensional linear subspace of
$H^{1}(\Omega)$, we get by the min-max principle (2.5b) that
$\mu_{k+1}^{b}(\Omega)\leq\lambda=\lambda_{k}^{b}(\Omega)$. Thus, the
inequality in the theorem is proved for convex polygons.

Step 3: general convex domains. For any bounded convex domain
$\Omega\subset{\mathbb{R}}^{2}$ one can construct by [Br08] a sequence of
convex polygons $\\{\Omega_{n}\\}_{n\in{\mathbb{N}}}$ such that the inclusion
$\Omega\subset\Omega_{n}$ holds for all $n\in{\mathbb{N}}$ and that
$|\Omega_{n}\setminus\Omega|\rightarrow 0$. Let us fix $k\in{\mathbb{N}}$. The
min-max principle for the magnetic Laplacian with Dirichlet boundary
conditions (2.5a) and the inclusion $\Omega\subset\Omega_{n}$ immediately
imply that

$\lambda_{k}^{b}(\Omega)\geq\lambda_{k}^{b}(\Omega_{n}),\qquad\text{for
all}\,\,n\in{\mathbb{N}}.$ (3.15)

It follows from Proposition A.2 in Appendix A that

$\liminf_{n\rightarrow\infty}\mu_{k+1}^{b}(\Omega_{n})\geq\mu_{k+1}^{b}(\Omega).$
(3.16)

Combining (3.15) and (3.16) with the eigenvalue inequality in the theorem
already shown for convex polygons, we get

$\mu_{k+1}^{b}(\Omega)\leq\liminf_{n\rightarrow\infty}\mu_{k+1}^{b}(\Omega_{n})\leq\liminf_{n\rightarrow\infty}\lambda_{k}^{b}(\Omega_{n})\leq\lambda_{k}^{b}(\Omega),$

by which the inequality in the theorem is proved also for general convex
domains.

##  4\. Proof of Theorem 1.2

The proof of this theorem follows similar strategy as the proof of Theorem
1.1. However, there will also be some new ideas related to the symmetry
assumption on the domain. We will prove the theorem first for convex
polyhedral domains in ${\mathbb{R}}^{3}$ satisfying the symmetry condition.
The eigenvalue inequality for general convex domains in ${\mathbb{R}}^{3}$
will be then derived via approximation. Assume until the last step of the
proof that $\Omega\subset{\mathbb{R}}^{3}$ is a convex polyhedral domain such
that ${\mathsf{J}}(\Omega)=\Omega$, where the mapping ${\mathsf{J}}$ is
defined as in (1.2).

Step 1: Symmetry of the eigenfunction. For the sake of brevity we use the
notation $\lambda=\lambda_{k}^{b}(\Omega)$ for the $k$-th eigenvalue of
${\mathsf{H}}_{b,\rm D}^{\Omega}$. Recall that this eigenvalue is assumed to
be simple. Let $v\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ be an eigenfunction
of ${\mathsf{H}}_{b,\rm D}^{\Omega}$ corresponding to the eigenvalue
$\lambda$. Let us consider the function $\hat{v}:=v\circ{\mathsf{J}}\in
H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$. We get that

$\displaystyle-(\nabla-{\mathsf{i}}{\bf A})^{2}\hat{v}$
$\displaystyle=-\Delta(v\circ{\mathsf{J}})+2{\mathsf{i}}bx_{1}\partial_{2}(v\circ{\mathsf{J}})+b^{2}x_{1}^{2}(v\circ{\mathsf{J}})$
$\displaystyle=-(\Delta
v)\circ{\mathsf{J}}-2{\mathsf{i}}bx_{1}(\partial_{2}v)\circ{\mathsf{J}}+b^{2}x_{1}^{2}(v\circ{\mathsf{J}})$
$\displaystyle=\big{(}-\Delta
v+2{\mathsf{i}}bx_{1}\partial_{2}v+b^{2}x_{1}^{2}v\big{)}\circ{\mathsf{J}}=0.$

Hence, $\hat{v}$ is also an eigenfunction of ${\mathsf{H}}_{b,\rm D}^{\Omega}$
corresponding to the eigenvalue $\lambda$. By simplicity of this eigenvalue we
conclude that $\hat{v}=zv$ with some $z\in{\mathbb{C}}$, $|z|=1$.

Step 2: construction of a trial subspace of $H^{1}(\Omega)$. By the min-max
principle for the magnetic Laplacian with Dirichlet boundary conditions (2.5a)
there exists a subspace ${\mathcal{L}}\subset H^{1}_{0}(\Omega)$ with
$\dim{\mathcal{L}}=k$ such that

$\mathfrak{h}_{b,{\rm
D}}^{\Omega}[u]\leq\lambda\|u\|^{2}_{L^{2}(\Omega)},\qquad\text{for
all}\,\,u\in{\mathcal{L}}.$ (4.1)

For the eigenfunction $v\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ of the
magnetic Dirichlet Laplacian on $\Omega$ corresponding to the eigenvalue
$\lambda$, we get $\partial_{2}v,\partial_{3}v\in H^{1}(\Omega)$. Let us
define the subspace ${\mathcal{M}}$ of $H^{1}(\Omega)$ by

${\mathcal{M}}:={\mathcal{L}}+{\rm span}\,\\{\partial_{2}v,\partial_{3}v\\}.$
(4.2)

Our next aim is to verify that $\dim{\mathcal{M}}=k+2$. In three dimensions
this argument is slightly more involved than in two dimensions. Clearly, we
have $\dim{\mathcal{M}}\leq k+2$. Suppose for the moment that
$\dim{\mathcal{M}}<k+2$. Hence, there would exist $c_{1},c_{2}\in{\mathbb{C}}$
(not both equal to zero) such that
$c_{1}\partial_{2}v+c_{2}\partial_{3}v\in{\mathcal{L}}\subset
H^{1}_{0}(\Omega)$. Without loss of generality we can assume that real parts
not of both $c_{1},c_{2}$ are equal to zero. Let $\Gamma\subset\partial\Omega$
be a face of the polyhedral domain $\Omega$ and let
$\tau_{1}\in{\mathbb{R}}^{3}$ and $\tau_{2}\in{\mathbb{R}}^{3}$ be linearly
independent unit tangential vectors to $\Gamma$. It is always possible to
choose the face $\Gamma$ so that

${\rm span}\,\\{\tau_{1},\tau_{2},(0,c_{1},c_{2})^{\top}\\}={\mathbb{C}}^{3},$
(4.3)

as otherwise, the non-trivial vector $(0,{\rm Re}\,c_{1},{\rm
Re}\,c_{2})^{\top}$ would be tangential to every face of the polyhedral domain
$\Omega$, which is not possible for clear geometric reasons. Let us assume
that the face $\Gamma$ is chosen so that (4.3) holds. It would follow from
$v,c_{1}\partial_{2}v+c_{2}\partial_{3}v\in H^{1}_{0}(\Omega)$ that

$\tau_{1}\cdot(\nabla v)|_{\Gamma}=\tau_{2}\cdot(\nabla
v)|_{\Gamma}=0,\qquad(0,c_{1},c_{2})^{\top}\cdot(\nabla v)|_{\Gamma}=0.$

Hence, we would get using (4.3) that $(\nabla v)|_{\Gamma}=0$ and, in
particular, we would have $\partial_{\nu}v|_{\Gamma}=0$. Thus, the function
$v\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ would satisfy the differential
equation $-(\nabla-{\mathsf{i}}{\bf A})^{2}v=\lambda v$ in $\Omega$ and
$\partial_{\nu}v|_{\Gamma}=0$. By Lemma 2.1 we would get that $v$ is
identically zero in $\Omega$ leading to a contradiction.

Step 3: application of the min-max principle. Any $w\in{\mathcal{M}}\subset
H^{1}(\Omega)$ can be represented as

$w=u+c_{1}\partial_{2}v+c_{2}\partial_{3}v$ (4.4)

with some $c_{1},c_{2}\in{\mathbb{C}}$ and $u\in{\mathcal{L}}$. Substituting
$w$ into the quadratic form for the magnetic Neumann Laplacian on the domain
$\Omega$ we get

$\displaystyle\mathfrak{h}_{b,\rm N}^{\Omega}[w]$
$\displaystyle=\|(\nabla-{\mathsf{i}}{\bf
A})u\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{3})}+2{\rm
Re}\,\left\\{\left((\nabla-{\mathsf{i}}{\bf A})u,(\nabla-{\mathsf{i}}{\bf
A})(c_{1}\partial_{2}v+c_{2}\partial_{3}v)\right)_{L^{2}(\Omega;{\mathbb{C}}^{3})}\right\\}$
(4.5) $\displaystyle\qquad\qquad\qquad+\|(\nabla-{\mathsf{i}}{\bf
A})(c_{1}\partial_{2}v+c_{2}\partial_{3}v)\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{3})}.$

We will analyse the three terms appearing on the right hand side in the above
equation separately. We derive from (4.1)

$\|(\nabla-{\mathsf{i}}{\bf
A})u\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{3})}\leq\lambda\|u\|^{2}_{L^{2}(\Omega)}.$
(4.6)

Note that

$\displaystyle-(\nabla-{\mathsf{i}}{\bf
A})^{2}(c_{1}\partial_{2}v+c_{2}\partial_{3}v)$ (4.7)
$\displaystyle\qquad=-\Delta(c_{1}\partial_{2}v+c_{2}\partial_{3}v)+2{\mathsf{i}}bx_{1}\partial_{2}(c_{1}\partial_{2}v+c_{2}\partial_{3}v)+b^{2}x_{1}^{2}(c_{1}\partial_{2}v+c_{2}\partial_{3}v)$
$\displaystyle\qquad=-(c_{1}\partial_{2}+c_{2}\partial_{3})(\nabla-{\mathsf{i}}{\bf
A})^{2}v=\lambda(c_{1}\partial_{2}v+c_{2}\partial_{3}v)\in L^{2}(\Omega).$

where the $\Delta\partial_{j}v$, $j=2,3$, should be understood first in the
distributional sense. As a direct consequence of the last computation we infer
that $\Delta\partial_{j}v\in L^{2}(\Omega)$ for $j=2,3$. By (2.4) we can
perform the integration by parts

$\displaystyle\left((\nabla-{\mathsf{i}}{\bf A})u,(\nabla-{\mathsf{i}}{\bf
A})(c_{1}\partial_{2}v+c_{2}\partial_{3}v)\right)_{L^{2}(\Omega;{\mathbb{C}}^{3})}$
(4.8) $\displaystyle\qquad=-\left(u,(\nabla-{\mathsf{i}}{\bf
A})^{2}(c_{1}\partial_{2}v+c_{2}\partial_{3}v)\right)_{L^{2}(\Omega)}=\lambda(u,c_{1}\partial_{2}v+c_{2}\partial_{3}v)_{L^{2}(\Omega)},$

where the boundary terms vanished since $u\in H^{1}_{0}(\Omega)$.

The analysis of the last term on the right hand side in (4.5) is the most
involved. We get

$\displaystyle\|(\nabla-{\mathsf{i}}{\bf
A})(c_{1}\partial_{2}v+c_{2}\partial_{3}v)\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{3})}$
(4.9) $\displaystyle\qquad=|c_{1}|^{2}\|(\nabla-{\mathsf{i}}{\bf
A})\partial_{2}v\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{3})}+2{\rm
Re}\,\big{\\{}((\nabla-{\mathsf{i}}{\bf
A})c_{1}\partial_{2}v,(\nabla-{\mathsf{i}}{\bf
A})c_{2}\partial_{3}v)_{L^{2}(\Omega;{\mathbb{C}}^{3})}\big{\\}}$
$\displaystyle\qquad\qquad+|c_{2}|^{2}\|(\nabla-{\mathsf{i}}{\bf
A})\partial_{3}v\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{3})}.$

For any face $\Gamma\subset\partial\Omega$ of the polyhedral domain $\Omega$
we derive from $v|_{\Gamma}=0$ that $\nabla
v|_{\Gamma}=\big{(}\partial_{\nu}v|_{\Gamma}\big{)}\nu$. Hence, we get that
$\partial_{i}v|_{\Gamma}=\nu_{i}\partial_{\nu}v|_{\Gamma}$, $i\in\\{1,2,3\\}$,
where $\nu=(\nu_{1},\nu_{2},\nu_{3})^{\top}$. In particular, we get for any
$i\in\\{2,3\\}$ using integration by parts

$\displaystyle 2{\rm
Re}\,\left({\mathsf{i}}\int_{\Omega}x_{1}\partial_{2}v\overline{\partial_{ii}v}{\,\mathrm{d}}x\right)$
$\displaystyle=-2{\rm
Re}\,\left({\mathsf{i}}\int_{\Omega}x_{1}\partial_{2i}v\overline{\partial_{i}v}{\,\mathrm{d}}x\right)+2{\rm
Re}\,\left(\int_{\partial\Omega}{\mathsf{i}}x_{1}\nu_{i}(\partial_{2}v|_{\partial\Omega})\overline{(\partial_{i}v|_{\partial\Omega})}{\,\mathrm{d}}\sigma\right)$
$\displaystyle=-2{\rm
Re}\,\left({\mathsf{i}}\int_{\Omega}x_{1}\partial_{2i}v\overline{\partial_{i}v}{\,\mathrm{d}}x\right)+{\rm
Re}\,\left(\int_{\partial\Omega}{\mathsf{i}}x_{1}\nu_{i}^{2}\nu_{2}\big{|}\partial_{\nu}v|_{\partial\Omega}\big{|}^{2}{\,\mathrm{d}}\sigma\right)$
$\displaystyle=-2{\rm
Re}\,\left({\mathsf{i}}\int_{\Omega}x_{1}\partial_{2i}v\overline{\partial_{i}v}{\,\mathrm{d}}x\right)$
$\displaystyle={\rm
Re}\,\left({\mathsf{i}}\int_{\Omega}x_{1}\big{(}\partial_{i}v\overline{\partial_{2i}v}-\partial_{2i}v\overline{\partial_{i}v}\big{)}{\,\mathrm{d}}x\right).$
(4.10)

Using that $v$ is an eigenfunction of ${\mathsf{H}}_{b,\rm D}^{\Omega}$
corresponding to the eigenvalue $\lambda$ and performing the integration by
parts with the help of (4.10) and Corollary 2.5 we get for $i\in\\{2,3\\}$

$\displaystyle\lambda\int_{\Omega}|\partial_{i}v|^{2}{\,\mathrm{d}}x$
$\displaystyle=-\lambda\int_{\Omega}v\overline{\partial_{ii}v}{\,\mathrm{d}}x$
$\displaystyle=\int_{\Omega}(\nabla-{\mathsf{i}}{\bf
A})^{2}v\overline{\partial_{ii}v}{\,\mathrm{d}}x$ $\displaystyle={\rm
Re}\,\int_{\Omega}(\nabla-{\mathsf{i}}{\bf
A})^{2}v\overline{\partial_{ii}v}{\,\mathrm{d}}x$ $\displaystyle={\rm
Re}\,\int_{\Omega}\big{(}\partial_{11}v+\partial_{22}v+\partial_{33}v-2{\mathsf{i}}bx_{1}\partial_{2}v-b^{2}x_{1}^{2}v\big{)}\overline{\partial_{ii}v}{\,\mathrm{d}}x$
$\displaystyle={\rm
Re}\,\int_{\Omega}\big{(}\partial_{1i}v\overline{\partial_{1i}v}+\partial_{2i}v\overline{\partial_{2i}v}+\partial_{3i}v\overline{\partial_{3i}v}-2{\mathsf{i}}bx_{1}\partial_{2}v\overline{\partial_{ii}v}+b^{2}x_{1}^{2}\partial_{i}v\overline{\partial_{i}v}\big{)}{\,\mathrm{d}}x$
$\displaystyle={\rm
Re}\,\int_{\Omega}\big{(}|\partial_{1i}v|^{2}+|\partial_{2i}v|^{2}\\!+\\!|\partial_{3i}v|^{2}-{\mathsf{i}}bx_{1}\partial_{i}v\overline{\partial_{2i}v}\\!+\\!{\mathsf{i}}bx_{1}\partial_{2i}v\overline{\partial_{i}v}+b^{2}x_{1}^{2}|\partial_{i}v|^{2}\big{)}{\,\mathrm{d}}x$
$\displaystyle={\rm Re}\,\big{(}(\nabla-{\mathsf{i}}{\bf
A})\partial_{i}v,(\nabla-{\mathsf{i}}{\bf
A})\partial_{i}v\big{)}_{L^{2}(\Omega;{\mathbb{C}}^{3})}=\|(\nabla-{\mathsf{i}}{\bf
A})\partial_{i}v\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{3})}.$ (4.11)

Moreover, we find via the integration by parts formula (2.4) and using
identity (4.7)

$\displaystyle\big{(}(\nabla-{\mathsf{i}}{\bf
A})\partial_{2}v,(\nabla-{\mathsf{i}}{\bf
A})\partial_{3}v\big{)}_{L^{2}(\Omega;{\mathbb{C}}^{3})}$ (4.12)
$\displaystyle\qquad=\big{(}\partial_{2}v,-(\nabla-{\mathsf{i}}{\bf
A})^{2}\partial_{3}v\big{)}_{L^{2}(\Omega)}+\big{\langle}\partial_{2}v|_{\partial\Omega},\partial_{\nu}(\partial_{3}v)|_{\partial\Omega}-{\mathsf{i}}\nu\cdot({\bf
A}|_{\partial\Omega})\partial_{3}v|_{\partial\Omega}\big{\rangle}_{\frac{1}{2},-\frac{1}{2}}$
$\displaystyle\qquad=\lambda\big{(}\partial_{2}v,\partial_{3}v\big{)}_{L^{2}(\Omega)}+\big{\langle}\partial_{2}v|_{\partial\Omega},\partial_{\nu}(\partial_{3}v)|_{\partial\Omega}-{\mathsf{i}}\nu\cdot({\bf
A}|_{\partial\Omega})\partial_{3}v|_{\partial\Omega}\big{\rangle}_{\frac{1}{2},-\frac{1}{2}}.$

Next, our aim is to show that under the imposed symmetry assumption the
duality product on the right hand side vanishes. For the function
$\hat{v}=v\circ{\mathsf{J}}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$
constructed in Step 1 we get

$\displaystyle\big{\langle}\partial_{2}v|_{\partial\Omega},\partial_{\nu}(\partial_{3}v)|_{\partial\Omega}-{\mathsf{i}}\nu\cdot({\bf
A}|_{\partial\Omega})\partial_{3}v|_{\partial\Omega}\big{\rangle}_{\frac{1}{2},-\frac{1}{2}}$
$\displaystyle\qquad=\big{\langle}\partial_{2}\hat{v}|_{\partial\Omega},\partial_{\nu}(\partial_{3}\hat{v})|_{\partial\Omega}-{\mathsf{i}}\nu\cdot({\bf
A}|_{\partial\Omega})\partial_{3}\hat{v}|_{\partial\Omega}\big{\rangle}_{\frac{1}{2},-\frac{1}{2}}$
$\displaystyle\qquad=((\nabla-{\mathsf{i}}{\bf
A})\partial_{2}\hat{v},(\nabla-{\mathsf{i}}{\bf
A})\partial_{3}\hat{v})_{L^{2}(\Omega;{\mathbb{C}}^{3})}-\lambda(\partial_{2}\hat{v},\partial_{3}\hat{v})_{L^{2}(\Omega)}$
$\displaystyle\qquad=-\big{(}((\nabla-{\mathsf{i}}{\bf
A})\partial_{2}v)\circ{\mathsf{J}},((\nabla-{\mathsf{i}}{\bf
A})\partial_{3}v)\circ{\mathsf{J}}\big{)}_{L^{2}(\Omega;{\mathbb{C}}^{3})}+\lambda\big{(}(\partial_{2}v)\circ{\mathsf{J}},(\partial_{3}v)\circ{\mathsf{J}}\big{)}_{L^{2}(\Omega)}$
$\displaystyle\qquad=-\big{(}((\nabla-{\mathsf{i}}{\bf
A})\partial_{2}v,((\nabla-{\mathsf{i}}{\bf
A})\partial_{3}v\big{)}_{L^{2}(\Omega;{\mathbb{C}}^{3})}+\lambda\big{(}\partial_{2}v,\partial_{3}v\big{)}_{L^{2}(\Omega)}$
$\displaystyle\qquad=-\big{\langle}\partial_{2}v|_{\partial\Omega},\partial_{\nu}(\partial_{3}v)|_{\partial\Omega}-{\mathsf{i}}\nu\cdot({\bf
A}|_{\partial\Omega})\partial_{3}v|_{\partial\Omega}\big{\rangle}_{\frac{1}{2},-\frac{1}{2}}=0,$

where in the first step we used $\hat{v}=zv$ with some $z\in{\mathbb{C}}$ such
that $|z|=1$, in the second step we applied the formula (4.12), in the third
step we used that $\hat{v}=v\circ{\mathsf{J}}$, in the fourth step we
implicitly used the change of variables $x^{\prime}={\mathsf{J}}x$ and used
the symmetry of $\Omega$, in the fifth step we again applied the formula
(4.12) in the opposite direction. Finally, the whole expression vanished,
since we have shown that the duality product is equal to minus itself. Hence,
we end up with the identity

${\rm Re}\,\big{\\{}\big{(}(\nabla-{\mathsf{i}}{\bf
A})c_{1}\partial_{2}v,(\nabla-{\mathsf{i}}{\bf
A})c_{2}\partial_{3}v\big{)}_{L^{2}(\Omega;{\mathbb{C}}^{3})}\big{\\}}=\lambda{\rm
Re}\,(c_{1}\partial_{2}v,c_{2}\partial_{3}v)_{L^{2}(\Omega)}$ (4.13)

where we implicitly used that
$(\partial_{2}v,\partial_{3}v)_{L^{2}(\Omega)}=-(v,\partial_{23}v)_{L^{2}(\Omega)}=-(\partial_{23}v,v)_{L^{2}(\Omega)}$
is a real number. Substituting (4.11) and (4.13) into (4.9) we arrive at

$\displaystyle\|(\nabla-{\mathsf{i}}{\bf
A})(c_{1}\partial_{2}v+c_{2}\partial_{3}v)\|^{2}_{L^{2}(\Omega;{\mathbb{C}}^{3})}$
(4.14)
$\displaystyle\qquad=\lambda|c_{1}|^{2}\|\partial_{2}v\|^{2}_{L^{2}(\Omega)}+2\lambda{\rm
Re}\,\big{\\{}(c_{1}\partial_{2}v,c_{2}\partial_{3}v)_{L^{2}(\Omega)}\big{\\}}+\lambda|c_{2}|^{2}\|\partial_{3}v\|^{2}_{L^{2}(\Omega)}$
$\displaystyle\qquad=\lambda\|c_{1}\partial_{2}v+c_{2}\partial_{3}v\|^{2}_{L^{2}(\Omega)}.$

Finally, we get from (4.6), (4.8) and (4.14) that

$\displaystyle\mathfrak{h}_{b,\rm N}^{\Omega}[w]$
$\displaystyle\leq\lambda\|u\|^{2}_{L^{2}(\Omega)}+2\lambda{\rm
Re}\,\big{\\{}(u,c_{1}\partial_{2}v+c_{2}\partial_{3}v)_{L^{2}(\Omega)}\big{\\}}+\lambda\|c_{1}\partial_{2}v+c_{2}\partial_{3}v\|^{2}_{L^{2}(\Omega)}$
$\displaystyle=\lambda\|u+c_{1}\partial_{2}v+c_{2}\partial_{3}v\|^{2}_{L^{2}(\Omega)}=\lambda\|w\|^{2}_{L^{2}(\Omega)}.$

We have shown that for any $w\in{\mathcal{M}}\subset H^{1}(\Omega)$ it holds
that $\mathfrak{h}_{b,\rm N}^{\Omega}[w]\leq\lambda\|w\|^{2}_{L^{2}(\Omega)}$.
Since ${\mathcal{M}}$ is a $(k+2)$-dimensional linear subspace of
$H^{1}(\Omega)$, we get by the min-max principle that
$\mu_{k+2}^{b}(\Omega)\leq\lambda=\lambda_{k}^{b}(\Omega)$. Thus, the
inequality in the theorem is proved for convex polyhedral domains invariant
under the mapping ${\mathsf{J}}$.

Step 4: general convex domains invariant under the mapping ${\mathsf{J}}$. For
any bounded convex domain $\Omega\subset{\mathbb{R}}^{3}$ one can construct by
[Br08] a sequence of convex polyhedral domains
$\\{\Omega_{n}\\}_{n\in{\mathbb{N}}}$ such that the inclusion
$\Omega\subset\Omega_{n}$ holds for all $n\in{\mathbb{N}}$ and that
$|\Omega_{n}\setminus\Omega|\rightarrow 0$. If the domain $\Omega$ satisfies,
in addition, $\Omega={\mathsf{J}}(\Omega)$, we can consider the sequence of
convex polyhedral domains
$\widehat{\Omega}_{n}:=\Omega_{n}\cap{\mathsf{J}}(\Omega_{n})$, which clearly
satisfies both conditions $\Omega\subset\widehat{\Omega}_{n}$ for all
$n\in{\mathbb{N}}$ and $|\widehat{\Omega}_{n}\setminus\Omega|\rightarrow 0$
and, in addition, we have
$\widehat{\Omega}_{n}={\mathsf{J}}(\widehat{\Omega}_{n})$.

Let us fix $k\in{\mathbb{N}}$. The min-max principle for the magnetic
Laplacian with Dirichlet boundary conditions (2.5a) and the inclusion
$\Omega\subset\widehat{\Omega}_{n}$ immediately imply that

$\lambda_{k}^{b}(\Omega)\geq\lambda_{k}^{b}(\widehat{\Omega}_{n}),\qquad\text{for
all}\,\,n\in{\mathbb{N}}.$ (4.15)

It follows from Proposition A.2 in Appendix A that

$\liminf_{n\rightarrow\infty}\mu_{k+2}^{b}(\widehat{\Omega}_{n})\geq\mu_{k+2}^{b}(\Omega).$
(4.16)

Combining (4.15) and (4.16) with the eigenvalue inequality in the theorem
already shown for convex polygons, we get

$\mu_{k+2}^{b}(\Omega)\leq\liminf_{n\rightarrow\infty}\mu_{k+2}^{b}(\widehat{\Omega}_{n})\leq\liminf_{n\rightarrow\infty}\lambda_{k}^{b}(\widehat{\Omega}_{n})\leq\lambda_{k}^{b}(\Omega),$

by which the inequality in the theorem is proved also for general convex
domains invariant under the mapping ${\mathsf{J}}$.

##  5\. The case of the disk

In this section, we will consider the magnetic Laplacian on the disk. Recall
that the Levine-Weinberger inequality in two dimensions reads as follows: the
$(k+2)$-th Neumann eigenvalue of the Laplacian on a planar convex domain is
not larger than its $k$-th Dirichlet eigenvalue. The index $k+2$ can not be
replaced by $k+3$, because the fourth Neumann eigenvalue of the Laplacian on
the disk is larger than its first Dirichlet eigenvalue. Thus, it is natural to
test the magnetic eigenvalue inequality in Theorem 1.1 also on the disk.

Let ${\mathcal{B}}\subset{\mathbb{R}}^{2}$ be the disk of unit radius centred
at the origin. In view of the scaling properties
$\lambda_{k}^{b}(t{\mathcal{B}})=t^{-2}\lambda_{k}^{bt^{2}}({\mathcal{B}})$
and $\mu_{k}^{b}(t{\mathcal{B}})=t^{-2}\mu_{k}^{bt^{2}}({\mathcal{B}})$ valid
for any $k\in{\mathbb{N}}$ and all $t>0$ (see [CLPS23, p. 456]), it suffices
to test the eigenvalue inequality on the unit disk. The eigenvalues of
${\mathsf{H}}_{b,\rm N}^{\mathcal{B}}$ and ${\mathsf{H}}_{b,\rm
D}^{\mathcal{B}}$ are continuous functions in $b$ (cf. [KL22, Appendix B] for
the Neumann case, the Dirichlet case is analogous). Hence, we know that
$\mu_{4}^{b}({\mathcal{B}})>\lambda_{1}^{b}({\mathcal{B}})$ for all
sufficiently small $b>0$, because the same inequality holds for the usual
Laplacian, which corresponds to $b=0$. Thus, the index $k+1$ in Theorem 1.1
can not be, in general, replaced by $k+3$. However, this observation does not
exclude the possibility that the index $k+1$ can be replaced by $k+2$.

It follows from [FFG+23, Proposition 2.1] that $\mu_{3}^{b}({\mathcal{B}})<b$
if and only if $b>4$, because the flux
$\Phi:=\frac{b|{\mathcal{B}}|}{2\pi}=\frac{b}{2}>2$ for $b>4$. On the other
hand, according to [FH, Lemma 1.4.1] we have
$\lambda_{1}^{b}({\mathcal{B}})\geq b$ for all $b>0$. Hence, we get that
$\mu_{3}^{b}({\mathcal{B}})<\lambda_{1}^{b}({\mathcal{B}})$ for all $b>4$. In
the remaining part of this appendix, we will analyse the case $b\in(0,4]$
numerically. The numerics is rather simple and reduces to solving explicit
transcendental equations in terms of generalized Laguerre functions.

The magnetic Neumann Laplacian ${\mathsf{H}}_{b,\rm N}^{{\mathcal{B}}}$ can be
decomposed by e.g. [KL22, Section 3] up to unitary equivalence as

${\mathsf{H}}_{b,\rm
N}^{{\mathcal{B}}}\simeq\bigoplus_{n\in{\mathbb{Z}}}{\mathsf{T}}_{b,n},$ (5.1)

where the self-adjoint fiber operator ${\mathsf{T}}_{b,n}$,
$n\in{\mathbb{Z}}$, acts in the Hilbert space $L^{2}((0,1);r{\,\mathrm{d}}r)$
and is associated with the quadratic form

$\displaystyle{\mathfrak{t}}_{b,n}[f]$
$\displaystyle:=\int_{0}^{1}|f^{\prime}(r)|^{2}r{\,\mathrm{d}}r+\int_{0}^{1}\left(\frac{n}{r}-\frac{br}{2}\right)^{2}|f(r)|^{2}r{\,\mathrm{d}}r,$
(5.2) $\displaystyle\qquad\mathrm{dom}\,{\mathfrak{t}}_{b,n}$
$\displaystyle:=\big{\\{}f\colon f,f^{\prime},nfr^{-1}\in
L^{2}((0,1);r{\,\mathrm{d}}r)\big{\\}}.$

Let us denote by $\mu_{n,1}=\mu_{n,1}(b)$, $n\in{\mathbb{Z}}$, the lowest
eigenvalue of the fiber operator ${\mathsf{T}}_{b,n}$. In view of the
decomposition (5.1) for any $\lambda>0$

$\\#\\{k\in{\mathbb{N}}\colon\mu_{k}^{b}({\mathcal{B}})\leq\lambda\\}\geq\\#\\{n\in{\mathbb{Z}}\colon\mu_{n,1}(b)\leq\lambda\\}.$
(5.3)

Let us denote by $L^{a}_{\gamma}$ the generalized Laguerre function (see, e.g.
, [AS, §22] for precise definitions and properties). By the analysis in
[CLPS23, Appendix B], $\mu_{n,1}$ is the smallest positive solution of the
equation

$F_{n,b}(\mu):=\frac{{\,\mathrm{d}}}{{\,\mathrm{d}}r}\left(e^{-\frac{br^{2}}{4}}r^{n}L^{n}_{\frac{1}{2}\big{(}\frac{\mu}{b}-1\big{)}}\left(\frac{r^{2}b}{2}\right)\right)\bigg{|}_{r=1}=0,$
(5.4)

and the associated eigenfunction is given by

$v_{n}(r)=e^{-\frac{br^{2}}{4}}r^{n}L^{n}_{\frac{1}{2}\big{(}\frac{\mu_{n,1}}{b}-1\big{)}}\left(\frac{r^{2}b}{2}\right).$

For the magnetic Dirichlet Laplacian on the disk ${\mathsf{H}}_{b,{\rm
D}}^{\mathcal{B}}$ we know according to [E96, Proposition 2.1] that
$\lambda_{1}^{b}({\mathcal{B}})$ is equal to the lowest eigenvalue
$\lambda_{0,1}=\lambda_{0,1}(b)$ of the one-dimensional self-adjoint fiber
operator in $L^{2}((0,1);r{\,\mathrm{d}}r)$ associated with the quadratic form

$\displaystyle{\mathfrak{t}}_{b}[f]$
$\displaystyle:=\int_{0}^{1}|f^{\prime}(r)|^{2}r{\,\mathrm{d}}r+\int_{0}^{1}\frac{b^{2}r^{3}}{4}|f(r)|^{2}{\,\mathrm{d}}r$
$\displaystyle\mathrm{dom}\,{\mathfrak{t}}_{b}$
$\displaystyle:=\big{\\{}f\colon f,f^{\prime}\in
L^{2}((0,1);r{\,\mathrm{d}}r),\,f(1)=0\big{\\}}.$

This is a direct consequence of the fact that, in the symmetric gauge of the
homogeneous magnetic field, the ground state of the magnetic Laplacian on the
disk is a radial function. Mimicking the analysis in [CLPS23, Appendix B] we
get that $\lambda_{0,1}$ is the smallest positive solution of the equation

$G_{b}(\lambda):=L^{0}_{\frac{1}{2}\big{(}\frac{\lambda}{b}-1\big{)}}\left(\frac{b}{2}\right)=0.$
(5.5)

In Figure 5.1, we have plotted the graphs of $\lambda_{0,1}$, $\mu_{0,1}$,
$\mu_{1,1}$, $\mu_{-1,1}$, $\mu_{2,1}$ as functions of $b$. In the plotted
range of $b$, for any $b>0$ among the values $\mu_{0,1}(b)$, $\mu_{1,1}(b)$,
$\mu_{-1,1}(b)$, $\mu_{2,1}(b)$ at least three do not exceed
$\lambda_{0,1}(b)$. We also remark that at $b=2$ the graphs of
$\lambda_{0,1}$, $\mu_{-1,1}$, and $\mu_{2,1}$ indeed cross at one point. This
property can be rigorously checked. By [GR, 8.970 (4)] we have the following
explicit expression for the generalized Laguerre functions:
$L^{0}_{1}(x)=1-x$, $L^{-1}_{1}(x)=-x$, and $L^{2}_{1}(x)=3-x$. Using these
expression we find

$\displaystyle G_{2}(6)$ $\displaystyle=L^{0}_{1}(1)=0,$ $\displaystyle
F_{-1,2}(6)$
$\displaystyle=\frac{{\,\mathrm{d}}}{{\,\mathrm{d}}r}\left(e^{-\frac{r^{2}}{2}}r^{-1}L_{1}^{-1}(r^{2})\right)\bigg{|}_{r=1}=-\frac{{\,\mathrm{d}}}{{\,\mathrm{d}}r}\left(e^{-\frac{r^{2}}{2}}r\right)\bigg{|}_{r=1}=\left(e^{-\frac{r^{2}}{2}}\big{[}r^{2}-1\big{]}\right)\bigg{|}_{r=1}=0,$
$\displaystyle F_{2,2}(6)$
$\displaystyle=\frac{{\,\mathrm{d}}}{{\,\mathrm{d}}r}\left(e^{-\frac{r^{2}}{2}}r^{2}L_{1}^{2}(r^{2})\right)\bigg{|}_{r=1}=\frac{{\,\mathrm{d}}}{{\,\mathrm{d}}r}\left(e^{-\frac{r^{2}}{2}}r^{2}(3-r^{2})\right)\bigg{|}_{r=1}$
$\displaystyle=\left(e^{-\frac{r^{2}}{2}}\big{[}r^{5}-7r^{3}+6r\big{]}\right)\bigg{|}_{r=1}=0.$

Figure 5.1. The lowest eigenvalues of the fiber operators are plotted as
functions of $b$: $\lambda_{0,1}$ – thick black curve; $\mu_{0,1}$ – thin red
curve; $\mu_{1,1}$ – thin green curve; $\mu_{-1,1}$ – thin blue curve;
$\mu_{2,1}$ – thin yellow curve.

This numerical test provides an evidence for the validity of the inequality
$\mu_{3}^{b}({\mathcal{B}})\leq\lambda_{1}^{b}({\mathcal{B}})$ for all $b>0$.
Therefore, a counterexample showing sharpness of Theorem 1.1 should either be
based on the comparison of higher eigenvalues of the disk or should be based
on a domain different from the disk. Thus, it remains an open problem whether
the inequality $\mu_{k+2}^{b}(\Omega)\leq\lambda_{k}^{b}(\Omega)$ for all
$b>0$ and any $k\in{\mathbb{N}}$ holds for any convex domain
$\Omega\subset{\mathbb{R}}^{2}$. We would like to refrain from making any
conjectures.

## Acknowledgement

The author acknowledges the support by the Czech Science Foundation (GAČR)
within the project 21-07129S.

##  Appendix A Lower semicontinuity of magnetic Neumann eigenvalues

In this appendix, we establish a lower semi-continuity result for the magnetic
Neumann eigenvalues on varying convex domains satisfying an inclusion
property. Although, we expect that under our assumptions the magnetic Neumann
eigenvalues are in fact continuous with respect to the variation of the
domain, lower semi-continuity turns out to be sufficient for our purposes.
General stability results for the eigenvalues of elliptic operators with
Neumann-type boundary conditions under variations of the domain can be found
e.g. in [BD02, BL07], see also the references therein. We expect that the
abstract results in [BL07] can be adjusted to the magnetic Neumann Laplacian,
but we prefer to perform a more explicit analysis.

In the proof of a lower semi-continuity result the following lemma is useful.

######  Lemma A.1 ([MKh, Lemma 4.9, Remark 4.2]).

Let $U,V\subset{\mathbb{R}}^{d}$, $d\geq 2$, be bounded convex domains such
that $U\subset V$. Then for any $u\in H^{1}(V)$ the following inequality holds

$\int_{V\setminus U}|u|^{2}{\,\mathrm{d}}x\leq\frac{2|V\setminus
U|}{|V|}\int_{V}|u|^{2}{\,\mathrm{d}}x+\frac{C(d)({\rm
diam}\,(V))^{d+1}|V\setminus U|^{1/d}}{|V|}\int_{V}|\nabla
u|^{2}{\,\mathrm{d}}x,$

where ${\rm diam}\,(V)>0$ is the diameter of $V$ and the constant $C(d)>0$
depends only on the dimension.

Now, we are ready to formulate and prove the key result of this appendix.

######  Proposition A.2.

Let $\Omega\subset{\mathbb{R}}^{d}$ with $d=2,3$ be a bounded convex domain.
Let $\\{\Omega_{n}\\}_{n\in{\mathbb{N}}}$ be a sequence of bounded convex
domains in ${\mathbb{R}}^{d}$ such that $\Omega_{n}\supset\Omega$ for all
$n\in{\mathbb{N}}$ and that $|\Omega_{n}\setminus\Omega|\rightarrow 0$. Then,
for all $b>0$ and any $k\in{\mathbb{N}}$

$\liminf_{n\rightarrow\infty}\mu_{k}^{b}(\Omega_{n})\geq\mu_{k}^{b}(\Omega).$

###### Proof.

For the convenience of the reader, we split the argument into two steps. In
the first step, we obtain estimates of the eigenfunctions of
${\mathsf{H}}_{b,{\rm N}}^{\Omega_{n}}$ inside $\Omega_{n}\setminus\Omega$.
Then, in the second step we apply the min-max principle to
${\mathsf{H}}_{b,{\rm N}}^{\Omega}$ using as trial functions the restrictions
to $\Omega$ of the eigenfunctions of ${\mathsf{H}}_{b,{\rm N}}^{\Omega_{n}}$.
This strategy is reminiscent of the one used by Levine and Weinberger in
[LW86, Lemma 4.6].

Step 1: estimates of the eigenfunctions in $\Omega_{n}\setminus\Omega$. Let us
fix $k\in{\mathbb{N}}$. Let $u_{1,n},u_{2,n},\dots,u_{k,n}\in
H^{1}(\Omega_{n})$ be the orthonormal eigenfunctions of ${\mathsf{H}}_{b,\rm
N}^{\Omega_{n}}$ corresponding to the lowest $k$ eigenvalues:
$\mu_{1}^{b}(\Omega_{n}),\mu_{2}^{b}(\Omega_{n}),\dots,\mu_{k}^{b}(\Omega_{n})$.
Note that the geometric assumptions on $\Omega_{n}$ and $\Omega$ yield

$\rho:=\sup_{n\in{\mathbb{N}}}{\rm
diam}\,(\Omega_{n})<\infty\qquad\alpha:=\sup_{n\in{\mathbb{N}}}|\Omega_{n}\setminus\Omega|<\infty.$

Combining Lemma A.1 and the diamagnetic inequality [LL, Theorem 7.21] we get
for any $l\in\\{1,2,\dots,k\\}$

$\displaystyle\int_{\Omega_{n}\setminus\Omega}|u_{l,n}|^{2}{\,\mathrm{d}}x$
$\displaystyle\leq\frac{2|\Omega_{n}\setminus\Omega|}{|\Omega_{n}|}\int_{\Omega_{n}}|u_{l,n}|^{2}{\,\mathrm{d}}x+\frac{C(d)({\rm
diam}\,(\Omega_{n}))^{d+1}|\Omega_{n}\setminus\Omega|^{1/d}}{|\Omega_{n}|}\int_{\Omega_{n}}|\nabla|u_{l,n}||^{2}{\,\mathrm{d}}x$
$\displaystyle\leq\frac{2|\Omega_{n}\setminus\Omega|}{|\Omega|}+\frac{C(d)\rho^{d+1}|\Omega_{n}\setminus\Omega|^{1/d}}{|\Omega|}\int_{\Omega_{n}}|(\nabla-{\mathsf{i}}{\bf
A})u_{l,n}|^{2}{\,\mathrm{d}}x$
$\displaystyle\leq\frac{1}{|\Omega|}\left(2|\Omega_{n}\setminus\Omega|+C(d)\rho^{d+1}|\Omega_{n}\setminus\Omega|^{1/d}\mu_{l}^{b}(\Omega_{n})\right).$

Applying the min-max principles (2.5) to the operators ${\mathsf{H}}_{b,\rm
N}^{\Omega_{n}}$, ${\mathsf{H}}_{b,\rm D}^{\Omega_{n}}$, and
${\mathsf{H}}_{b,\rm D}^{\Omega}$ we get for $l\in\\{1,2,\dots,k\\}$ the
following chain of inequalities

$\mu_{l}^{b}(\Omega_{n})\leq\mu_{k}^{b}(\Omega_{n})\leq\lambda_{k}^{b}(\Omega_{n})\leq\lambda_{k}^{b}(\Omega),$

where for the last inequality we employed $\Omega\subset\Omega_{n}$. Thus,
there exists a constant $c>0$, which depends only on $\Omega$, $\alpha$,
$\rho$, $b$, and $k$ such that

$\int_{\Omega_{n}\setminus\Omega}|u_{l,n}|^{2}{\,\mathrm{d}}x\leq
c|\Omega_{n}\setminus\Omega|^{1/d}$

for all $l\in\\{1,2,\dots,k\\}$. In particular, for any $\varepsilon>0$ there
exists a sufficiently large $N\in{\mathbb{N}}$ such that for any
$l\in\\{1,2,\dots,k\\}$ and all $n\geq N$

$\int_{\Omega_{n}\setminus\Omega}|u_{l,n}|^{2}{\,\mathrm{d}}x\leq\varepsilon.$
(A.1)

Step 2: application of the min-max principle. Consider the subspace

${\mathcal{L}}_{k,n}:={\rm
span}\,\\{u_{1,n}|_{\Omega},u_{2,n}|_{\Omega},\dots,u_{k,n}|_{\Omega}\\}\subset
H^{1}(\Omega).$

For any $v=\sum_{l=1}^{k}c_{l}u_{l,n}|_{\Omega}\in{\mathcal{L}}_{k,n}$ with
arbitrary $c_{l}\in{\mathbb{C}}$ ($l=1,2,\dots,k$) we get

$\displaystyle\int_{\Omega}|v|^{2}{\,\mathrm{d}}x$
$\displaystyle=\sum_{i=1}^{k}\sum_{j=1}^{k}c_{i}\overline{c_{j}}\int_{\Omega}u_{i,n}\overline{u_{j,n}}{\,\mathrm{d}}x$
(A.2)
$\displaystyle\geq\sum_{l=1}^{k}|c_{l}|^{2}\int_{\Omega}|u_{l,n}|^{2}{\,\mathrm{d}}x-\sum_{\begin{smallmatrix}i,j=1\\\
i\neq
j\end{smallmatrix}}^{k}|c_{i}||c_{j}|\left|\int_{\Omega_{n}\setminus\Omega}u_{i,n}\overline{u_{j,n}}{\,\mathrm{d}}x\right|\geq\sum_{l=1}^{k}|c_{l}|^{2}(1-k\varepsilon).$

Hence, for all sufficiently small $\varepsilon>0$ we get that $v\neq 0$
provided that not all of the coefficients $\\{c_{l}\\}_{l=1}^{k}$ are equal to
zero. Hence, it holds that $\dim{\mathcal{L}}_{k,n}=k$ for all sufficiently
large $n\in{\mathbb{N}}$.

By (2.5b) we conclude that there exist complex coefficients
$\\{c_{l}^{\prime}\\}_{l=1}^{k}$ such that for
$v=\sum_{l=1}^{k}c_{l}^{\prime}u_{l,n}|_{\Omega}\in H^{1}(\Omega)$ one has

$\mu_{k}^{b}(\Omega)\leq\frac{\displaystyle\int_{\Omega}|(\nabla-{\mathsf{i}}{\bf
A})v|^{2}{\,\mathrm{d}}x}{\displaystyle\int_{\Omega}|v|^{2}{\,\mathrm{d}}x}.$

Let us define the function $u=\sum_{l=1}^{k}c_{l}^{\prime}u_{l,n}\in
H^{1}(\Omega_{n})$. Choosing $N\in{\mathbb{N}}$ such that the estimate in
(A.1) holds with $\varepsilon<\frac{1}{k}$ for all $n\geq N$ and applying the
lower bound in (A.2) we end up with

$\mu_{k}^{b}(\Omega)\leq\frac{1}{1-k\varepsilon}\frac{\displaystyle\int_{\Omega_{n}}|(\nabla-{\mathsf{i}}{\bf
A})u|^{2}{\,\mathrm{d}}x}{\displaystyle\int_{\Omega_{n}}|u|^{2}{\,\mathrm{d}}x}\leq\frac{\mu_{k}^{b}(\Omega_{n})}{1-k\varepsilon}.$

From this inequality we get

$\liminf_{n\rightarrow\infty}\mu_{k}^{b}(\Omega_{n})\geq\mu_{k}^{b}(\Omega)(1-k\varepsilon).$

The claim follows from the fact that we can take $\varepsilon>0$ arbitrarily
small. ∎

## References

  * [AS] M. Abramowitz and I. Stegun, _Handbook of mathematical functions with formulas, graphs, and mathematical tables_ , Washington, D.C., 1964.
  * [A86] P. Aviles, Symmetry theorems related to Pompeiu’s problem, Amer. J. Math. 108 (1986), 1023–1036.
  * [BGM22] J. Behrndt, F. Gesztesy, and M. Mitrea, Sharp boundary trace theory and Schrödinger operators on bounded Lipschitz domains, to appear in Mem. Amer. Math. Soc., arXiv:2209.09230.
  * [BR12] J. Behrndt and J. Rohleder, An inverse problem of Calderón type with partial data, Commun. Partial Differ. Equations 37 (2012), 1141–1159.
  * [BR20] J. Behrndt and J. Rohleder, Inverse problems with partial data for elliptic operators on unbounded Lipschitz domains, Inverse Probl. 36 (2020), 035009.
  * [BD02] V. I. Burenkov and E. B. Davies, Spectral stability of the Neumann Laplacian, J. Differ. Equations 186 (2002), 485–508.
  * [BL07] V. Burenkov and P. D. Lamberti, Spectral stability of general non-negative selfadjoint operators with applications to Neumann-type operators, J. Differ. Equations 233 (2007), 345–379.
  * [Br08] E. Bronstein, Approximation of convex sets by polytopes, J. Math. Sci. 153 (2008), 727–762.
  * [CLPS23] B. Colbois, C. Léna, L. Provenzano, and A. Savo, Geometric bounds for the magnetic Neumann eigenvalues in the plane, J. Math. Pures Appl. (9) 179 (2023), 454–497.
  * [CLPS24] B. Colbois, C. Léna, L. Provenzano, and A. Savo, A reverse Faber-Krahn inequality for the magnetic Laplacian, arXiv:2403.11336.
  * [DT24] C. Denis and A. F. M. ter Elst, The Stokes Dirichlet-to-Neumann operator, J. Evol. Equ. 24 (2024), 22.
  * [EKP16] T. Ekholm, H. Kovařík, and F. Portmann, Estimates for the lowest eigenvalue of magnetic Laplacians, J. Math. Anal. Appl. 439 (2016), 330–346.
  * [E96] L. Erdős, Rayleigh-type isoperimetric inequality with a homogeneous magnetic field. Calc. Var. PDE 4 (1996), 283–292.
  * [F05] N. Filonov, On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator, St. Petersbg. Math. J. 16 (2005), 413–416.
  * [FH] S. Fournais and B. Helffer, Spectral methods in surface superconductivity, Birkhäuser, Basel, 2010.
  * [FFG+23] S. Fournais, R. L. Frank, M. Goffeng, A. Kachmar, and M. Sundqvist, Counting negative eigenvalues for the magnetic Pauli Operator, to appear in Duke. Math. J., arXiv:2307.16079.
  * [FL10] R. L. Frank and A. Laptev, Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group, Int. Math. Res. Not. (2010), 2889–2902.
  * [FLW23] R. L. Frank, A. Laptev, and T. Weidl, _Schrödinger operators: eigenvalues and Lieb-Thirring inequalities_ , Cambridge University Press, Cambridge, 2023.
  * [F91] L. Friedlander, Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Ration. Mech. Anal. 116 (1991), 153–160.
  * [GR] I. S. Gradshteyn and I. M. Ryzhik, _Table of integrals, series, and products,_ Elsevier/Academic Press, Amsterdam, 2015.
  * [Gr] P. Grisvard, _Elliptic problems in nonsmooth domains_ , Pitman Publishing, Boston-London-Melbourne, 1985.
  * [KL22] A. Kachmar and V. Lotoreichik, On the isoperimetric inequality for the magnetic Robin Laplacian with negative boundary parameter, J. Geom. Anal. 32 (2022), 182.
  * [KL23] A. Kachmar and V. Lotoreichik, A geometric bound on the lowest magnetic Neumann eigenvalue via the torsion function, to appear in SIAM J. Math. Anal., arXiv:2312.06161.
  * [LS15] R. S. Laugesen and B. A. Siudeja, Magnetic spectral bounds on starlike plane domains, ESAIM, Control Optim. Calc. Var. 21 (2015), 670–689
  * [LW86] H. Levine and H. F. Weinberger, Inequalities between Dirichlet and Neumann eigenvalues, Arch. Ration. Mech. Anal. 94 (1986), 193–208.
  * [LMP] M. Levitin, D. Mangoubi, and I. Polterovich, _Topics in spectral geometry_ , American Mathematical Society, Providence, 2024.
  * [LL] E. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, 2001.
  * [L24] V. Lotoreichik, Improved inequalities between Dirichlet and Neumann eigenvalues of the biharmonic operator, Proc. Amer. Math. Soc. 152 (2024), 2571–2584.
  * [LR17] V. Lotoreichik and J. Rohleder, Eigenvalue inequalities for the Laplacian with mixed boundary conditions, J. Differ. Equations 263 (2017), 491–508.
  * [MKh] V. A. Marchenko and E. Ya. Khruslov, Homogenization of partial differential equations, Birkhäuser, Boston, 2006.
  * [McL] W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
  * [P55] L. E. Payne, Inequalities for eigenvalues of membranes and plates, J. Rational Mech. Anal. 4 (1955), 517–529.
  * [P91] L. E. Payne, Some comments on the past fifty years of isoperimetric inequalities, in: _Inequalities (Birmingham 1987)_ , Lect. Notes Pure Appl. Math. 129 (1991), 143–161.
  * [P52] G. Pólya, Remarks on the foregoing paper, J. Math. Phys. 31 (1952), 55–57.
  * [Pr19] L. Provenzano, Inequalities between Dirichlet and Neumann eigenvalues of the polyharmonic operators, Proc. Amer. Math. Soc. 147 (2019), 4813–4821.
  * [R21] J. Rohleder, Inequalities between Neumann and Dirichlet eigenvalues of Schrödinger operators, J. Spectr. Theory 11 (2021), 915–933.
  * [R23] J. Rohleder, Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains, arXiv:2306.12922.


#  Dressed quark-gluon vertex form factors from gauge symmetry

Bruno El-Bennich1 Fernando E. Serna1,2 and Roberto Correa da Silveira1 1LFTC,
Universidade Cidade de São Paulo, Rua Galvão Bueno 868, 01506-000 São Paulo,
SP, Brazil
2Departamento de Física, Universidad de Sucre, Carrera 28 No. 5-267, Barrio
Puerta Roja, Sincelejo, Colombia  Luis Albino3 and Adnan Bashir3 3Instituto de
Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo,
Morelia, Michoacán 58040, México  Eduardo Rojas4 4Departamento de Física,
Universidad de Nariño, A.A. 1175, San Juan de Pasto, Colombia

###### Abstract

We present preliminary results on the longitudinal _and_ transverse form
factors of the quark-gluon vertex as functions of the incoming and outgoing
quark momenta and an angle $\theta=2\pi/3$ between them. The expressions for
these form factors were previously derived from Slavnov-Taylor identities,
gauge covariance and multiplicative renormalizability that firmly constrain
the fermion-boson vertex.

###### pacs:

12.38.-t 11.15.Tk 02.30.Rz 11.30.Qc 14.65.Bt

day month yearday month year Quantum Chromodynamics, Strong Interactions,
Nonperturbative Techniques, Gap Equation, Quark-Gluon Vertex, Dynamical Chiral
Symmetry Breaking

The origin of _dynamical chiral symmetry breaking_ (DCSB), the mass-generating
mechanism responsible for the overwhelming contribution to the nuclei’s
masses, lies in the non-Abelian nature of the theory of strong interactions
known as Quantum Chromodynamics (QCD). Starting with the seminal work by Nambu
and Jona-Lasinio [1], this mechanism has been gradually elucidated in QCD over
the past decades. While its role in generating hadron masses two orders of
magnitude larger than those of the light current quarks is nowadays widely
recognized, its likely deeper connection to _confinement_ still remains
speculative.

A common approach to investigate DCSB is to study the gap equation of the
quark, i.e. its two-point Green function, and its nonperturbative formulation
in terms of the Dyson-Schwinger equation (DSE) [2]. The latter is a Euclidean-
space description of the quark’s equation of motion in relativistic quantum
field theory and can be derived from the generating functional in QCD [3]. The
self-energy term in this integral equation involves other Green functions,
namely the gluon propagator and the quark-gluon vertex, which are irreducible
two- and three-point functions, respectively. Both contribute, along with the
strong coupling $\alpha_{s}$, to the integral kernel’s “strength” in the DSE.

Indeed, it is this strength that controls the emergence of a mass gap. While
the strong coupling and the form factor associated with the gluon dressing
form an overall strength factor, the twelve tensor structures of the quark-
gluon vertex reveal a more intricate story. Their contributions are codified
in so-called longitudinal and transverse form factors and their convolution
with the strong coupling and the gluon dressing function gives rise to a
constituent quark mass scale.

Since the full structure of the dressed quark-gluon vertex, and in particular
of the associated form factors, still poses a serious computational challenge
in functional as well as in lattice-QCD approaches, a common expedient in
applications to hadron physics is to retain merely its perturbative
$\gamma_{\mu}$ term. Folding its form factor with that of the gluon propagator
and the strong coupling, one arrives at the nowadays well known _rainbow-
ladder_ truncation of the gap equation, in which a single analytic function
mimics the infrared and ultraviolet behavior and the strength of the strong
interaction in an effective manner [4, 5].

While this approach has certainly proven to be successful in the computation
of the light meson and baryon spectrum and their electromagnetic properties
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15], it fails to correctly describe the
scalar and axialvector meson masses and does not produce a satisfying mass
ordering of higher radially excited mesons. It also gives rise to a spurious
spectrum of unobserved light mesons [6] with “exotic” quantum numbers, and
admits $\bar{3}_{c}$ colored diquark bound-states. Still, the latter feature
can be favorably used to derive approximate Faddeev wave functions of baryons.
The shortcoming of this leading approximation is also observed in solving the
Bethe-Salpeter equation for pseudoscalar and vector $D$ and $B$ mesons [6, 9],
but can be overcome introducing a flavor dependence in the quark-gluon
interaction [16, 17, 18]. However, this comes at the cost of additional
parameters for the charm and bottom mesons.

Important improvements, based on the three-particle irreducible (3PI)
effective QCD action [19] or on a model ansatz for the quark-gluon vertex [20,
21] amongst others, have been obtained over the past decade, and one may
assert that functional QCD approaches to light and flavored mesons, heavy
quarkonia and baryons based on the DSE in conjunction with either the Bethe-
Salpeter equation (BSE) or Faddeev equation are overall very successful. This
includes the mass spectrum of light and heavy mesons, the nucleon and $\Delta$
baryons, their parity partners and radial excitations, as well as Compton
scattering, elastic and transition form factors. Extensions to tetraquark
states have also been studied within this approach [22].

Nonetheless, a less model dependent interaction kernel, based on _calculated_
QCD Green functions, of the quark DSE and related bound-state equations is
desirable. Only a detailed construction of the interaction kernel, which
involves the fully dressed quark-gluon vertex, will allow to verify whether
the known hadron spectrum can be completely described with functional methods
in QCD. In addition, there are not merely phenomenological but also formal,
field-theoretical motivations to study the analytic behavior of the fermion-
boson vertex. After all, this vertex plays a pivotal role for DCSB in QED and
in QCD. Its contribution to the infrared behavior of the quark propagator and
to fragmentation functions, and therefore to the elucidation of the
confinement mechanism, cannot be appreciated enough.

In this contribution we extend recent studies on the transverse quark-gluon
vertex, which we derived from transverse Slavnov-Taylor identities and
multiplicative renormalizability in Refs. [23, 24]. In those studies we did
not present figures of the different vertex form factors, so we here take the
opportunity to fill this gap.

The dressed quark-gluon vertex is the essential three-point function which
describes the nonperturbative coupling of a dressed quark to a dressed gluon.
As such, it is the source of nonperturbative radiative gluon corrections to
the current quark’s relativistic motion. As in other relativistic quantum
field theories, the related equation of motion can be expressed by a DSE
conveniently derived within a functional approach to QCD [3]:

$\displaystyle S^{-1}(p)$ $\displaystyle=\,Z_{2}\,i\gamma\cdot p+Z_{4}\,m$
$\displaystyle\,+Z_{1}g^{2}\\!\int^{\Lambda}\\!\\!\frac{d^{4}k}{(2\pi)^{4}}\
\Delta^{ab}_{\mu\nu}(q)\,\gamma_{\mu}t^{a}S(k)\,\Gamma_{\nu}^{b}(k,p)\ .$ (1)

In this integral equation, $m$ is the renormalized current-quark mass and
$Z_{4}(\mu,\Lambda)$ is its renormalization constant in the QCD Lagrangian,
while $Z_{1}(\mu,\Lambda)$ and $Z_{2}(\mu,\Lambda)$ are vertex and wave-
function renormalization constants, respectively. The integral in Eq. (1)
expresses the quark’s self-energy $\Sigma(p^{2})$, where $\Lambda$ is an
ultraviolet Poincaré invariant cut-off and $\mu$ is the renormalization scale
imposed, so that $S^{-1}(p)|_{p^{2}=\mu^{2}}=\gamma\cdot p+m(\mu)$, with the
common choice $\Lambda\gg\mu$. Furthermore, $\Delta_{\mu\nu}(q)$ is the
dressed gluon propagator in Landau gauge and
$\Gamma^{a}_{\mu}(k,p)=\Gamma_{\mu}(k,p)\,t^{a}$ is the dressed quark-gluon
vertex, where $t^{a}=\lambda^{a}/2$ are the SU(3)c group generators and $a,b$
represent color indices.

The solutions of the DSE (1) can be most generally decomposed into scalar and
vector pieces,

$S(p)=\frac{1}{i\gamma\cdot
p\,A(p^{2})+B(p^{2})}=\,\frac{Z(p^{2})}{i\gamma\cdot p+M(p^{2})}\ ,$ (2)

where $Z(p^{2},\Lambda^{2},\mu^{2})=1/A(p^{2},\Lambda^{2},\mu^{2})$ and
$M(p^{2})=B(p^{2},\mu^{2},\Lambda^{2})/A(p^{2},\mu^{2},\Lambda^{2})$ are the
flavor dependent, running wave renormalization and mass functions,
respectively.

The complete vertex, $\Gamma_{\mu}(k,p)$, can be expanded in terms of four
non-transverse and eight transverse covariant vector structures [25],

$\displaystyle\Gamma_{\mu}(k,p)$
$\displaystyle=\,\Gamma^{L}_{\mu}(k,p)+\Gamma^{T}_{\mu}(k,p)$ $\displaystyle=$
$\displaystyle\
\sum_{i=1}^{4}\,\lambda_{i}(k,p)L^{i}_{\mu}(k,p)+\sum_{i=1}^{8}\tau_{i}(k,p)T^{i}_{\mu}(k,p)\,,$
(3)

in which $p$ is the incoming and $k$ the outgoing quark momentum and the gluon
momentum, $q=k-p$, flows into the vertex. The transverse vertex is naturally
defined by $q\cdot\Gamma^{T}(k,p)=0$. We work with the vector base for
$L^{i}_{\mu}(k,p)$ and $T^{i}_{\mu}(k,p)$ defined in Ref. [26].

The form factors of the quark-gluon vertex, in particular of the longitudinal
components, have been explored in pQCD and in nonperturbative approaches, see
for instance Refs. [27, 28, 29, 30, 31, 32, 33, 21, 34, 35, 36, 37] or Ref.
[24] for a more detailed bibliography on the fermion-boson vertex. Recently,
we employed two transverse Slavnov-Taylor identities [38], which express color
gauge invariance and Lorentz covariance and constrain the transverse quark-
gluon vertex, to derive the eight $\tau_{i}(k,p)$ form factors in QCD [24].
Along with the known expressions for $\lambda_{i}(k,p)$ [31, 32] we found that
$\Gamma_{\mu}(k,p)$ in Eq. (3) can be described by the following set of form
factors:

$\displaystyle\lambda_{1}(k,p)$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\,G(q^{2})X_{0}(q^{2})\left[A(k^{2})+A(p^{2})\right]\
,$ (4) $\displaystyle\lambda_{2}(k,p)$ $\displaystyle=$ $\displaystyle
G(q^{2})X_{0}(q^{2})\,\frac{A(k^{2})-A(p^{2})}{k^{2}-p^{2}}\ ,$ (5)
$\displaystyle\lambda_{3}(k,p)$ $\displaystyle=$ $\displaystyle
G(q^{2})X_{0}(q^{2})\,\frac{B(k^{2})-B(p^{2})}{k^{2}-p^{2}}\ ,$ (6)
$\displaystyle\lambda_{4}(k,p)$ $\displaystyle=$ $\displaystyle\ 0\ ,$ (7)
$\displaystyle\tau_{1}(k,p)$ $\displaystyle=$
$\displaystyle-\frac{Y_{1}}{2(k^{2}-p^{2})\nabla(k,p)}\ ,$ (8)
$\displaystyle\tau_{2}(k,p)$ $\displaystyle=$
$\displaystyle-\frac{Y_{5}-3Y_{3}}{4(k^{2}-p^{2})\nabla(k,p)}\ ,$ (9)
$\displaystyle\tau_{3}(k,p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,G(q^{2})X_{0}(q^{2})\left[\frac{A(k^{2})-A(p^{2})}{k^{2}-p^{2}}\right]$
(10) $\displaystyle+$
$\displaystyle\frac{Y_{2}}{4\nabla(k,p)}-\frac{(k+p)^{2}(Y_{3}-Y_{5})}{8(k^{2}-p^{2})\nabla(k,p)}\
,$ $\displaystyle\tau_{4}(k,p)$ $\displaystyle=$
$\displaystyle-\frac{6Y_{4}+Y_{6}^{A}}{8\nabla(k,p)}-\frac{(k+p)^{2}Y_{7}^{S}}{8(k^{2}-p^{2})\nabla(k,p)}\
,\hskip 14.22636pt$ (11) $\displaystyle\tau_{5}(k,p)$ $\displaystyle=$
$\displaystyle-G(q^{2})X_{0}(q^{2})\left[\frac{B(k^{2})-B(p^{2})}{k^{2}-p^{2}}\right]$
(12) $\displaystyle-$ $\displaystyle\frac{2Y_{4}+Y_{6}^{A}}{2(k^{2}-p^{2})}\
,$ $\displaystyle\tau_{6}(k,p)$ $\displaystyle=$
$\displaystyle\frac{(k-p)^{2}Y_{2}}{4(k^{2}-p^{2})\nabla(k,p)}-\frac{Y_{3}-Y_{5}}{8\nabla(k,p)}\
,$ (13)

Figure 1: Form factors of the longitudinal vertex $\Gamma^{L}_{\mu}(k,p)$ (3)
as functions of the quark momenta, $k^{2}$ and $p^{2}$ [in GeV2], and for the
angle $\theta=2\pi/3$.

$\displaystyle\tau_{7}(k,p)$ $\displaystyle=$
$\displaystyle\frac{q^{2}(6Y_{4}+Y_{6}^{A})}{4(k^{2}-p^{2})\nabla(k,p)}+\frac{Y_{7}^{S}}{4\nabla(k,p)}\
,$ (14) $\displaystyle\tau_{8}(k,p)$ $\displaystyle=$
$\displaystyle-G(q^{2})X_{0}(q^{2})\left[\frac{A(k^{2})-A(p^{2})}{k^{2}-p^{2}}\right]$
(15) $\displaystyle-$ $\displaystyle\frac{2Y_{8}^{A}}{k^{2}-p^{2}}\ .$

Figure 2: Form factors of the transverse vertex $\Gamma^{T}_{\mu}(k,p)$ (3) as
functions of the quark momenta, $k^{2}$ and $p^{2}$ [in GeV2], and for the
angle $\theta=2\pi/3$.

In Eqs. (8) to (15) the Gram determinant is defined by
$\nabla(k,p)=k^{2}p^{2}-(k\cdot p)^{2}$. The form factors $\lambda_{i}(k,p)$,
$i=1,2,3$, and $\tau_{i}(k,p)$, $i=3,5,8$, are proportional to the ghost-
dressing function $G(q^{2})$ which is renormalized as $G(\mu^{2})=1$.
Moreover, $X_{0}(q^{2})$ is the leading form factor of the quark-ghost
scattering amplitude, $H^{a}(k,p)=H(k,p)t^{a}$; see, e.g., Refs. [32, 33] for
details. The a priori unknown scalar functions, $Y_{i}^{A,S}$, are form
factors we introduce to decompose a four-point function that appears in the
transverse Slavnov-Taylor identities and which involves a non-local vector
vertex and a Wilson line to preserve gauge invariance. We refer to the
discussion in Ref. [38] and merely stress that the $Y_{i}^{A,S}$ functions
have been constrained by us in Ref. [39] with the vertex ansatz of Ref. [29]
and insisting on multiplicative renormalizability.

We solve the DSE (1) with numerical input from lattice QCD [40, 41] for the
gluon and ghost dressing functions, $\Delta(q^{2})$ and $G(q^{2}$), and with
the vertex defined by Eqs. (3) to (15). The solution for the vector and scalar
components of the quark propagator, $A(p^{2})$ and $B(p^{2})$ respectively,
are then used to compute the form factors $\lambda_{i}(k,p)$ and
$\tau_{i}(k,p)$. We present them in Figs. 1 and 2 as functions of the momenta
squared $k^{2}$ and $p^{2}$ and for the kinematic configuration:
$\cos\theta=k\cdot p/|k||p|=-1/2$, $\theta=2\pi/3$.

The functional form of the $\lambda_{i}(k,p)$ form factors is similar to that
found in Refs. [32, 33] though with differences in magnitude, as those studies
exclusively concentrated on $\Gamma^{L}_{\mu}(k,p)$. Therefore, some strength
is shifted from the transverse to the longitudinal vertex and a direct
comparison is difficult. As observed in Ref. [24], the dominating contribution
of the transverse vertex in the gap equation (1), and therefore to DCSB, is
due to the form factors $\tau_{4}(k,p)$ and $\tau_{7}(k,p)$. We multiply them
and also $\tau_{1}(k,p)$ by $q^{2}$ in Fig. 2 in order to regularize a
singular behavior at the origin. Note that this poses no problem in the
numerical treatment of the quark DSE since kinematic factors in the integral
kernel have an analogous regularizing effect.

A more comprehensive treatment of the these form factors taking into account
other angles $\theta$, which include the soft-gluon and symmetric quark limit,
is underway. This also requires the contributions of the sub-leading form
factors $X_{1}(k,p)$, $X_{2}(k,p)$ and $X_{3}(k,p)$ that parametrize the
quark-ghost scattering amplitude.

Acknowledgments

B.E.’s research activity is funded by FAPESP grant no. 2018/20218-4 and CNPq
grant no. 428003/2018-4. F.E.S. is a CAPES-PNPD postdoctoral fellow, grant no.
88882.314890/2013-01, L.A. was a FAPESP postdoctoral fellow supported by grant
no. 2018/17643-5, and R.C.S. is a CAPES PhD fellow. A.B. acknowledges funding
from the Coordinación de la Investigación Científica (CIC) of the University
of Michoacán and CONACyT, Mexico, through grant nos. 4.10 and CB2014-22117,
respectively.

## References

  * $1$. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345-358 (1961) doi:10.1103/PhysRev.122.345
  * $2$. A. Bashir, L. Chang, I. C. Cloët, B. El-Bennich, Y. X. Liu, C. D. Roberts and P. C. Tandy, Commun. Theor. Phys. 58, 79-134 (2012) doi:10.1088/0253-6102/58/1/16
  * $3$. C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33, 477-575 (1994) doi:10.1016/0146-6410(94)90049-3
  * $4$. P. Maris and P. C. Tandy, Phys. Rev. C 60, 055214 (1999) doi:10.1103/PhysRevC.60.055214
  * $5$. S. x. Qin, L. Chang, Y.-x. Liu, C. D. Roberts and D. J. Wilson, Phys. Rev. C 84, 042202 (2011) doi:10.1103/PhysRevC.84.042202
  * $6$. E. Rojas, B. El-Bennich and J. P. B. C. de Melo, Phys. Rev. D 90, 074025 (2014) doi:10.1103/PhysRevD.90.074025
  * $7$. K. Raya, L. Chang, A. Bashir, J. J. Cobos-Martínez, L. X. Gutiérrez-Guerrero, C. D. Roberts and P. C. Tandy, Phys. Rev. D 93, no.7, 074017 (2016) doi:10.1103/PhysRevD.93.074017
  * $8$. B. El-Bennich, G. Krein, E. Rojas and F. E. Serna, Few Body Syst. 57, no.10, 955-963 (2016) doi:10.1007/s00601-016-1133-x
  * $9$. F. F. Mojica, C. E. Vera, E. Rojas and B. El-Bennich, Phys. Rev. D 96, no.1, 014012 (2017) doi:10.1103/PhysRevD.96.014012
  * $10$. I. C. Cloët, G. Eichmann, B. El-Bennich, T. Klähn and C. D. Roberts, Few Body Syst. 46, 1-36 (2009) doi:10.1007/s00601-009-0015-x
  * $11$. G. Eichmann, R. Alkofer, A. Krassnigg and D. Nicmorus, Phys. Rev. Lett. 104, 201601 (2010) doi:10.1103/PhysRevLett.104.201601
  * $12$. J. Segovia, B. El-Bennich, E. Rojas, I. C. Cloët, C. D. Roberts, S. S. Xu and H. S. Zong, Phys. Rev. Lett. 115, no.17, 171801 (2015) doi:10.1103/PhysRevLett.115.171801
  * $13$. C. Chen, B. El-Bennich, C. D. Roberts, S. M. Schmidt, J. Segovia and S. Wan, Phys. Rev. D 97, no.3, 034016 (2018) doi:10.1103/PhysRevD.97.034016
  * $14$. G. Eichmann, C. S. Fischer and H. Sanchis-Alepuz, Phys. Rev. D 94, no.9, 094033 (2016) doi:10.1103/PhysRevD.94.094033
  * $15$. G. Eichmann, H. Sanchis-Alepuz, R. Williams, R. Alkofer and C. S. Fischer, Prog. Part. Nucl. Phys. 91, 1-100 (2016) doi:10.1016/j.ppnp.2016.07.001
  * $16$. F. E. Serna, B. El-Bennich and G. Krein, Phys. Rev. D 96, no.1, 014013 (2017) doi:10.1103/PhysRevD.96.014013
  * $17$. F. E. Serna, R. C. da Silveira, J. J. Cobos-Martínez, B. El-Bennich and E. Rojas, Eur. Phys. J. C 80, no.10, 955 (2020) doi:10.1140/epjc/s10052-020-08517-3
  * $18$. B. El-Bennich and F. E. Serna, PoS CHARM2020, 025 (2021) doi:10.22323/1.385.0025
  * $19$. R. Williams, C. S. Fischer and W. Heupel, Phys. Rev. D 93, no.3, 034026 (2016) doi:10.1103/PhysRevD.93.034026
  * $20$. L. Chang and C. D. Roberts, Phys. Rev. C 85, 052201 (2012) doi:10.1103/PhysRevC.85.052201
  * $21$. D. Binosi, L. Chang, J. Papavassiliou, S. X. Qin and C. D. Roberts, Phys. Rev. D 95, no.3, 031501 (2017) doi:10.1103/PhysRevD.95.031501
  * $22$. W. Heupel, G. Eichmann and C. S. Fischer, Phys. Lett. B 718, 545-549 (2012) doi:10.1016/j.physletb.2012.11.009
  * $23$. F. E. Serna, C. Chen and B. El-Bennich, Phys. Rev. D 99, no.9, 094027 (2019) doi:10.1103/PhysRevD.99.094027
  * $24$. L. Albino, A. Bashir, B. El-Bennich, E. Rojas, F. E. Serna and R. C. da Silveira, JHEP 11, 196 (2021) doi:10.1007/JHEP11(2021)196
  * $25$. J. S. Ball and T. W. Chiu, Phys. Rev. D 22, 2542 (1980) doi:10.1103/PhysRevD.22.2542
  * $26$. A. Kizilersu, M. Reenders and M. R. Pennington, Phys. Rev. D 52 (1995), 1242-1259 doi:10.1103/PhysRevD.52.1242
  * $27$. R. Alkofer, C. S. Fischer, F. J. Llanes-Estrada and K. Schwenzer, Annals Phys. 324, 106-172 (2009) doi:10.1016/j.aop.2008.07.001
  * $28$. R. Williams, Eur. Phys. J. A 51, no.5, 57 (2015) doi:10.1140/epja/i2015-15057-4
  * $29$. A. Bashir, R. Bermudez, L. Chang and C. D. Roberts, Phys. Rev. C 85, 045205 (2012) doi:10.1103/PhysRevC.85.045205
  * $30$. R. Bermudez, L. Albino, L. X. Gutiérrez-Guerrero, M. E. Tejeda-Yeomans and A. Bashir, Phys. Rev. D 95, no.3, 034041 (2017) doi:10.1103/PhysRevD.95.034041
  * $31$. E. Rojas, J. P. B. C. de Melo, B. El-Bennich, O. Oliveira and T. Frederico, JHEP 10, 193 (2013) doi:10.1007/JHEP10(2013)193
  * $32$. A. C. Aguilar, J. C. Cardona, M. N. Ferreira and J. Papavassiliou, Phys. Rev. D 96, no.1, 014029 (2017) doi:10.1103/PhysRevD.96.014029
  * $33$. A. C. Aguilar, J. C. Cardona, M. N. Ferreira and J. Papavassiliou, Phys. Rev. D 98, no.1, 014002 (2018) doi:10.1103/PhysRevD.98.014002
  * $34$. O. Oliveira, T. Frederico and W. de Paula, Eur. Phys. J. C 80, no.5, 484 (2020) doi:10.1140/epjc/s10052-020-8037-0
  * $35$. M. Peláez, M. Tissier and N. Wschebor, Phys. Rev. D 92, no.4, 045012 (2015) doi:10.1103/PhysRevD.92.045012
  * $36$. J. I. Skullerud, P. O. Bowman, A. Kızılersü, D. B. Leinweber and A. G. Williams, JHEP 04, 047 (2003) doi:10.1088/1126-6708/2003/04/047
  * $37$. A. Kızılersü, O. Oliveira, P. J. Silva, J. I. Skullerud and A. Sternbeck, Phys. Rev. D 103, no.11, 114515 (2021) doi:10.1103/PhysRevD.103.114515
  * $38$. H. x. He, Phys. Rev. D 80, 016004 (2009) doi:10.1103/PhysRevD.80.016004
  * $39$. L. Albino, A. Bashir, L. X. G. Guerrero, B. E. Bennich and E. Rojas, Phys. Rev. D 100, no.5, 054028 (2019) doi:10.1103/PhysRevD.100.054028
  * $40$. D. Dudal, O. Oliveira and P. J. Silva, Annals Phys. 397, 351-364 (2018) doi:10.1016/j.aop.2018.08.019
  * $41$. A. G. Duarte, O. Oliveira and P. J. Silva, Phys. Rev. D 94 (2016) no.1, 014502 doi:10.1103/PhysRevD.94.014502