# The fate of Landau levels under $\delta$-interactions

Jussi Behrndt  ,  Markus Holzmann  ,  Vladimir Lotoreichik  and  Georgi
Raikov† Institut für Angewandte Mathematik
Technische Universität Graz
Steyrergasse 30
8010 Graz
Austria<EMAIL_ADDRESS>Institut für Angewandte Mathematik
Technische Universität Graz
Steyrergasse 30
8010 Graz
Austria<EMAIL_ADDRESS>Department of Theoretical Physics
Nuclear Physics Institute
Czech Academy of Sciences
250 68 Řež near Prague
Czech Republic<EMAIL_ADDRESS>Facultad de Matemáticas
Pontificia Universidad Católica de Chile
Av. Vicuña Mackenna 4860
Santiago de Chile
Chile  Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev Str., bl. 8,
1113 Sofia, Bulgaria

###### Abstract.

We consider the self-adjoint Landau Hamiltonian $H_{0}$ in
$L^{2}({\mathbb{R}}^{2})$ whose spectrum consists of infinitely degenerate
eigenvalues $\Lambda_{q}$, $q\in{\mathbb{Z}}_{+}$, and the perturbed operator
$H_{\upsilon}=H_{0}+\upsilon\delta_{\Gamma}$, where
$\Gamma\subset{\mathbb{R}}^{2}$ is a regular Jordan $C^{1,1}$-curve and
$\upsilon\in L^{p}(\Gamma;{\mathbb{R}})$, $p>1$, has a constant sign. We
investigate ${\rm ker}(H_{\upsilon}-\Lambda_{q})$, $q\in{\mathbb{Z}}_{+}$, and
show that generically

$0\leq{\rm dim\,ker}(H_{\upsilon}-\Lambda_{q})-{\rm
dim\,ker}(T_{q}(\upsilon\delta_{\Gamma}))<\infty,$

where $T_{q}(\upsilon\delta_{\Gamma})=p_{q}(\upsilon\delta_{\Gamma})p_{q}$, is
an operator of Berezin-Toeplitz type, acting in
$p_{q}L^{2}({\mathbb{R}}^{2})$, and $p_{q}$ is the orthogonal projection on
${\rm ker}\,(H_{0}-\Lambda_{q})$. If $\upsilon\neq 0$ and $q=0$, we prove that
${\rm ker}\,(T_{0}(\upsilon\delta_{\Gamma}))=\\{0\\}$. If $q\geq 1$ and
$\Gamma={\mathcal{C}}_{r}$ is a circle of radius $r$, we show that ${\rm
dim\,ker}(T_{q}(\delta_{{\mathcal{C}}_{r}}))\leq q$, and the set of
$r\in(0,\infty)$ for which ${\rm
dim\,ker}(T_{q}(\delta_{{\mathcal{C}}_{r}}))\geq 1$ is infinite and discrete.

AMS 2010 Mathematics Subject Classification: 81Q10, 47A10, 47A55, 47N50

Keywords: Landau Hamiltonian, $\delta$-interactions, perturbations of
eigenspaces,
Berezin-Toeplitz operators, Laguerre polynomials

##  1\. Introduction

The aim of this article is to study the spectral type of the Landau levels of
the singularly perturbed Landau Hamiltonian

(1.1) $H_{\upsilon}=(-i\nabla-A)^{2}+\upsilon\delta_{\Gamma},$

where $A(x):=\tfrac{b}{2}(-x_{2},x_{1})$,
$x=(x_{1},x_{2})\in{\mathbb{R}}^{2}$, is a magnetic potential which generates
a constant scalar magnetic field $b>0$, and the singular perturbation is
supported on a $C^{1,1}$-smooth Jordan curve $\Gamma\subset{\mathbb{R}}^{2}$
and has strength $\upsilon\in L^{p}(\Gamma;{\mathbb{R}})$, $p>1$. The
expression (1.1) is of formal nature here and the self-adjoint operator
$H_{\upsilon}$ will be defined rigorously via the corresponding quadratic form
in Section 2. If the singular perturbation is absent, that is, $\upsilon=0$ in
(1.1), then the operator reduces to the usual self-adjoint Landau Hamiltonian
$H_{0}=(-i\nabla-A)^{2}$. It is well known that

$\sigma(H_{0})=\sigma_{\rm
ess}(H_{0})=\bigcup_{q\in{\mathbb{Z}}_{+}}\left\\{\Lambda_{q}\right\\},$

where the Landau levels $\Lambda_{q}:=b(2q+1)$,
$q\in{\mathbb{Z}}_{+}=\\{0,1,2,\dots\\}$, are eigenvalues of $H_{0}$ of
infinite multiplicity. Under our assumption on $\Gamma$ and $\upsilon$ it
turns out that $H_{\upsilon}$ is a compact perturbation of $H_{0}$ in the
resolvent sense and hence the essential spectrum remains invariant, that is,

$\sigma_{\rm ess}(H_{\upsilon})=\sigma_{\rm
ess}(H_{0})=\bigcup_{q\in{\mathbb{Z}}_{+}}\left\\{\Lambda_{q}\right\\}.$

In the spectral gaps $(\Lambda_{q-1},\Lambda_{q})$, where
$q\in{\mathbb{Z}}_{+}$ and $\Lambda_{-1}:=-\infty$, of $H_{0}$ there may
appear discrete eigenvalues of $H_{\upsilon}$ which can only accumulate at the
Landau levels $\Lambda_{q}$, $q\in{\mathbb{Z}}_{+}$. Some results on the
asymptotic distribution near any fixed $\Lambda_{q}$ of these discrete
eigenvalues were obtained in [7]. In particular, it was shown that if either
$\upsilon\geq 0$ or $\upsilon\leq 0$ on $\Gamma$, $v\not\equiv 0$, and certain
additional regularity assumptions hold, then in a neighbourhood of any
$\Lambda_{q}$ there are infinitely many discrete eigenvalues of $H_{\upsilon}$
and their accumulation rate to the Landau levels is described in terms of the
logarithmic capacity of the interaction support; cf. [10, 20, 35, 37] for
similar results on the clustering of eigenvalues of Landau Hamiltonians on
unbounded domains with Dirichlet, Neumann, and Robin boundary conditions.

Our main objective in this article is to obtain a deeper understanding of the
spectral points $\Lambda_{q}$, $q\in{\mathbb{Z}}_{+}$, of the perturbed
operator $H_{\upsilon}$; in other words, we are interested in the fate of the
Landau levels $\Lambda_{q}$ under $\delta$-potentials of strength $\upsilon$.
In particular, we would like to know what part of the infinite-dimensional
eigenspace ${\rm ker}\,(H_{0}-\Lambda_{q})$ is transformed into an eigenspace
${\rm ker}\,(H_{\upsilon}-\Lambda_{q})$ under the singular perturbation
$\upsilon\delta_{\Gamma}$.

The analogous problem on the fate of Landau levels under regular perturbations
of the Landau Hamiltonian $H_{0}$ was investigated earlier in [28]. Roughly
speaking, it was shown that for any non-negative potential $V\in
L^{\infty}({\mathbb{R}}^{2};{\mathbb{R}})$, $V\not\equiv 0$, with
$\|V\|_{L^{\infty}({\mathbb{R}}^{2})}<2b$ one has

(1.2) ${\rm ker}\,(H_{0}\pm V-\Lambda_{q})=\left\\{0\right\\}.$

The assumption that $V$ is sign-definite is essential here. In fact, in [28]
it was also shown that for every $q\in{\mathbb{Z}}_{+}$, there exists a
compactly supported $V\in L^{\infty}({\mathbb{R}}^{2};{\mathbb{R}})$ with
$\|V\|_{L^{\infty}({\mathbb{R}}^{2})}<b$ of non-constant sign, such that

${\rm dim\,ker}\,(H_{0}+V-\Lambda_{q})=\infty.$

The key idea in the proof of (1.2) is to show ${\rm ker}\,(H_{0}\pm
V-\Lambda_{q})\subset{\rm ker}\,(\hat{T}_{q}(V))$ and ${\rm
ker}\,(\hat{T}_{q}(V))=\\{0\\}$ if $\|V\|_{L^{\infty}({\mathbb{R}}^{2})}<2b$,
where $\hat{T}_{q}(V):=p_{q}Vp_{q}$ is a Berezin-Toeplitz type operator and
$p_{q}$ denotes the orthogonal projection onto the eigenspace ${\rm
ker}\,(H_{0}-\Lambda_{q})$.

In our treatment of the perturbed Landau Hamiltonian with a $\delta$-potential
in (1.1) a singular analogue of the Berezin-Toeplitz operator plays a key
role; cf. the discussion below (2.11) for more details and references. More
precisely, if $\tau$ is the restriction operator onto $\Gamma$ we consider the
operator $T_{q}(\upsilon\delta_{\Gamma}):=(\tau p_{q})^{*}\upsilon(\tau
p_{q})$ and in our main results we show that the analysis of ${\rm
ker}\,(H_{\pm\upsilon}-\Lambda_{q})$ can be reduced to that of ${\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$. Namely, under the definiteness
assumption $\upsilon\geq 0$ we prove in Theorem 3.1 that

${\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))\subset{\rm
ker}\,(H_{\pm\upsilon}-\Lambda_{q}),\quad q\in{\mathbb{Z}}_{+},$

and $0\leq{\rm dim\,ker}\,(H_{\pm\upsilon}-\Lambda_{q})-{\rm
dim\,ker}\,(T_{q}(\upsilon\delta_{\Gamma}))<\infty$ for all
$q\in{\mathbb{Z}}_{+}$. Furthermore, if $\|\upsilon\|_{L^{p}(\Gamma)}$ is not
too large it turns out that

${\rm ker}\,(H_{\pm\upsilon}-\Lambda_{q})={\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma})),\quad q\in{\mathbb{Z}}_{+}.$

As we will see, for $\upsilon\geq 0$ the kernel of
$T_{q}(\upsilon\delta_{\Gamma})$ consists of eigenfunctions of $H_{0}$ for
$\Lambda_{q}$ which vanish on the support of $\upsilon$. Intuitively, it is
clear that such functions $u\in{\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$
are also eigenfunctions of $H_{\upsilon}$, as in this case one formally has
$\upsilon\delta_{\Gamma}u=0$, i.e. the singular interaction does not have an
effect on $u$, and hence $H_{\upsilon}u=H_{0}u=\Lambda_{q}u$. This allows one
to show with the help of [7, Lemma 3.7] that the kernel of
$T_{q}(\upsilon\delta_{\Gamma})$ is finite-dimensional under the assumption
that $\upsilon$ is strictly positive. Moreover, this connection provides a
direct link to nodal sets for eigenfunctions of $H_{0}$ and the non-emptiness
of ${\rm ker}\,(H_{\upsilon}-\Lambda_{q})$. The above observation means, in
particular, that for all $\upsilon\in L^{p}(\Gamma)$ one has ${\rm
ker}\,(H_{\pm\upsilon}-\Lambda_{q})\neq\\{0\\}$, whenever $\Gamma$ is
contained in a nodal set of an eigenfunction of $H_{0}$. This is in strong
contrast to the case of regular potentials; cf. (1.2). Additionally, for the
first Landau level $\Lambda_{0}$ we find ${\rm
ker}\,(T_{0}(\upsilon\delta_{\Gamma}))=\\{0\\}$ in Theorem 3.6, which leads to

${\rm ker}\,(H_{\upsilon}-\Lambda_{0})=\\{0\\}.$

At present it is not clear if ${\rm
dim\,ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$ can be further estimated for
higher Landau levels and general curves $\Gamma$. However, we find it
worthwhile to discuss the special case that $\Gamma={\mathcal{C}}_{r}$ is a
circle of radius $r\in(0,\infty)$. In this situation we find that

${\rm dim\,ker}\,(T_{q}(\delta_{{\mathcal{C}}_{r}}))\leq q,\quad q=1,2\ldots,$

and the radii $r\in(0,\infty)$ for which ${\rm
dim\,ker}(T_{q}(\delta_{{\mathcal{C}}_{r}}))\geq 1$ form an infinite and
discrete set; cf. Theorem 3.9. The idea to prove these results is again to
study when a circle is a nodal set for an eigenfunction of $H_{0}$ and use the
fact, that this can be characterized explicitly in terms of zeros of Laguerre
polynomials. Translating this observation to the spectral points $\Lambda_{q}$
of the perturbed Landau operator (1.1) leads to a precise understanding of the
fate of Landau levels under $\delta$-perturbations supported on circles. For
example, if $\upsilon\not\equiv 0$ and $\upsilon\geq 0$ on some non-empty open
subset of $\Gamma$, then $\Lambda_{q}$ can only be an eigenvalue of finite
multiplicity; cf. Section 3.2 for details.

The article is organized as follows. In the next section we introduce the
Landau Hamiltonian perturbed by singular $\delta$-interactions. In Section 3
we formulate our main results. Section 4 contains some auxiliary facts from
the spectral theory of the Landau Hamiltonian. Finally, in Section 5 we prove
our main theorems.

Note by J. Behrndt, M. Holzmann, and V. Lotoreichik. Our coauthor Georgi
Raikov passed away unexpectedly on March 9, 2021, while the work on this
manuscript was in its active phase. The topics in the present paper result
from various discussions with Georgi dating back to 2018 and the first draft
of this paper was written by him. When preparing the final text it was our aim
to preserve Georgis original handwriting and genuine style. This paper is a
tribute to the memory of Georgi Raikov, an influential mathematician,
respected colleague, and good friend. We will miss him.

Acknowledgements. We are indebted to Vincent Bruneau and Grigori Rozenblum for
very helpful discussions and literature hints. We are also particularly
grateful to Lilia Simeonova who kindly provided us various handwritten notes
and additional comments by Georgi on earlier versions of this manuscript.
Furthermore, we would like to thank the anonymous referee for a very careful
reading of our manuscript and for pointing out various improvements. J.
Behrndt and M. Holzmann gratefully acknowledge financial support by the
Austrian Science Fund (FWF): P 33568-N. V. Lotoreichik was supported by the
Czech Science Foundation project 21-07129S. This publication is based upon
work from COST Action CA 18232 MAT-DYN-NET, supported by COST (European
Cooperation in Science and Technology), www.cost.eu.

##  2\. Landau Hamiltonians with $\delta$-interactions supported on curves

Let $b>0$ be a constant scalar magnetic field. Then

$A(x):=\frac{b}{2}(-x_{2},x_{1}),\quad x=(x_{1},x_{2})\in{\mathbb{R}}^{2},$

is a magnetic potential which generates $b$, i.e.

$b=\frac{\partial A_{2}}{\partial x_{1}}-\frac{\partial A_{1}}{\partial
x_{2}}.$

Denote by

$\Pi(A)=(\Pi_{1}(A),\Pi_{2}(A)):=-i\nabla-A$

the magnetic gradient. In the following, for
$\ell=(\ell_{1},\ell_{2})\in{\mathbb{Z}}_{+}^{2}$ with
${\mathbb{Z}}_{+}=\\{0,1,2,\dots\\}$ the notations $|\ell|:=\ell_{1}+\ell_{2}$
and $\Pi(A)^{\ell}:=\Pi_{1}(A)^{\ell_{1}}\Pi_{2}(A)^{\ell_{2}}$ are used. For
an open non-empty set $\Omega\subset{\mathbb{R}}^{2}$ and an index
$s\in{\mathbb{Z}}_{+}$ introduce the magnetic Sobolev spaces

${\rm
H}_{A}^{s}(\Omega):=\left\\{u\in{\mathcal{D}}^{\prime}(\Omega)\,|\,\Pi(A)^{\ell}u\in
L^{2}({\mathbb{R}}^{2}),\;\ell\in{\mathbb{Z}}_{+}^{2},\;0\leq|\ell|\leq
s\right\\}$

with a norm defined by

$\|u\|^{2}_{{\rm
H}_{A}^{s}(\Omega)}:=\sum_{\ell\in{\mathbb{Z}}_{+}^{2}:0\leq|\ell|\leq
s}\int_{\Omega}|\Pi(A)^{\ell}u|^{2}dx.$

Throughout this paper it is assumed that $\Gamma\subset{\mathbb{R}}^{2}$ is a
$C^{1,1}$-smooth Jordan curve, i.e. a closed simple curve which is mapped onto
the unit circle by a $C^{1,1}$-smooth diffeomorphism. Let ${\rm
H}^{1/2}(\Gamma)$ be the $L^{2}$-based Sobolev space of order $1/2$ on
$\Gamma$.

The Dirichlet trace operator $\tau:{\rm H}^{1}_{A}({\mathbb{R}}^{2})\to{\rm
H}^{1/2}(\Gamma)$ is the continuous extension of the restriction map

${\rm H}^{1}_{A}({\mathbb{R}}^{2})\cap C({\mathbb{R}}^{2})\ni u\mapsto
u_{|\Gamma}.$

Assume that $\upsilon\in L^{p}(\Gamma;{\mathbb{R}})$ with $p>1$. Denote by
$H_{\upsilon}$ the self-adjoint operator generated in
$L^{2}({\mathbb{R}}^{2})$ by the symmetric, densely defined, lower-bounded,
and closed quadratic form

(2.1) $\int_{{\mathbb{R}}^{2}}|\Pi(A)u|^{2}dx+\int_{\Gamma}\upsilon|\tau
u|^{2}\,ds,\quad u\in{\rm H}_{A}^{1}({\mathbb{R}}^{2});$

cf. Appendix A. In particular, for $\upsilon=0$ one obtains

$H_{0}=\Pi_{1}(A)^{2}+\Pi_{2}(A)^{2}=(-i\nabla-A)^{2},$

which is the Landau Hamiltonian, self-adjoint on ${\rm
H}_{A}^{2}({\mathbb{R}}^{2})$ (see, e.g., [18, Appendix A]), and essentially
self-adjoint on $C_{0}^{\infty}({\mathbb{R}}^{2})$ (see [30, Theorem 2]). As
mentioned in the introduction one has

$\sigma(H_{0})=\sigma_{\rm
ess}(H_{0})=\bigcup_{q\in{\mathbb{Z}}_{+}}\left\\{\Lambda_{q}\right\\},$

where $\Lambda_{q}:=b(2q+1)$, $q\in{\mathbb{Z}}_{+}$, are the Landau levels
which are eigenvalues of $H_{0}$ of infinite multiplicity (see [4, 17, 29]).
In particular,

$\inf\sigma(H_{0})=\Lambda_{0}=b>0.$

Note that integration by parts allows to find an explicit characterization of
$H_{\upsilon}$. Denote by $\Omega_{\rm in}$ and $\Omega_{\rm ex}$ the interior
and the exterior of $\Gamma$, respectively, and by $\nu$ the unit normal
vector on $\Gamma$ pointing outwards of $\Omega_{\rm in}$. Then one can show
in the same way as in [7, Section 4] that

(2.2)
$\begin{split}(H_{\upsilon}u)_{\natural}&=(-i\nabla-A)^{2}u_{\natural}\quad\text{
for }\natural={\rm in},{\rm ex}\\\
{\mathfrak{D}}(H_{\upsilon})&=\bigg{\\{}u\in{\rm
H}_{A}^{1}({\mathbb{R}}^{2})|(-i\nabla-A)^{2}u_{\natural}\in
L^{2}(\Omega_{\natural})\text{ for }\natural={\rm in},{\rm ex},\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\frac{\partial u_{\rm
ex}}{\partial\nu}-\frac{\partial u_{\rm in}}{\partial\nu}=\upsilon
u\,\,\text{on}\,\,\Gamma\bigg{\\}}.\end{split}$

In the next lemma it is shown that the difference of the resolvents of $H_{0}$
and $H_{\upsilon}$ is compact, which implies that the essential spectra of
$H_{0}$ and $H_{\upsilon}$ coincide. In order to formulate the lemma, define
for $\lambda>-b$ the operator

(2.3)
$G_{\upsilon}(\lambda):=|\upsilon|^{1/2}\tau(H_{0}+\lambda)^{-1/2}:L^{2}({\mathbb{R}}^{2})\to
L^{2}(\Gamma).$

######  Lemma 2.1.

Let $\lambda>-b$ and set $J_{\upsilon}:={\rm sign}\,\upsilon$. Then
$G_{\upsilon}(\lambda)$ is compact and there exists $\lambda_{0}>-b$ such that
for all $\lambda>\lambda_{0}$ the resolvent difference of $H_{0}$ and
$H_{\upsilon}$ is a compact operator in $L^{2}({\mathbb{R}}^{2})$ and admits
the factorization

$\begin{split}&(H_{\upsilon}+\lambda)^{-1}-(H_{0}+\lambda)^{-1}\\\
&\quad=-(H_{0}+\lambda)^{-1/2}G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda)(I+G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda))^{-1}(H_{0}+\lambda)^{-1/2}.\end{split}$

In particular, one has

(2.4) $\sigma_{\rm ess}(H_{0})=\sigma_{\rm
ess}(H_{\upsilon})=\bigcup_{q\in{\mathbb{Z}}_{+}}\left\\{\Lambda_{q}\right\\}.$

###### Proof.

Note first that the operator $G_{\upsilon}(\lambda)$ in (2.3) depends only on
$|\upsilon|$ but not on the sign of $\upsilon$. The assumption $\upsilon\in
L^{p}(\Gamma;{\mathbb{R}})$ with $p>1$, and the compactness of the trace ${\rm
H}^{1}_{A}({\mathbb{R}}^{2})\rightarrow L^{r}(\Gamma)$ for any $r>1$ (see [33,
Section 2.6, Theorem 6.2]), easily imply that $G_{\upsilon}(\lambda)$ is
compact, and

$\begin{split}\|G_{\upsilon}(\lambda)\|^{2}&=\sup_{0\neq w\in
L^{2}({\mathbb{R}}^{2})}\frac{\||\upsilon|^{1/2}\tau(H_{0}+\lambda)^{-1/2}w\|^{2}_{L^{2}(\Gamma)}}{\|w\|^{2}_{L^{2}({\mathbb{R}}^{2})}}\\\
&=\sup_{0\neq u\in{\rm
H}_{A}^{1}({\mathbb{R}}^{2})}\frac{\||\upsilon|^{1/2}\tau
u\|^{2}_{L^{2}(\Gamma)}}{\|(H_{0}+\lambda)^{1/2}u\|^{2}_{L^{2}({\mathbb{R}}^{2})}}\\\
&=\sup_{0\neq u\in{\rm
H}_{A}^{1}({\mathbb{R}}^{2})}\frac{\int_{\Gamma}|\upsilon||\tau
u|^{2}ds}{\int_{{\mathbb{R}}^{2}}(|\Pi(A)u|^{2}+\lambda|u|^{2})\,dx},\end{split}$

so that the Hölder inequality leads to the estimate

(2.5) $\|G_{\upsilon}(\lambda)\|^{2}\leq
C_{p}(\lambda)\|\upsilon\|_{L^{p}(\Gamma)}$

with

(2.6) $C_{p}(\lambda):=\sup_{0\neq u\in{\rm
H}_{A}^{1}({\mathbb{R}}^{2})}\frac{\left(\int_{\Gamma}|\tau
u|^{2p^{\prime}}ds\right)^{1/p^{\prime}}}{\int_{{\mathbb{R}}^{2}}(|\Pi(A)u|^{2}+\lambda|u|^{2})\,dx},\quad
p^{\prime}:=\frac{p}{p-1},\quad\lambda>-b.$

Set

$J_{\upsilon}={\rm sign}\,{\upsilon}:=\left\\{\begin{aligned}
&\upsilon|\upsilon|^{-1}&&\text{if }\;\upsilon\neq 0,\\\ &0&&\text{if
}\;\upsilon=0.\end{aligned}\right.$

Let $\lambda>-b$. Then one has for $u\in L^{2}(\mathbb{R}^{2})$,
$w:=(H_{0}+\lambda)^{-1/2}u\in{\rm H}^{1}_{A}({\mathbb{R}}^{2})$, and the
self-adjoint operator
$G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda)$

(2.7)
$\big{\langle}(I+G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda))u,u\big{\rangle}_{L^{2}(\mathbb{R}^{2})}=\int_{{\mathbb{R}}^{2}}\big{(}|\Pi(A)w|^{2}+\lambda|w|^{2}\big{)}dx+\int_{\Gamma}\upsilon|\tau
w|^{2}\,ds$

and hence, with (2.1) one concludes that

(2.8) $\lambda>-\inf\sigma(H_{\upsilon})$

is equivalent to

(2.9) $I+G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda)>0.$

Assume in the following that $\lambda>-\inf\sigma(H_{\upsilon})$ is fixed. Due
to the compactness of
$G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda)$ it follows from
(2.9) that the operator

$I+G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda):L^{2}({\mathbb{R}}^{2})\to
L^{2}({\mathbb{R}}^{2})$

is boundedly invertible. Thus, we have

(2.10) $H_{\upsilon}+\lambda=M_{\upsilon}(\lambda)^{*}M_{\upsilon}(\lambda)$

where the operator

$M_{\upsilon}(\lambda):=(I+G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda))^{1/2}(H_{0}+\lambda)^{1/2},\quad{\mathfrak{D}}(M_{\upsilon}(\lambda))={\rm
H}^{1}_{A}(\mathbb{R}^{2}),$

is closed in $L^{2}({\mathbb{R}}^{2})$, as it is a product of a bijective
operator in $L^{2}(\mathbb{R}^{2})$ and $(H_{0}+\lambda)^{1/2}$, which is
bijective from ${\rm H}^{1}_{A}(\mathbb{R}^{2})$ to $L^{2}(\mathbb{R}^{2})$.
The representation in (2.10) can be seen with the help of the quadratic form
in (2.1) associated with $H_{\upsilon}$ and a similar calculation as in (2.7).
Therefore, the operators $H_{\upsilon}+\lambda$ and, hence, $H_{\upsilon}$ are
self-adjoint on

${\mathfrak{D}}(H_{\upsilon}):=\left\\{u\in{\rm
H}^{1}_{A}({\mathbb{R}}^{2})\,|\,M_{\upsilon}(\lambda)u\in{\mathfrak{D}}(M_{\upsilon}(\lambda)^{*})\right\\}$

(see [40, Theorem X.25]). In the above construction we have obtained an
alternative characterisation of the operator domain of $H_{\upsilon}$; cf.
(2.2). Moreover,

$\begin{split}(H_{\upsilon}&+\lambda)^{-1}-(H_{0}+\lambda)^{-1}\\\
&=(H_{0}+\lambda)^{-1/2}\big{(}I+G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda)\big{)}^{-1}(H_{0}+\lambda)^{-1/2}-(H_{0}+\lambda)^{-1}\\\
&=-(H_{0}+\lambda)^{-1/2}G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda)(I+G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda))^{-1}(H_{0}+\lambda)^{-1/2}.\end{split}$

Bearing in mind the compactness of the operator
$G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda)$, and applying a
suitable version of the Weyl theorem on the invariance of the essential
spectrum (see, e.g., [9, Chapter 9, Section 1, Theorem 4]), we obtain (2.4). ∎

Consider the operator

(2.11) $T_{q}(\upsilon\delta_{\Gamma}):=(\tau p_{q})^{*}\upsilon(\tau
p_{q}),\quad q\in{\mathbb{Z}}_{+},$

which can be viewed as a singular analogue of a Berezin-Toeplitz operator. The
relation of Landau Hamiltonians coupled with regular potentials $V$ and the
Berezin-Toeplitz type operators $\hat{T}_{q}(V)=p_{q}Vp_{q}$ was discovered in
[38] and further studied in many publications. Singular Toeplitz operators as
in (2.11) play an important role in modern operator theory and are also of
independent interest. They were already considered in [2], and in connection
with magnetic Laplacians with different types of boundary conditions these
types of operators appear in [20, 21, 37]; we also refer the reader to [3, 10,
11, 13, 16, 32, 36, 41, 42] for some other recent related works in this
context. Note that the operator $T_{q}(\upsilon\delta_{\Gamma})$ corresponds
to the quadratic form

(2.12)
$t_{q}(\upsilon\delta_{\Gamma})[u]:=\int_{\Gamma}\upsilon(x)|u(x)|^{2}ds,\quad
u\in p_{q}L^{2}(\mathbb{R}^{2}).$

######  Lemma 2.2.

The operator $T_{q}(\upsilon\delta_{\Gamma})$, $q\in{\mathbb{Z}}_{+}$, is a
compact self-adjoint operator in $p_{q}L^{2}(\mathbb{R}^{2})$. For
$\lambda>-b$ and $G_{\upsilon}(\lambda)$ in (2.3) one has

(2.13)
$T_{q}(\upsilon\delta_{\Gamma})=(\Lambda_{q}+\lambda)p_{q}G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda)p_{q}.$

Moreover, if $\upsilon$ has a constant sign, then there exists
$\lambda_{0}>-b$ such that for all $\lambda>\lambda_{0}$

(2.14) ${\rm ker}\,(p_{q}Q_{\upsilon}(\lambda)p_{q})={\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma})),$

where
$Q_{\upsilon}(\lambda):=(H_{\upsilon}+\lambda)^{-1}-(H_{0}+\lambda)^{-1}$.

###### Proof.

Recall that $J_{\upsilon}$ denotes the sign of $\upsilon$. A simple
calculation involving the form $t_{q}(\upsilon\delta_{\Gamma})$ shows for any
$u\in p_{q}L^{2}(\mathbb{R}^{2})$ and $\lambda>-b$ that

$\begin{split}t_{q}(\upsilon\delta_{\Gamma})[u]&=\int_{\Gamma}\upsilon(x)|u(x)|^{2}ds\\\
&=(\Lambda_{q}+\lambda)\int_{\Gamma}\upsilon(x)|(\tau(H_{0}+\lambda)^{-1/2}u)(x)|^{2}ds\\\
&=(\Lambda_{q}+\lambda)\big{\langle}J_{\upsilon}G_{\upsilon}(\lambda)p_{q}u,G_{\upsilon}(\lambda)p_{q}u\big{\rangle}_{L^{2}(\Gamma)}.\end{split}$

Thus, we get the representation (2.13), which also shows with Lemma 2.1 that
$T_{q}(\upsilon\delta_{\Gamma})$ is compact and self-adjoint. Moreover, (2.13)
immediately implies

(2.15) ${\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma}))=\ker(p_{q}G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda)p_{q}).$

It remains to prove equation (2.14). For this, assume that $\upsilon$ has a
constant sign, i.e. $\pm\upsilon\geq 0$, and fix $\lambda>-b$ sufficiently
large such that $H_{\upsilon}+\lambda$ is strictly positive. Then the
equivalence of (2.8) and (2.9) shows that
$I+G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda)$ is strictly
positive and bounded. Thus, we get for
$K_{\upsilon}(\lambda):=G_{\upsilon}(\lambda)^{*}G_{\upsilon}(\lambda)\geq 0$
by Lemma 2.1 that

$\begin{split}\big{\langle}p_{q}&Q_{\upsilon}(\lambda)p_{q}u,u\big{\rangle}_{L^{2}(\mathbb{R}^{2})}=-J_{\upsilon}(\Lambda_{q}+\lambda)^{-1}\big{\langle}K_{\upsilon}(\lambda)(I+J_{\upsilon}K_{\upsilon}(\lambda))^{-1}p_{q}u,p_{q}u\big{\rangle}_{L^{2}(\mathbb{R}^{2})}\\\
&=-J_{\upsilon}(\Lambda_{q}+\lambda)^{-1}\big{\langle}(I+J_{\upsilon}K_{\upsilon}(\lambda))^{-1}(K_{\upsilon}(\lambda))^{1/2}p_{q}u,(K_{\upsilon}(\lambda))^{1/2}p_{q}u\big{\rangle}_{L^{2}(\mathbb{R}^{2})}.\end{split}$

Taking the bijectivity of
$I+G_{\upsilon}(\lambda)^{*}J_{\upsilon}G_{\upsilon}(\lambda)$ and (2.15) into
account, this leads to (2.14). ∎

Via the form $t_{q}(\upsilon\delta_{\Gamma})$ one gets also another
interesting characterization of ${\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$
in the case that $\upsilon\geq 0$ almost everywhere on $\Gamma$, as then it
follows from (2.12) that $T_{q}(\upsilon\delta_{\Gamma})$ is a non-negative
operator and that ${\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))\neq\\{0\\}$ if
and only if there exists a $u\in p_{q}L^{2}(\mathbb{R}^{2})$ such that
$\upsilon u=0$ on $\Gamma$. This yields

(2.16) ${\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))=\\{u\in
p_{q}L^{2}(\mathbb{R}^{2}):u=0\text{ on }\rm{supp}\,\upsilon\\},$

where $\rm{supp}\,\upsilon$ denotes the essential support of $\upsilon$. In
other words, this means that ${\rm
ker}\;(T_{q}(\upsilon\delta_{\Gamma}))\neq\\{0\\}$ if and only if the
essential support of $\upsilon$ is contained in a nodal set of an
eigenfunction of $H_{0}$ for $\Lambda_{q}$. Furthermore, the dimension of
${\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$ is equal to the number of
linearly independent eigenfunctions $u$ of $H_{0}$ for $\Lambda_{q}$ such that
$u=0$ on $\rm{supp}\,\upsilon$. For studies on nodal sets of eigenfunctions we
refer the reader to, e.g., [24, 25, 26, 34]. Clearly, a similar consideration
is true if $\upsilon\leq 0$ almost everywhere on $\Gamma$.

##  3\. Main results

In this section we formulate our main results on the fate of Landau levels
under $\delta$-perturbations supported on curves. The case of general
$C^{1,1}$-smooth Jordan curves is treated first and, roughly speaking, we show
that the analysis of the eigenspaces ${\rm
ker}\,(H_{\pm\upsilon}-\Lambda_{q})$ of the perturbed Landau Hamiltonian can
be reduced to the analysis of the kernels ${\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$ of the Berezin-Toeplitz type operators
defined in (2.11). This connection is of independent interest, but also turns
out to be useful for a more explicit analysis of the Landau levels. We
illustrate this for the special case of $\delta$-perturbations supported on
circles.

###  3.1. Singular interactions supported on $C^{1,1}$-smooth Jordan curves

Throughout this subsection let $\Gamma\subset{\mathbb{R}}^{2}$ be a
$C^{1,1}$-smooth Jordan curve and assume that $\upsilon\in
L^{p}(\Gamma;{\mathbb{R}})$ for some $p>1$ is such that $\upsilon\geq 0$ on
$\Gamma$ and $\upsilon\not\equiv 0$. Our first theorem contains two
independent statements which concern the operators $H_{\upsilon}$ and
$H_{-\upsilon}$ respectively. The proof of Theorem 3.1 can be found in
Subsection 5.1.

######  Theorem 3.1.

Let $q\in{\mathbb{Z}}_{+}$ and let $T_{q}(\upsilon\delta_{\Gamma})$ be the
operator of Berezin-Toeplitz type in (2.11).

  * (i)

There holds

${\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))\subset{\rm
ker}\,(H_{\pm\upsilon}-\Lambda_{q}).$

  * (ii)

There exist $n_{q}^{\pm}\in{\mathbb{Z}}_{+}$ depending on $\upsilon$ such that

(3.1) ${\rm dim\,ker}\,(H_{\pm\upsilon}-\Lambda_{q})\leq{\rm
dim\,ker}\,(T_{q}(\upsilon\delta_{\Gamma}))+n_{q}^{\pm}$

and for $q=0$ one can choose $n_{0}^{+}=0$.

  * (iii)

There exist $\upsilon^{\pm}_{q}>0$ such that
$\|\upsilon\|_{L^{p}(\Gamma)}<\upsilon^{\pm}_{q}$ implies

(3.2) ${\rm ker}\,(H_{\pm\upsilon}-\Lambda_{q})={\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$

and for $q=0$ one can choose $\upsilon_{0}^{+}=\infty$. Moreover, there is a
constant $c>0$ independent of $b$ and $q$ such that

(3.3)
$\upsilon_{q}^{+}\geq\frac{2bc}{(\Lambda_{q}+1)(\Lambda_{q-1}+1)}\quad\text{and}\quad\upsilon_{q}^{-}\geq\frac{2bc}{2b+(\Lambda_{q}+1)(\Lambda_{q+1}+1)}.$

######  Remark 3.2.

Writing (3.2), we mean that $u\in{\rm ker}\,(H_{\pm\upsilon}-\Lambda_{q})$
implies $u=p_{q}u$, and

$u\in{\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))\subset
p_{q}L^{2}({\mathbb{R}}^{2}),$

and vice versa. A similar remark applies to all further inclusions of the same
kind. We also mention that the inequality (3.1) will be obtained by showing
that ${\rm ker}\,(H_{\pm\upsilon}-\Lambda_{q})$ is the sum of ${\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$ and a finite dimensional space.

######  Remark 3.3.

The definitions of the numbers $\upsilon^{\pm}_{q}$ in Theorem 3.1 (iii) are
given in (5.36) and (5.45), respectively. Their precise value is not obvious.
However, equation (3.2) is also true, if one replaces $\upsilon_{q}^{\pm}$ by
the lower bounds in (3.3). Since $\Lambda_{q}=b(2q+1)$, we see that these
lower bounds are decreasing in $q$; the same is true for $\upsilon^{\pm}_{q}$
defined in (5.36) and (5.45).

######  Remark 3.4.

The operator $H_{\upsilon}$ can be introduced as a self-adjoint extension of
the symmetric operator $S$ given by

(3.4) $Su=(-i\nabla-A)^{2}u,\qquad{\mathfrak{D}}(S)=\big{\\{}u\in{\rm
H}^{2}_{A}(\mathbb{R}^{2})\,|\,u_{|\Gamma}=0\big{\\}},$

i.e. $S$ is the restriction of $H_{0}$ onto functions in ${\rm
H}^{2}_{A}({\mathbb{R}}^{2})$ that vanish on $\Gamma$; cf. [7] for the case
$\upsilon\in L^{\infty}(\Gamma;\mathbb{R})$. In view of (2.16), if $u\in{\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$ for $\upsilon>0$ or $\upsilon<0$, then
$u\in{\rm ker}\,(S-\Lambda_{q})$ and thus, as $H_{\upsilon}$ was defined as an
extension of $S$, $u\in{\rm ker}(H_{\upsilon}-\Lambda_{q})$. Thus, the result
of Theorem 3.1 (i) can also be interpreted from an extension theoretic point
of view.

######  Remark 3.5.

Considering (3.1) the question arises, if ${\rm
dim\,ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$ is finite, as then ${\rm
dim\,ker}\,(H_{\pm\upsilon}-\Lambda_{q})<\infty$. If $\upsilon$ is strictly
positive, i.e. if $\upsilon\geq c>0$ everywhere on $\Gamma$, then by [7, Lemma
3.7] and (2.16) one indeed has ${\rm
dim\,ker}\,(T_{q}(\upsilon\delta_{\Gamma}))<\infty$.

Theorem 3.1 reduces the analysis of ${\rm ker}\,(H_{\upsilon}-\Lambda_{q})$ to
that of ${\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$. That is why our further
results concern ${\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$. The situation
is particularly simple for the Berezin-Toeplitz operator
$T_{0}(\upsilon\delta_{\Gamma})$ as the next theorem shows. Its proof can be
found in Subsection 5.2.

######  Theorem 3.6.

For $q=0$ we have

(3.5) ${\rm ker}\,(T_{0}(\upsilon\delta_{\Gamma}))=\\{0\\}.$

Combining (3.5) and (3.1)–(3.2) with $q=0$ we obtain the following corollary.

######  Corollary 3.7.

We have

${\rm ker}\,(H_{\upsilon}-\Lambda_{0})=\\{0\\}\quad\text{and}\quad{\rm
dim\,ker}\,(H_{-\upsilon}-\Lambda_{0})<\infty.$

Moreover, if $\|\upsilon\|_{L^{p}(\Gamma)}$ is sufficiently small, then ${\rm
ker}\,(H_{-\upsilon}-\Lambda_{0})=\\{0\\}.$

The next remark concerns regular perturbations of the Landau Hamiltonian and
the fate of the Landau levels as investigated earlier in [28].

######  Remark 3.8.

Assume that $V\in L^{\infty}({\mathbb{R}}^{2};\mathbb{R})$ satisfies
$V\not\equiv 0$, $V\geq 0$, and $\lim_{|x|\to\infty}V(x)=0$. Then, applying
the general scheme of the proof of Theorem 3.1 below, one can verify that

(3.6) ${\rm dim\,ker}\,(H_{0}\pm V-\Lambda_{q})<\infty$

and

(3.7) ${\rm ker}\,(H_{0}\pm V-\Lambda_{q})={\rm ker}\,(\hat{T}_{q}(V)),$

if $\|V\|_{L^{\infty}(\mathbb{R}^{2})}<V_{q}^{\pm}$ for some constants
$V_{q}^{\pm}>0$; here $\hat{T}_{q}(V)=p_{q}Vp_{q},$ and $p_{q}$ is the
orthogonal projection onto ${\rm ker}\,(H_{0}-\Lambda_{q})$.

The main idea of the proof of [28, Theorem 1] is to show that, under the
assumptions $\|V\|_{L^{\infty}({\mathbb{R}}^{2})}<2b$ and $V\geq 0$, one has

(3.8) ${\rm ker}\,(H_{0}\pm V-\Lambda_{q})\subset{\rm
ker}\,(\hat{T}_{q}(V)),\quad q\in{\mathbb{Z}}_{+}.$

On the other hand, if additionally $V\not\equiv 0$, then the results of [28,
39] imply

(3.9) ${\rm ker}\,(\hat{T}_{q}(V))=\left\\{0\right\\},\quad
q\in{\mathbb{Z}}_{+},$

and (1.2) follows from (3.8) and (3.9).

If one compares (3.6)–(3.7) (following our approach) with (3.8)–(3.9)
(following the approach in [28]), then the upper bounds $V_{q}^{\pm}$ for
$\|V\|_{L^{\infty}({\mathbb{R}}^{2})}$ in (3.7) are not as explicit as the
upper bound $2b$ in (3.8), but one gets in (3.6) also results for potentials
with $\|V\|_{L^{\infty}({\mathbb{R}}^{2})}\geq 2b$.

###  3.2. Singular interactions supported on circles

Now we illustrate the above results for the special case that
$\Gamma={\mathcal{C}}_{r}$ is a circle of radius $r\in(0,\infty)$. In this
situation more explicit results on the structure of ${\rm
ker}\,(T_{q}(\delta_{{\mathcal{C}}_{r}}))$ are obtained, as one can use an
explicit basis of ${\rm ker}(H_{0}-\Lambda_{q})$ in polar coordinates. Using
this and the simple characterization of ${\rm
ker}(T_{q}(\delta_{{\mathcal{C}}_{r}}))$ from (2.16), it will turn out that
${\rm ker}(T_{q}(\delta_{{\mathcal{C}}_{r}}))\neq\\{0\\}$ is equivalent to the
fact that $br^{2}/2$ is a zero of a suitable Laguerre polynomial. Since (2.16)
implies ${\rm ker}\,(T_{q}(\delta_{{\mathcal{C}}_{r}}))={\rm
ker}(T_{q}(\upsilon\delta_{\mathcal{C}_{r}}))$, if $\upsilon$ is strictly
positive or negative everywhere on $\Gamma$, this yields further results on
${\rm ker}\,(H_{\pm\upsilon}-\Lambda_{q})$. The obtained result on ${\rm
ker}\,(T_{q}(\delta_{{\mathcal{C}}_{r}}))$ is the following:

######  Theorem 3.9.

Let $q\in{\mathbb{N}}$, assume that $\Gamma={\mathcal{C}}_{r}$ is a circle of
radius $r\in(0,\infty)$, and let $T_{q}(\delta_{{\mathcal{C}}_{r}})$ be the
corresponding operator of Berezin-Toeplitz type in (2.11).

  * (i)

For any $r\in(0,\infty)$ we have

(3.10) ${\rm dim\,ker}\,(T_{q}(\delta_{{\mathcal{C}}_{r}}))\leq q.$

  * (ii)

The set

${\mathcal{D}}_{q}:=\left\\{r\in(0,\infty)\,|\,{\rm
dim\,ker}\,(T_{q}(\delta_{{\mathcal{C}}_{r}}))\geq 1\right\\}$

is infinite and discrete.

Theorem 3.9 will be proved in Subsection 5.3. Since (2.16) implies the
relation ${\rm ker}\,(T_{q}(\delta_{{\mathcal{C}}_{r}}))={\rm
ker}(T_{q}(\upsilon\delta_{\mathcal{C}_{r}}))$, if $\upsilon$ is strictly
positive, a combination of (3.1) and (3.10) leads to the following corollary.

######  Corollary 3.10.

Let $q\in{\mathbb{N}}$, let $\Gamma={\mathcal{C}}_{r}$ be a circle of radius
$r\in(0,\infty)$, and assume that $\upsilon\in
L^{p}({\mathcal{C}}_{r};{\mathbb{R}})$ with $p>1$ satisfies $\upsilon\geq c$
on ${\mathcal{C}}_{r}$ with some constant $c>0$. Then

${\rm dim\,ker}\,(H_{\pm\upsilon}-\Lambda_{q})<\infty.$

######  Remark 3.11.

For $q\in{\mathbb{N}}$ set

${\mathcal{D}}_{q,j}:=\left\\{r\in(0,\infty)\,|\,{\rm
dim\,ker}\,(T_{q}(\delta_{{\mathcal{C}}_{r}}))=j\right\\},\quad j=1,\ldots,q,$

so that ${\mathcal{D}}_{q}=\cup_{j=1}^{q}{\mathcal{D}}_{q,j}$; note that the
union stops at $j=q$ by Theorem 3.9 (i). In the proof of Theorem 3.9, we will
describe the dimension of ${\rm ker}\,(T_{q}(\delta_{{\mathcal{C}}_{r}}))$ in
terms of the zeros of Laguerre polynomials of $q$-th degree (see (5.50)–(5.51)
below). If $q=1,2$, these zeros can be easily calculated, and we obtain
explicitly the sets ${\mathcal{D}}_{q}$ and their components
${\mathcal{D}}_{q,j}$, namely

(3.11) ${\mathcal{D}}_{1}={\mathcal{D}}_{1,1}=\ \sqrt{(2/b){\mathbb{N}}},$
(3.12)
${\mathcal{D}}_{2}=\sqrt{(2/b)(({\mathbb{N}}+1)-\sqrt{({\mathbb{N}}+1)})}\cup\sqrt{(2/b)({\mathbb{N}}+\sqrt{{\mathbb{N}}})},$
(3.13)
${\mathcal{D}}_{2,2}=\sqrt{(2/b)({\mathbb{N}}^{2}+{\mathbb{N}})},\quad{\mathcal{D}}_{2,1}={\mathcal{D}}_{2}\setminus{\mathcal{D}}_{2,2},$

where $b>0$ is the magnetic field.

######  Remark 3.12.

Taking the explicit representation (4.6) of the orthonormal basis of ${\rm
ran}\,(p_{q})={\rm ker}\,(H_{0}-\Lambda_{q})$, $q\in\mathbb{N}$, into account,
we conclude that $\Lambda_{q}$ is still an eigenvalue of the symmetric
operator $S$ defined in (3.4) provided that $br^{2}/2$ is a root of a Laguerre
polynomial of $q$-th degree. Since $H_{\upsilon}$ is a self-adjoint extension
of $S$ for all $\upsilon\in L^{p}(\Gamma;\mathbb{R})$, $\Lambda_{q}$ remains
an eigenvalue of $H_{\upsilon}$ under the same assumption on $r>0$.

##  4\. Auxiliary results from the spectral theory of $H_{0}$

In this section we recall several known facts about $H_{0}$ that are necessary
for our considerations; the first part follows [37, Section 4.2], while the
second part on the basis of ${\rm ker}\,(H_{0}-\Lambda_{q})$ can be found in
[39, Section 3.1], see also [23]. In the following, we describe a suitable
spectral representation of $H_{0}$. Set

$\phi(x):=\frac{b|x|^{2}}{4},\quad x=(x_{1},x_{2})\in{\mathbb{R}}^{2}.$

Introduce the magnetic creation operator

(4.1) $a^{*}=\Pi_{1}(A)-i\Pi_{2}(A)=-2ie^{\phi}\frac{\partial}{\partial
z}e^{-\phi},\quad z=x_{1}+ix_{2},$

and the magnetic annihilation operator

(4.2)
$a=\Pi_{1}(A)+i\Pi_{2}(A)=-2ie^{-\phi}\frac{\partial}{\partial\bar{z}}e^{\phi},\quad\bar{z}=x_{1}-ix_{2}.$

The operators $a$ and $a^{*}$ are closed on
${\mathfrak{D}}(a)={\mathfrak{D}}(a^{*})={\rm H}_{A}^{1}({\mathbb{R}}^{2})$,
and are mutually adjoint in $L^{2}({\mathbb{R}}^{2})$. Moreover,

(4.3) $[a,a^{*}]=2b,$

and

$H_{0}=a^{*}a+b=aa^{*}-b.$

Further,

(4.4) ${\rm ker}\,(H_{0}-\Lambda_{q})=(a^{*})^{q}\,{\rm ker}\,(a),\quad
q\in{\mathbb{Z}}_{+}.$

By (4.2), we have

(4.5) ${\rm ker}\;(a)=\left\\{u\in
L^{2}({\mathbb{R}}^{2})\,|\,u=e^{-\phi}\,g,\quad\frac{\partial
g}{\partial\bar{z}}=0\right\\}.$

Thus, $e^{\phi}\,{\rm ker}\,(H_{0}-\Lambda_{0})=e^{\phi}\,{\rm ker}\,(a)$
coincides with the Fock-Segal-Bargmann space of entire functions $g\in
L^{2}({\mathbb{R}}^{2};e^{-2\phi}dx)$ (see, e.g., [23, Section 3.2]). Assume
now that

$u\in{\rm ker}\,(H_{0}-\Lambda_{q}),\quad q\in{\mathbb{N}}.$

By (4.4) and (4.1), there exists an entire function $g\in
L^{2}({\mathbb{R}}^{2};e^{-2\phi}dx)$ such that

$\begin{split}(e^{\phi}u)(x)&=:f(x)=((e^{\phi}(a^{*})^{q}\,e^{-\phi})g)(x)=(-2i)^{q}\left(\left(e^{2\phi}\frac{\partial^{q}}{\partial
z^{q}}\,e^{-2\phi}\right)g\right)(x)\\\
&=(-2i)^{q}\sum_{\ell=0}^{q}\binom{q}{\ell}\left(-\frac{b\bar{z}}{2}\right)^{\ell}g^{(q-\ell)}(z),\quad
x\in{\mathbb{R}}^{2},\,\,z=x_{1}+ix_{2}.\end{split}$

Evidently, $f\in L^{2}({\mathbb{R}}^{2};e^{-2\phi}dx)$ is a polyanalytic
function of order $q+1$, i.e. $f$ is a solution of the equation

$\frac{\partial^{q+1}f}{\partial\bar{z}^{q+1}}(z)=0,\quad z\in{\mathbb{C}},$

(see [1, 5, 6] and also [42, Section 2.2]). We have

$\left\\{h\in
L^{2}({\mathbb{R}}^{2};e^{-2\phi}dx)\,\big{|}\,\frac{\partial^{q+1}h}{\partial\bar{z}^{q+1}}=0\right\\}=\bigoplus_{j=0}^{q}e^{\phi}\,{\rm
ker}\,(H_{0}-\Lambda_{j})$

and the spaces $e^{\phi}\,{\rm ker}\,(H_{0}-\Lambda_{j})$, $j=0,\ldots,q$, are
called sometimes true poly-Fock spaces of order $j$ (see [1, 44]).

Next, we introduce an explicit orthonormal basis of every ${\rm
ker}\,(H_{0}-\Lambda_{q})$, $q\in{\mathbb{Z}}_{+}$, called sometimes the
angular-momentum basis. Let at first $q=0$. Then the functions

$\widetilde{\varphi}_{k,0}(x)=z^{k}e^{-\phi(x)},\quad
x\in{\mathbb{R}}^{2},\quad z=x_{1}+ix_{2},\quad k\in{\mathbb{Z}}_{+},$

form an orthogonal basis of ${\rm ker}\;(a)={\rm ran}\,(p_{0})$ (see, e.g.,
[23, Sections 3.1–3.2]). Normalizing, we obtain an orthonormal basis of ${\rm
ran}\,(p_{0})$, consisting of the functions

${\varphi}_{k,0}(x):=\frac{\widetilde{\varphi}_{k,0}(x)}{\|\widetilde{\varphi}_{k,0}\|_{L^{2}({\mathbb{R}}^{2})}}=\sqrt{\frac{b}{2\pi}}\sqrt{\frac{1}{k!}}\left(\sqrt{\frac{b}{2}}\,z\right)^{k}\,e^{-\phi(x)},\quad
x\in{\mathbb{R}}^{2},\quad k\in{\mathbb{Z}}_{+}.$

Let now $q\geq 1$. Set

$\widetilde{{\varphi}}_{k,q}=(a^{*})^{q}\,\varphi_{k,0},\quad
k\in{\mathbb{Z}}_{+}.$

The commutation relation (4.3) easily implies

$\langle\widetilde{{\varphi}}_{k,q},\widetilde{{\varphi}}_{\ell,q}\rangle_{L^{2}({\mathbb{R}}^{2})}=(2b)^{q}q!\delta_{k\ell},\quad
k,\ell\in{\mathbb{Z}}_{+}.$

Therefore, the functions

${\varphi}_{k,q}:=\frac{\widetilde{\varphi}_{k,q}}{\|\widetilde{\varphi}_{k,q}\|_{L^{2}({\mathbb{R}}^{2})}}=\frac{\widetilde{\varphi}_{k,q}}{\sqrt{(2b)^{q}q!}},\quad
k\in{\mathbb{Z}}_{+},$

form an orthonormal basis of ${\rm ran}\,(p_{q})$, $q\in{\mathbb{N}}$. They
admit a more explicit representation (see [39, Subsection 3.1]), namely

(4.6)
$\begin{split}\varphi_{k,q}(x)=\frac{1}{i^{q}}\sqrt{\frac{b}{2\pi}}\sqrt{\frac{q!}{k!}}\left(\sqrt{\frac{b}{2}}\,z\right)^{k-q}&\,{\rm
L}_{q}^{(k-q)}\left(\frac{b|x|^{2}}{2}\right)e^{-\phi(x)},\\\
&x\in{\mathbb{R}}^{2},\quad z=x_{1}+ix_{2},\quad
k\in{\mathbb{Z}}_{+},\end{split}$

where

${\rm
L}_{q}^{(\alpha)}(t):=\frac{t^{-\alpha}e^{t}}{q!}\frac{d^{q}}{dt^{q}}\left(t^{q+\alpha}e^{-t}\right),\quad
t>0,\quad\alpha\in{\mathbb{R}},\quad q\in{\mathbb{Z}}_{+},$

are the (generalized) Laguerre polynomials (see [22, Eq. 8.970 (1)]). In
particular,

(4.7) ${\rm L}_{1}^{(\alpha)}(t)=-t+\alpha+1,$ (4.8) ${\rm
L}_{2}^{(\alpha)}(t)=\frac{1}{2}\left(t^{2}-2(\alpha+2)t+(\alpha+2)(\alpha+1)\right).$

The Laguerre polynomials ${\rm L}_{q}^{(\alpha)}$ with $q\in{\mathbb{Z}}_{+}$
and $\alpha>-1$ satisfy

(4.9) $\int_{0}^{\infty}e^{-t}t^{\alpha}{\rm L}_{q}^{(\alpha)}(t){\rm
L}_{p}^{(\alpha)}(t)dt=\Gamma(\alpha+1)\binom{q+\alpha}{q}\delta_{qp},\quad
q,p\in{\mathbb{Z}}_{+},$

(see [43, Eq. (5.1.1)]).

Finally, for $y\in{\mathbb{R}}^{2}$, introduce the magnetic translations

(4.10) $({\mathcal{T}}_{y}u)(x):=e^{-i\frac{b}{2}(x\wedge y)}\,u(x-y),\quad
x\in{\mathbb{R}}^{2},$

where

$x\wedge y:=x_{1}y_{2}-x_{2}y_{1}.$

Evidently, for each $y\in{\mathbb{R}}^{2}$, the operator ${\mathcal{T}}_{y}$
is unitary in $L^{2}({\mathbb{R}}^{2})$. A direct calculation yields

${\mathcal{T}}_{y}^{*}\,\Pi_{j}(A)\,{\mathcal{T}}_{y}=\Pi_{j}(A),\quad j=1,2,$

and, therefore,

${\mathcal{T}}_{y}^{*}\,H_{0}\,{\mathcal{T}}_{y}=H_{0},\quad
y\in{\mathbb{R}}^{2}.$

Then the spectral theorem implies

(4.11) ${\mathcal{T}}_{y}^{*}\,p_{q}\,{\mathcal{T}}_{y}=p_{q},\quad
y\in{\mathbb{R}}^{2},\quad q\in{\mathbb{Z}}_{+}.$

##  5\. Proofs of the main results

###  5.1. Proof of Theorem 3.1

We start with the proof of item (i). Denote by $\Omega_{\rm in}$ and
$\Omega_{\rm ex}$ the interior and the exterior of $\Gamma$ respectively, and
by $\nu$ the unit normal vector on $\Gamma$ pointing outwards of $\Omega_{\rm
in}$. For $w\in L^{2}({\mathbb{R}}^{2})$ set

$w_{\natural}:=w_{|\Omega_{\natural}},\quad\natural={\rm in},{\rm ex}.$

In view of (2.2) we have that $w\in{\mathfrak{D}}(H_{\pm\upsilon})$ is
equivalent to the following conditions:
(a) $w\in{\rm H}_{A}^{1}({\mathbb{R}}^{2})$;
(b) $(-i\nabla-A)^{2}w_{\natural}\in L^{2}(\Omega_{\natural})$, $\natural={\rm
in},{\rm ex}$;
(c) $\left(\frac{\partial w_{\rm ex}}{\partial\nu}-\frac{\partial w_{\rm
in}}{\partial\nu}\mp\upsilon w\right)_{\Gamma}=0$.
Moreover, if $w\in{\mathfrak{D}}(H_{\pm\upsilon})$, then

(5.1)
$(H_{\pm\upsilon}w)_{\natural}=(-i\nabla-A)^{2}w_{\natural},\quad\quad\natural={\rm
in},{\rm ex}.$

Assume $u\in{\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$,
$q\in{\mathbb{Z}}_{+}$. Since $\upsilon\geq 0$, by (2.16) this is equivalent
to

(5.2) $u\in{\rm ker}\,(H_{0}-\Lambda_{q})\subset{\mathfrak{D}}(H_{0})={\rm
H}_{A}^{2}({\mathbb{R}}^{2})$

and

(5.3) $\upsilon u=0\quad\mbox{on}\quad\Gamma.$

By $u\in{\rm H}_{A}^{2}({\mathbb{R}}^{2})$ conditions (a)–(b) are fulfilled
and, moreover,

(5.4) $\frac{\partial u_{\rm ex}}{\partial\nu}=\frac{\partial u_{\rm
in}}{\partial\nu}\quad\mbox{on}\quad\Gamma.$

Combining (5.3) with (5.4) we find that also (c) also holds, i.e.
$u\in{\mathfrak{D}}(H_{\pm\upsilon})$. By (5.2) we have

$H_{0}u=(-i\nabla-A)^{2}u=\Lambda_{q}u\quad\mbox{in}\quad{\mathbb{R}}^{2}$

and, hence,

(5.5)
$(-i\nabla-A)^{2}u_{\natural}=\Lambda_{q}u_{\natural}\quad\mbox{in}\quad\Omega_{\natural},\quad\natural={\rm
in},{\rm ex}.$

Bearing in mind (5.1), we now find that (5.5) implies
$H_{\pm\upsilon}u=\Lambda_{q}u$, i.e.

$u\in{\rm ker}\,(H_{\pm\upsilon}-\Lambda_{q}).$

The remaining items (ii) and (iii) will be proved together. To make the proof
accessible in an easier way, we have split it into several steps. First, the
case $\upsilon\geq 0$ is treated. In Step 1 the claims for the first Landau
level $\Lambda_{0}$ are shown. In Step 2–5 the claims for $\Lambda_{q}$,
$q\in\mathbb{N}$, are verified. More precisely, in Step 2 the eigenvalue
equation is reduced to equations for operators which are easier accessible for
our purposes. Then, in Step 3 a representation of ${\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$ involving these new operators is
proved. In Step 4 we are putting all this together to verify assertion (ii)
for $\upsilon\geq 0$, while in Step 5 the proof of item (iii) in this case is
concluded. Finally, in Step 6 the case $\upsilon\leq 0$ is treated.

First let us introduce the notations which will be used throughout the proof.
Assume, as usual, that $\upsilon\in L^{p}(\Gamma;{\mathbb{R}})$ with $p>1$,
$\upsilon\geq 0$, $\upsilon\neq 0$, and (2.9) holds true. Set

$Q^{+}_{\upsilon}(\lambda):=(H_{0}+\lambda)^{-1}-(H_{\upsilon}+\lambda)^{-1},\quad
Q^{-}_{\upsilon}(\lambda):=-(H_{0}+\lambda)^{-1}+(H_{-\upsilon}+\lambda)^{-1}.$

By Lemma 2.1, we have $Q^{\pm}_{\upsilon}(\lambda)\geq 0$, and the operators
$Q^{\pm}_{\upsilon}(\lambda)$ are compact in $L^{2}({\mathbb{R}}^{2})$. Note
that Lemma 2.1 also implies

(5.6)
$Q^{\pm}_{\upsilon}(\lambda)=(H_{0}+\lambda)^{-1/2}G_{\upsilon}(\lambda)^{*}(I\pm
G_{\upsilon}(\lambda)G_{\upsilon}(\lambda)^{*})^{-1}G_{\upsilon}(\lambda)(H_{0}+\lambda)^{-1/2}.$

Further, put

$P_{q}^{+}:=\sum_{j=q}^{\infty}p_{j}\quad\text{and}\quad
P_{q}^{-}:=I-P_{q}^{+},$

so that $P_{0}^{+}=I$, and $P_{0}^{-}=0$. For $q\geq 1$ the projections
$P_{q}^{\pm}$ have infinite rank. Finally, set

$\mu_{q}(\lambda):=\left(\Lambda_{q}+\lambda\right)^{-1},\quad
q\in{\mathbb{Z}}_{+},\quad\lambda>-b.$

Step 1: We first prove the part of Theorem 3.1 (ii) and (iii) concerning
positive perturbations $H_{\upsilon}$ and start with the case $q=0$. Assume
that

$u\in{\rm ker}(H_{\upsilon}-\Lambda_{0}).$

Then $u$ satisfies

$(H_{\upsilon}+\lambda)^{-1}u=\mu_{0}(\lambda)u$

or, equivalently,

(5.7)
$\left((H_{0}+\lambda)^{-1}-\mu_{0}(\lambda)\right)u-Q^{+}_{\upsilon}(\lambda)u=0,\quad\lambda>-b.$

Thus,

(5.8)
$\big{\langle}\left((H_{0}+\lambda)^{-1}-\mu_{0}(\lambda)\right)u,u\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}-\big{\langle}Q^{+}_{\upsilon}(\lambda)u,u\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}=0.$

Both terms on the left-hand side of (5.8) are non-positive, and hence they
both should vanish. Since $(H_{0}+\lambda)^{-1}-\mu_{0}(\lambda)$ is non-
positive, the equality

$\big{\langle}\left((H_{0}+\lambda)^{-1}-\mu_{0}(\lambda)\right)u,u\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}=0$

and the min-max principle imply $u=p_{0}u$. Then,

$\big{\langle}Q^{+}_{\upsilon}(\lambda)u,u\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}=\big{\langle}p_{0}Q^{+}_{\upsilon}(\lambda)p_{0}u,u\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}=0,$

where the operator $p_{0}Q^{+}_{\upsilon}(\lambda)p_{0}$ is self-adjoint and
non-negative in the space $p_{0}L^{2}({\mathbb{R}}^{2})={\rm ran}\,(p_{0})$.
Hence, by Lemma 2.2

(5.9) $u\in{\rm ker}\,(p_{0}Q^{+}_{\upsilon}(\lambda)p_{0})={\rm
ker}\,(T_{0}(\upsilon\delta_{\Gamma})).$

Thus, we obtain the inclusion $\subset$ in (3.2) for $H_{\upsilon}$ and $q=0$.
The remaining inclusion $\supset$ in (3.2) is clear by (i). Therefore, item
(iii) is shown in the case of positive perturbations and $q=0$, which implies
also assertion (ii) in the same case.

Step 2: Assume now

$u\in{\rm ker}\,(H_{\upsilon}-\Lambda_{q}),\quad q\in{\mathbb{N}}.$

In this step we reduce the eigenvalue equation for $u$ to equations for
operators which are easier accessible for our purposes. Similarly to (5.7) we
have

(5.10)
$\left((H_{0}+\lambda)^{-1}-\mu_{q}(\lambda)\right)u-Q^{+}_{\upsilon}(\lambda)u=0,\quad\lambda>-b.$

Set

(5.11) $u^{+}:=P_{q}^{+}u\quad\text{and}\quad u^{-}:=P_{q}^{-}u,$

so that

(5.12) $u=u^{+}+u^{-}.$

Since $P_{q}^{\pm}$ are functions of $H_{0}$, they commute with
$(H_{0}+\lambda)^{-1}$ and thus, their application to (5.10) implies

(5.13)
$\left((H_{0}+\lambda)^{-1}-\mu_{q}(\lambda)\right)u^{+}=P_{q}^{+}Q^{+}_{\upsilon}(\lambda)u^{+}+P_{q}^{+}Q^{+}_{\upsilon}(\lambda)u^{-},$
(5.14)
$\left((H_{0}+\lambda)^{-1}-\mu_{q}(\lambda)\right)u^{-}=P_{q}^{-}Q^{+}_{\upsilon}(\lambda)u^{+}+P_{q}^{-}Q^{+}_{\upsilon}(\lambda)u^{-}.$

Let

(5.15) $S_{q}^{+}(\lambda):=P_{q}^{-}Q^{+}_{\upsilon}(\lambda)P_{q}^{-}$

and observe that by Lemma 2.1 the operator $S_{q}^{+}(\lambda)$ is compact,
self-adjoint, and non-negative in $P_{q}^{-}L^{2}({\mathbb{R}}^{2})$. Set

(5.16)
$m_{q}^{+}(\lambda):=\inf\sigma\left(\left((H_{0}+\lambda)^{-1}-\mu_{q}(\lambda)\right)_{|P_{q}^{-}L^{2}({\mathbb{R}}^{2})}\right)=\frac{2b}{(\Lambda_{q}+\lambda)(\Lambda_{q-1}+\lambda)},$

and

$S_{q,>}^{+}(\lambda):=S_{q}^{+}(\lambda)\,\mathds{1}_{[m_{q}^{+}(\lambda),\infty)}\left(S_{q}^{+}(\lambda)\right),\quad
S_{q,<}^{+}(\lambda):=S_{q}^{+}(\lambda)-S_{q,>}^{+}(\lambda);$

here and in the sequel $\mathds{1}_{{\mathcal{J}}}(T)$ denotes the spectral
projection of the operator $T=T^{*}$ associated with the Borel set
${\mathcal{J}}\subset{\mathbb{R}}$. Note that

${\rm rank}\,(S_{q,>}^{+}(\lambda))<\infty.$

Moreover, if

$\|S_{q}^{+}(\lambda)\|<m_{q}^{+}(\lambda),$

then $S_{q,>}^{+}(\lambda)=0$. Now, (5.14) is equivalent to

(5.17)
$\left((H_{0}+\lambda)^{-1}-\mu_{q}(\lambda)-S_{q,<}^{+}(\lambda)\right)u^{-}=P_{q}^{-}Q^{+}_{\upsilon}(\lambda)u^{+}+S_{q,>}^{+}(\lambda)u_{>}^{-},$

where

(5.18) $u_{>}^{-}:=P_{q,>}^{-}u$

and

(5.19)
$P_{q,>}^{-}=P_{q,>}^{-}(\lambda):=\mathds{1}_{[m_{q}^{+}(\lambda),\infty)}(S_{q}^{+}(\lambda))\,P_{q}^{-}.\\\
$

Note that

(5.20) ${\rm rank}\,(P_{q,>}^{-}(\lambda))={\rm
rank}\,(S_{q,>}^{+}(\lambda))<\infty.$

Since $S_{q}^{+}(\lambda)$ is compact and $m_{q}^{+}(\lambda)>0$, there exists
$\varepsilon>0$ such that

$\sigma(S_{q}^{+}(\lambda))\cap(m_{q}^{+}(\lambda)-\varepsilon,m_{q}^{+}(\lambda))=\emptyset$

and hence $\|S_{q,<}^{+}(\lambda)\|\leq
m_{q}^{+}(\lambda)-\varepsilon<m_{q}^{+}(\lambda)$. By definition of
$m_{q}^{+}(\lambda)$, we have

$\big{(}(H_{0}+\lambda)^{-1}-\mu_{q}(\lambda)\big{)}_{|P_{q}^{-}L^{2}({\mathbb{R}}^{2})}\geq
m_{q}^{+}(\lambda)$

and thus, the operator

$\left((H_{0}+\lambda)^{-1}-\mu_{q}(\lambda)\right)_{|P_{q}^{-}L^{2}({\mathbb{R}}^{2})}-S_{q,<}^{+}(\lambda)\\\
$

is positive and boundedly invertible on $P_{q}^{-}L^{2}({\mathbb{R}}^{2})$.
Set

$R_{q}^{+}(\lambda):=\left(\left((H_{0}+\lambda)^{-1}-\mu_{q}(\lambda)\right)_{|P_{q}^{-}L^{2}({\mathbb{R}}^{2})}-S_{q,<}^{+}(\lambda)\right)^{-1}.$

Then (5.17) implies

(5.21)
$u^{-}=R_{q}^{+}(\lambda)P_{q}^{-}Q^{+}_{\upsilon}(\lambda)u^{+}+R_{q}^{+}(\lambda)S_{q,>}^{+}(\lambda)u_{>}^{-}.$

Inserting (5.21) into (5.13) we get

(5.22)
$P_{q}^{+}X_{q}^{+}(\lambda)P_{q}^{+}u^{+}=-P_{q}^{+}Y_{q}^{+}(\lambda)u_{>}^{-},$

where

$X_{q}^{+}(\lambda):=-(H_{0}+\lambda)^{-1}+\mu_{q}(\lambda)+Q^{+}_{\upsilon}(\lambda)+Q^{+}_{\upsilon}(\lambda)R_{q}^{+}(\lambda)P_{q}^{-}Q^{+}_{\upsilon}(\lambda)$

and

$Y_{q}^{+}(\lambda):=Q^{+}_{\upsilon}(\lambda)R_{q}^{+}(\lambda)\,S_{q,>}^{+}(\lambda).$

Step 3: Define the operator

(5.23) $K_{q}^{+}:=P_{q}^{+}X_{q}^{+}(\lambda)P_{q}^{+},$

which is self-adjoint and non-negative in $P_{q}^{+}L^{2}({\mathbb{R}}^{2})$.
In this step we show

(5.24) ${\rm ker}\,(K_{q}^{+})={\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma})).$

In order to check (5.24), assume first that $w\in{\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$. Then $w=p_{q}w$, see Remark 3.2, and

(5.25) $\begin{split}\langle
K_{q}^{+}&w,w\rangle_{L^{2}({\mathbb{R}}^{2})}=\big{\langle}(-(H_{0}+\lambda)^{-1}+\mu_{q}(\lambda))p_{q}w,p_{q}w\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}\\\
&+\big{\langle}Q^{+}_{\upsilon}(\lambda)p_{q}w,p_{q}w\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}+\big{\langle}Q^{+}_{\upsilon}(\lambda)R_{q}^{+}(\lambda)P_{q}^{-}Q^{+}_{\upsilon}(\lambda)p_{q}w,p_{q}w\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}.\end{split}$

Using Lemma 2.2 and ${\rm ker}\,(A^{*}A)={\rm ker}\,(A)$, one has

(5.26) ${\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))={\rm
ker}\,(p_{q}Q^{+}_{\upsilon}(\lambda)p_{q})={\rm
ker}\,(Q^{+}_{\upsilon}(\lambda)^{1/2}p_{q}).$

Moreover, the definitions of $\mu_{q}(\lambda)$ and $p_{q}$ yield

(5.27) $\quad\big{(}(H_{0}+\lambda)^{-1}-\mu_{q}(\lambda)\big{)}p_{q}=0.$

Thus, (5.25)–(5.27) imply

$\langle K_{q}^{+}w,w\rangle_{L^{2}({\mathbb{R}}^{2})}=0.$

Since $K_{q}^{+}\geq 0$ we conclude that $w\in{\rm ker}\,(K_{q}^{+})$, i.e.

(5.28) ${\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma}))\subset{\rm
ker}\,(K_{q}^{+}).$

Let now $w\in{\rm ker}\,(K_{q}^{+})$. Then

$\big{\langle}(-(H_{0}+\lambda)^{-1}+\mu_{q}(\lambda))P_{q}^{+}w,P_{q}^{+}w\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}+\big{\langle}Q^{+}_{\upsilon}(\lambda)P_{q}^{+}w,P_{q}^{+}w\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}$
(5.29)
$+\big{\langle}Q^{+}_{\upsilon}(\lambda)R_{q}^{+}(\lambda)P_{q}^{-}Q^{+}_{\upsilon}(\lambda)P_{q}^{+}w,P_{q}^{+}w\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}=0.$

The three terms on the left-hand side of (5.29) are non-negative and therefore
all of them vanish. The equality

$\big{\langle}((H_{0}+\lambda)^{-1}-\mu_{q}(\lambda))P_{q}^{+}w,P_{q}^{+}w\big{\rangle}_{L^{2}({\mathbb{R}}^{2})}=0$

implies that $P_{q}^{+}w=p_{q}w$. Inserting this into

$\langle
Q^{+}_{\upsilon}(\lambda)P_{q}^{+}w,P_{q}^{+}w\rangle_{L^{2}({\mathbb{R}}^{2})}=0$

we obtain

$\langle
Q^{+}_{\upsilon}(\lambda)p_{q}w,p_{q}w\rangle_{L^{2}({\mathbb{R}}^{2})}=0,$

i.e. with (5.26)

$w\in{\rm ker}\,(p_{q}Q^{+}_{\upsilon}(\lambda)p_{q})={\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma})).$

Therefore,

${\rm ker}\,(K_{q}^{+})\subset{\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma})),$

which combined with (5.28) yields (5.24).

Step 4: Now we have everything in hands to finish the proof of assertion (ii)
for $\upsilon\geq 0$. For this purpose we show that the element $u\in{\rm
ker}\,(H_{\upsilon}-\Lambda_{q})$ can be written as
$u=u_{0}^{+}+W_{q}^{+}u_{>}^{-}$, where $u_{0}^{+}\in{\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$, $u_{>}^{-}\in{\rm
ran}\,(P_{q,>}^{-}(\lambda))$, and $W_{q}^{+}$ is a suitable operator; this
implies (3.1) as ${\rm rank}\,(P_{q,>}^{-}(\lambda))<\infty$ by (5.20). Let
$\pi_{0}^{+}$ be the orthogonal projection onto ${\rm ker}\,(K_{q}^{+})$ and
$\pi_{\perp}^{+}:=P_{q}^{+}-\pi_{0}^{+}$. Set

$u_{0}^{+}:=\pi_{0}^{+}u^{+}\quad\text{and}\quad
u_{\perp}^{+}:=\pi_{\perp}^{+}u^{+},$

where $u^{+}$ is the function defined in (5.11). Thus,

(5.30) $u^{+}=u_{0}^{+}+u_{\perp}^{+}.$

Denote by $K_{q,\perp}^{+}$ the operator
$\pi_{\perp}^{+}K_{q}^{+}\pi_{\perp}^{+}$, which is self-adjoint and positive
on ${\rm ran}\,(\pi_{\perp}^{+})$. Then, as $u_{0}^{+}\in{\rm
ker}\,(K_{q}^{+})$, (5.22) and (5.23) imply

(5.31) $K_{q,\perp}^{+}u_{\perp}^{+}=-P_{q}^{+}Y_{q}^{+}(\lambda)u^{-}_{>}.$

Since we started with an arbitrary $u\in{\rm ker}\,(H_{\upsilon}-\Lambda_{q})$
we have by (5.18)

$u_{>}^{-}\in P_{q,>}^{-}{\rm ker}\,(H_{\upsilon}-\Lambda_{q}).$

Then (5.31) implies

(5.32) $P_{q}^{+}Y_{q}^{+}(\lambda)P_{q,>}^{-}{\rm
ker}\,(H_{\upsilon}-\Lambda_{q})\subset{\rm ran}\,(K_{q,\perp}^{+}).$

Recall that $K_{q,\perp}^{+}$ is positive and hence invertible. Therefore, we
can define on $P_{q,>}^{-}{\rm ker}\,(H_{\upsilon}-\Lambda_{q})$ the operator

$L_{q}^{+}:=-(K_{q,\perp}^{+})^{-1}P_{q}^{+}Y_{q}^{+}(\lambda).$

By (5.31) we have

(5.33) $u_{\perp}^{+}=L_{q}^{+}u_{>}^{-}.$

Putting together (5.12), (5.21), (5.30), and (5.33), we find that for any
$u\in{\rm ker}\,(H_{\upsilon}-\Lambda_{q})$ we have

(5.34) $u=u_{0}^{+}+W_{q}^{+}u_{>}^{-},$

where

$u_{0}^{+}\in{\rm ker}\,(K_{q}^{+})={\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma})),\quad u_{>}^{-}\in P_{q,>}^{-}{\rm
ker}\,(H_{\upsilon}-\Lambda_{q}),$

and

$W_{q}^{+}=R_{q}^{+}(\lambda)S_{q,>}^{+}(\lambda)+(I+R_{q}^{+}(\lambda)P_{q}^{-}Q^{+}_{\upsilon}(\lambda))L_{q}^{+}.$

Deriving (5.34) we have also taken into account that

$R_{q}^{+}(\lambda)P_{q}^{-}Q^{+}_{\upsilon}(\lambda)u_{0}^{+}=0$

due to $u_{0}^{+}\in{\rm ker}\,(K_{q}^{+})$, (5.24), and (5.26). Now, (5.34)
and (5.32) entail (3.1) for the case of $H_{\upsilon}$ with

$n_{q}^{+}:=\inf_{\lambda\in(-b,\infty)}{\rm
rank}\,(P_{q,>}^{-}(\lambda)),\quad q\in{\mathbb{N}}.$

Step 5: Let us prove item (iii) for $H_{\upsilon}$ and $q\in{\mathbb{N}}$.
Note that we have already shown assertion (i) and hence it suffices to verify
the inclusion $\subset$ in (3.2). Since
$\|(I+G_{\upsilon}(\lambda)^{*}G_{\upsilon}(\lambda))^{-1}\|\leq 1$ we get by
(5.15) and (2.6),

(5.35)
$\|S_{q}^{+}(\lambda)\|\leq\|G_{\upsilon}(\lambda)\|^{2}\leq\|\upsilon\|_{L^{p}(\Gamma)}C_{p}(\lambda),\quad\lambda>-b+1.$

Let

(5.36)
$\|\upsilon\|_{L^{p}(\Gamma)}<\upsilon^{+}_{q}:=\sup_{\lambda\in(-b+1,\infty)}\frac{m_{q}^{+}(\lambda)}{C_{p}(\lambda)},$

where $m_{q}^{+}(\lambda)$ is the quantity defined in (5.16). Let $R>0$ be
such that $\Gamma$ is contained in the open ball $B_{R}$ with radius $R$.
Observe that by the diamagnetic inequality

(5.37) $\displaystyle C_{p}(1)$ $\displaystyle=\sup_{0\neq u\in{\rm
H}^{1}_{A}({\mathbb{R}}^{2})}\frac{\left(\int_{\Gamma}|\tau
u|^{2p^{\prime}}ds\right)^{1/p^{\prime}}}{\int_{{\mathbb{R}}^{2}}\big{(}|\Pi(A)u|^{2}+|u|^{2}\big{)}dx}$
$\displaystyle\leq\sup_{0\neq u\in{\rm
H}^{1}_{A}({\mathbb{R}}^{2})}\frac{\left(\int_{\Gamma}|\tau
u|^{2p^{\prime}}ds\right)^{1/p^{\prime}}}{\int_{{\mathbb{R}}^{2}}\big{(}\big{|}\nabla|u|\big{|}^{2}+|u|^{2}\big{)}dx}$
$\displaystyle\leq\sup_{0\neq u\in{\rm
H}^{1}_{A}({\mathbb{R}}^{2})}\frac{\left(\int_{\Gamma}|\tau
u|^{2p^{\prime}}ds\right)^{1/p^{\prime}}}{\int_{B_{R}}\big{(}\big{|}\nabla|u|\big{|}^{2}+|u|^{2}\big{)}dx}$
$\displaystyle=\sup_{0\neq u\in{\rm
H}^{1}(B_{R})}\frac{\left(\int_{\Gamma}|\tau
u|^{2p^{\prime}}ds\right)^{1/p^{\prime}}}{\int_{B_{R}}\big{(}\big{|}\nabla|u|\big{|}^{2}+|u|^{2}\big{)}dx}=:c^{-1},\qquad
p^{\prime}=\frac{p}{p-1},$

where the constant $c>0$ does not depend on the magnetic field. Hence, the
constant $\upsilon^{+}_{q}$ can be estimated from below as

$\upsilon^{+}_{q}\geq\frac{m_{q}^{+}(1)}{C_{p}(1)}\geq\frac{2bc}{(\Lambda_{q}+1)(\Lambda_{q-1}+1)},$

which is the bound in (3.3). Furthermore, (5.35) implies that there exist
$\lambda>-b+1$ such that

$\|S_{q}^{+}(\lambda)\|<m_{q}^{+}(\lambda),$

so that $P_{q,>}^{-}(\lambda)=0$ by (5.19). By (5.34), we conclude that if
$u\in{\rm ker}\,(H_{\upsilon}-\Lambda_{q})$, then $u\in{\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$, as $u_{>}^{-}\in{\rm
ran}\,(P_{q,>}^{-}(\lambda))=\\{0\\}$. Therefore, together with (i) we
conclude that (3.2) holds.

Step 6: Let us now consider $H_{-\upsilon}$, i.e. the Landau Hamiltonian
perturbed by a negative $\delta$-potential. The proof of Theorem 3.1 (ii) and
(iii) in this case is quite similar to the one concerning $H_{\upsilon}$, so
that we omit certain details. Assume

$u\in{\rm ker}\,(H_{-\upsilon}-\Lambda_{q}),\quad q\in{\mathbb{Z}}_{+}.$

Then, similarly to (5.10), we have

(5.38)
$\left((H_{0}+\lambda)^{-1}-\mu_{q}(\lambda)\right)u+Q^{-}_{\upsilon}(\lambda)u=0,\quad\lambda>-\inf\sigma(H_{-\upsilon}).$

Put

$u^{+}:=P_{q+1}^{+}u\quad\text{and}\quad\quad u^{-}:=P_{q+1}^{-}u,$

so that again

$u=u^{+}+u^{-}.$

Then (5.38) implies

(5.39)
$\left(\mu_{q}(\lambda)-(H_{0}+\lambda)^{-1}\right)u^{-}=P_{q+1}^{-}Q^{-}_{\upsilon}(\lambda)u^{-}+P_{q+1}^{-}Q^{-}_{\upsilon}(\lambda)u^{+},$
(5.40)
$\left(\mu_{q}(\lambda)-(H_{0}+\lambda)^{-1}\right)u^{+}=P_{q+1}^{+}Q^{-}_{\upsilon}(\lambda)u^{-}+P_{q+1}^{+}Q^{-}_{\upsilon}(\lambda)u^{+}.$

Let

$S_{q}^{-}(\lambda):=P_{q+1}^{+}Q^{-}_{\upsilon}(\lambda)P_{q+1}^{+}$

and observe that by Lemma 2.1 the operator $S_{q}^{-}(\lambda)$ is compact,
self-adjoint, and non-negative in $P_{q+1}^{+}L^{2}({\mathbb{R}}^{2})$. Set

(5.41)
$m_{q}^{-}(\lambda):=\inf\sigma\left(\left(\mu_{q}(\lambda)-(H_{0}+\lambda)^{-1}\right)_{|P_{q+1}^{+}L^{2}({\mathbb{R}}^{2})}\right)=\frac{2b}{(\Lambda_{q}+\lambda)(\Lambda_{q+1}+\lambda)}$

and

$S_{q,>}^{-}(\lambda):=S_{q}^{-}(\lambda)\,\mathds{1}_{[m_{q}^{-}(\lambda),\infty)}\left(S_{q}^{-}(\lambda)\right),\quad
S_{q,<}^{-}(\lambda):=S_{q}^{-}(\lambda)-S_{q,>}^{-}(\lambda).$

Now (5.40) is equivalent to

(5.42)
$\left(\left(\mu_{q}(\lambda)-(H_{0}+\lambda)^{-1}\right)-S_{q,<}^{-}(\lambda)\right)u^{+}=P_{q+1}^{+}Q^{-}_{\upsilon}(\lambda)u_{-}+S_{q,>}^{-}(\lambda)u^{+}_{>},$

where

$u_{>}^{+}:=P_{q+1,>}^{+}u$

and

$P_{q+1,>}^{+}=P_{q+1,>}^{+}(\lambda):=\mathds{1}_{[m_{q}^{-}(\lambda),\infty)}(S_{q}^{-}(\lambda))\,P_{q+1}^{+}.\\\
$

Note that

${\rm rank}\,(P_{q+1,>}^{+}(\lambda))={\rm
rank}\,(S_{q,>}^{-}(\lambda))<\infty.$

The operator

$\left(\mu_{q}(\lambda)-(H_{0}+\lambda)^{-1}\right)_{|P_{q+1}^{+}L^{2}({\mathbb{R}}^{2})}-S_{q,<}^{-}(\lambda)\\\
$

is positive and boundedly invertible in $P_{q+1}^{+}L^{2}({\mathbb{R}}^{2})$.
Set

$R_{q}^{-}(\lambda):=\left(\left(\mu_{q}(\lambda)-(H_{0}+\lambda)^{-1}\right)_{|P_{q+1}^{+}L^{2}({\mathbb{R}}^{2})}-S_{q,<}^{-}(\lambda)\right)^{-1}.$

Then (5.42) implies

(5.43)
$u^{+}=R_{q}^{-}(\lambda)\,P_{q+1}^{+}\,Q^{-}_{\upsilon}(\lambda)\,u^{-}+R_{q}^{-}(\lambda)\,S_{q,>}^{-}(\lambda)\,u_{>}^{+}.$

Inserting (5.43) into (5.39), we obtain

$P_{q+1}^{-}X_{q}^{-}(\lambda)P_{q+1}^{-}u^{-}=-P_{q+1}^{-}Y_{q}^{-}(\lambda)u_{>}^{+},$

with

$X_{q}^{-}(\lambda):=(H_{0}+\lambda)^{-1}-\mu_{q}(\lambda)+Q^{-}_{\upsilon}(\lambda)+Q^{-}_{\upsilon}(\lambda)R_{q}^{-}(\lambda)P_{q+1}^{+}Q^{-}_{\upsilon}(\lambda)$

and

$Y_{q}^{-}(\lambda):=Q^{-}_{\upsilon}(\lambda)R_{q}^{-}(\lambda)\,S_{q,>}^{-}(\lambda).$

Then, similarly to (5.34), we find that for any $u\in{\rm
ker}\,(H_{\upsilon}-\Lambda_{q})$ we have

(5.44) $u=u_{0}^{-}+W_{q}^{-}u_{>}^{+},$

with

$u_{0}^{-}\in{\rm ker}\,(T_{q}(\upsilon\delta_{\Gamma})),\quad u_{>}^{+}\in
P_{q+1,>}^{+}{\rm ker}\,(H_{-\upsilon}-\Lambda_{q}),$

and an appropriate operator

$W_{q}^{-}:P_{q+1,>}^{+}{\rm ker}\,(H_{-\upsilon}-\Lambda_{q})\to
L^{2}({\mathbb{R}}^{2}).$

Now, (5.44) entails (3.1) for $H_{-\upsilon}$ with

$n_{q}^{-}:=\inf_{\lambda\in(-\inf\sigma(H_{\upsilon}),\infty)}{\rm
rank}\,(P_{q+1,>}^{+}(\lambda))<\infty.$

Finally, we prove (3.2) for $H_{-\upsilon}$. Assume that

(5.45)
$\|\upsilon\|_{L^{p}(\Gamma)}<\upsilon_{q}^{-}:=\sup_{\lambda\in(-b+1,\infty)}\frac{m_{q}^{-}(\lambda)}{C_{p}(\lambda)(1+m_{q}^{-}(\lambda))},$

where $C_{p}(\lambda)$ and $m_{q}^{-}(\lambda)$ are the quantities defined in
(2.6) and (5.41), respectively. Note that using (5.37) we can estimate
$\upsilon_{q}^{-}$ from below as

$\upsilon_{q}^{-}\geq\frac{m_{q}^{-}(1)}{C_{p}(1)(1+m_{q}^{-}(1))}\geq\frac{2bc}{2b+(\Lambda_{q}+1)(\Lambda_{q+1}+1)}.$

Furthermore, there exists $\lambda\in(-b+1,\infty)$ such that

$\|\upsilon\|_{L^{p}(\Gamma)}\,C_{p}(\lambda)<\frac{m_{q}^{-}(\lambda)}{1+m_{q}^{-}(\lambda)}<1.$

By (2.5)

(5.46) $\|G_{\upsilon}(\lambda)\|^{2}\leq
C_{p}(\lambda)\|\upsilon\|_{L^{p}(\Gamma)}\leq\frac{m_{q}^{-}(\lambda)}{1+m_{q}^{-}(\lambda)}<1.$

On the other hand, since
$\|(I-G_{\upsilon}(\lambda)^{*}G_{\upsilon}(\lambda))^{-1}\|\leq(1-\|G_{\upsilon}(\lambda)\|^{2})^{-1}$,
similarly to (5.35) we have

(5.47)
$\|S_{q}^{-}(\lambda)\|\leq\frac{\|G_{\upsilon}(\lambda)\|^{2}}{1-\|G_{\upsilon}(\lambda)\|^{2}}.$

Putting together (5.47) and (5.46) we get

$\|S_{q}^{-}(\lambda)\|<m_{q}^{-}(\lambda),$

so that $P_{q+1,>}^{+}(\lambda)=0$, and (5.44) implies that for $u\in{\rm
ker}\,(H_{-\upsilon}-\Lambda_{q})$ we have $u\in{\rm
ker}\,(T_{q}(\upsilon\delta_{\Gamma}))$, i.e. (3.2) holds for $H_{-\upsilon}$.

###  5.2. Proof of Theorem 3.6

Assume that

$u=p_{0}u\in{\rm ker}\,(T_{0}(\upsilon\delta_{\Gamma})),$

which is equivalent to $\upsilon^{1/2}(u_{|\Gamma})=0$ as an element of
$L^{2}(\Gamma)$. Since we assume that $\upsilon\neq 0$ as an element of
$L^{p}(\Gamma;{\mathbb{R}})$, $p>1$, we have $u=0$ on a subset of $\Gamma$ of
a positive measure. Since $e^{\phi}u$ is entire, cf. (4.5), and its zeros are
isolated if $u\neq 0$, we easily find that $u=0$, i.e. (3.5) holds.

######  Remark 5.1.

The above argument is not applicable in the case $q\geq 1$, because there
exist poly-analytic functions $u$ which do not vanish identically on
${\mathbb{C}}$ but vanish on certain regular Jordan curves (see [6, Section
5]).

###  5.3. Proof of Theorem 3.9

Due to the invariance of $p_{q}$ under the magnetic translations (see (4.10)
and (4.11)) we may assume without loss of generality that ${\mathcal{C}}_{r}$
is centered at the origin.

Let $u\in{\rm ker}(H_{0}-\Lambda_{q})={\rm ran}\,(p_{q})$, $q\in{\mathbb{N}}$.
Then we have

(5.48) $u(x)=\sum_{k\in{\mathbb{Z}}_{+}}c_{k}\varphi_{k,q}(x),\quad
x\in{\mathbb{R}}^{2},$

with ${\bf
c}:=\left\\{c_{k}\right\\}_{k\in{\mathbb{Z}}_{+}}\in\ell^{2}({\mathbb{Z}}_{+})$,
where $\left\\{\varphi_{k,q}\right\\}_{k\in{\mathbb{Z}}_{+}}$ is the
orthonormal basis of ${\rm ker}(H_{0}-\Lambda_{q})$ defined in (4.6). Hence,
the representation in (5.48) generates a unitary operator
${\mathcal{U}}_{q}:{\rm ker}(H_{0}-\Lambda_{q})\to\ell^{2}({\mathbb{Z}}_{+})$
which maps $u$ to ${\bf c}$. On the other hand we have

$\langle
T_{q}(\delta_{{\mathcal{C}}_{r}})\varphi_{k,q},\varphi_{\ell,q}\rangle_{L^{2}({\mathbb{R}}^{2})}=\lambda_{k,q}(r)\delta_{k\ell},$

where

(5.49) $\begin{split}\lambda_{k,q}(r):&=\langle
T_{q}(\delta_{{\mathcal{C}}_{r}})\varphi_{k,q},\varphi_{k,q}\rangle_{L^{2}({\mathbb{R}}^{2})}\\\
&=b\frac{q!}{k!}\left(br^{2}/2\right)^{k-q}\,{\rm
L}_{q}^{(k-q)}\left(br^{2}/2\right)^{2}e^{-br^{2}/2},\quad
r\in(0,\infty).\end{split}$

Then we have

${\mathcal{U}}_{q}T_{q}(\delta_{{\mathcal{C}}_{r}}){\mathcal{U}}_{q}^{*}=\tau_{q,r},$

where $\tau_{q,r}:\ell^{2}({\mathbb{Z}}_{+})\to\ell^{2}({\mathbb{Z}}_{+})$ is
a compact self-adjoint operator defined by

$(\tau_{q,r}{\bf c})_{k}=\lambda_{k,q}(r)c_{k},\quad k\in{\mathbb{Z}}_{+},$

with ${\bf
c}=\left\\{c_{k}\right\\}_{k\in{\mathbb{Z}}_{+}}\in\ell^{2}({\mathbb{Z}}_{+})$.
In particular, the functions $\varphi_{k,q}$ are eigenfunctions of
$T_{q}(\delta_{{\mathcal{C}}_{r}})$ with eigenvalues equal to
$\lambda_{k,q}(r)$. For $r\in(0,\infty)$ set

(5.50) $m_{q}(r):=\\#\left\\{k\in{\mathbb{Z}}_{+}\,|\,{\rm
L}_{q}^{(k-q)}\left(br^{2}/2\right)=0\right\\}.$

Then (5.49) implies

(5.51) ${\rm
dim\,ker}\,(T_{q}(\delta_{{\mathcal{C}}_{r}}))=\\#\left\\{k\in{\mathbb{Z}}_{+}\,|\,\lambda_{k,q}(r)=0\right\\}=m_{q}(r),\quad
r\in(0,\infty),$

and

(5.52) ${\mathcal{D}}_{q}=\left\\{r\in(0,\infty)\,|\,{\rm
dim\,ker}\,(T_{q}(\delta_{{\mathcal{C}}_{r}}))\geq
1\right\\}=\left\\{r\in(0,\infty)\,|\,m_{q}(r)\geq 1\right\\}.$

Bearing in mind the expressions for the Laguerre polynomials ${\rm
L}_{q}^{(k-q)}$ with $q=1,2$, given in (4.7)–(4.8), we find that the zero of
${\rm L}_{1}^{(k-1)}$ is equal to $k$, while the zeros of ${\rm
L}_{2}^{(k-2)}$ are equal to $k\pm\sqrt{k}$, $k\in{\mathbb{Z}}_{+}$. Thus,
(5.51)–(5.52) easily entail the explicit description of the sets
${\mathcal{D}}_{q}$, $q=1,2$, and their components ${\mathcal{D}}_{q,j}$,
available in (3.11)–(3.13).

Let us estimate $m_{q}(r)$ and describe ${\mathcal{D}}_{q}$ in the general
case. Note that the polynomial ${\rm L}_{q}^{(\alpha)}$ with $\alpha>-1$ has
exactly $q$ simple strictly positive zeros (see [43, Theorem 3.3.1] and
(4.9)). Denote by $\left\\{\zeta_{\ell}(\alpha)\right\\}_{\ell=1}^{q}$,
$\alpha\in[0,\infty)$, the set of the zeros of ${\rm L}_{q}^{(\alpha)}$,
enumerated in decreasing order. The functions $\zeta_{\ell}$,
$\ell=1,\ldots,q$, are smooth strictly increasing functions (see [43,
Subsection 6.21 (4)]) which tend to infinity as $\alpha\to\infty$ (see [12]).
Thus we can classify the zeros of ${\rm L}_{q}^{(k-q)}$ with $k\geq q$. In
order to handle the polynomials ${\rm L}_{q}^{(k-q)}$ with $0\leq k<q$ we note
that

(5.53) ${\rm L}_{q}^{(k-q)}(t)=\frac{k!}{q!}(-t)^{q-k}{\rm
L}_{k}^{(q-k)}(t),\quad t\in{\mathbb{R}},$

(see [43, Eq.(5.2.1)]), so that if $k=0$ the polynomial ${\rm
L}_{q}^{(-q)}(t)$ is proportional to $t^{q}$, while if $q\geq 2$ and $1\leq
k<q-1$ the polynomial ${\rm L}_{q}^{(k-q)}$ has $k$ simple positive zeros and
a null root of order $q-k$. If $k=1,\ldots,q$, denote by
$\left\\{z_{m,k}\right\\}_{m=1}^{k}$ the set of the positive zeros of ${\rm
L}_{q}^{(k-q)}$, enumerated in decreasing order. Note that

$z_{\ell,q}=\zeta_{\ell}(0),\quad\ell=1,\ldots,q.$

Moreover, we have

(5.54) $\frac{d}{dt}{\rm L}_{k}^{(q-k)}(t)=-{\rm L}_{k-1}^{(q-k+1)}(t),\quad
t\in{\mathbb{R}},$

(see [43, Eq.(5.1.14)]), so that (5.53), (5.54), and Rolle’s theorem imply
that the zeros $z_{m,k}$ interlace, i.e.

$z_{m+1,k}<z_{m,k-1}<z_{m,k}$

(see [14, 15] for further details). If $q\geq 2$, let us extend the functions
$\zeta_{\ell}$, $\ell=1,\ldots,q-1$, to the interval $[-q+\ell,\infty)$. To
this end, set

$\zeta_{\ell}(-n)=z_{\ell,q-n},\quad n=1,\ldots,q-\ell,$

and interpolate by linear functions on the intervals $(-n,-n+1)$. Thus we
obtain a family of $q$ increasing Lipschitz functions $\zeta_{\ell}(\alpha)$,
$\ell=1,\ldots,q$, defined on $\alpha\in[-q+\ell,\infty)$, which tend to
infinity as $\alpha\to\infty$ and, if $q\geq 2$, we have

$\zeta_{\ell+1}(\alpha)<\zeta_{\ell}(\alpha),\quad\alpha\in[-q+\ell,\infty),\quad\ell=1,\ldots,q-1.$

Set

$\eta_{\ell}(\alpha):=\sqrt{2\zeta_{\ell}(\alpha)/b},\quad\alpha\in[-q+\ell+1,\infty),\quad\ell=1,\ldots,q.$

Thus we find that for any $r\in(0,\infty)$ the quantity $m_{q}(r)$ defined in
(5.50) is equal to the number of integers $\ell\in\\{1,\ldots,q\\}$ for which
$r\in{\rm ran}(\eta_{\ell})$ and $\eta_{\ell}^{-1}(r)\in{\mathbb{N}}-\\{q\\}$.
Then, evidently, $m_{q}(r)\leq q$ and combined with (5.51) this implies
(3.10).

Finally, the set ${\mathcal{D}}_{q}$ is infinite since it contains, for
example, all the points $r=\eta_{1}(k-q)$ with $k\in{\mathbb{N}}$. On the
other hand, ${\mathcal{D}}_{q}$ is discrete because it is locally finite.

##  Appendix A Closedness and semiboundedness of the quadratic form in (2.1)

Recall that we consider for $\upsilon\in L^{p}(\Gamma;\mathbb{R})$ the
following quadratic form

(A.1) $\int_{{\mathbb{R}}^{2}}|\Pi(A)u|^{2}dx+\int_{\Gamma}\upsilon|\tau
u|^{2}\,ds,\quad u\in{\rm H}_{A}^{1}({\mathbb{R}}^{2}).$

The function $\upsilon$ can be decomposed as
$\upsilon=\upsilon_{1}+\upsilon_{2}$, where $\upsilon_{1}\in
L^{\infty}(\Gamma)$ and where $\|\upsilon_{2}\|_{L^{p}(\Gamma)}\leq\delta$ for
arbitrarily small $\delta>0$. First, we get the following elementary estimate

(A.2) $\displaystyle\left|\int_{\Gamma}\upsilon|\tau
u|^{2}\,ds\right|\leq\int_{\Gamma}|\upsilon_{1}||\tau
u|^{2}\,ds+\int_{\Gamma}|\upsilon_{2}||\tau u|^{2}\,ds.$

Next, we estimate the two terms on the right hand side separately. Combining
[8, Lemma 2.6], the diamagnetic inequality [31, Theorem 7.21], and that
$\upsilon_{1}$ is a bounded function we obtain that for any $\varepsilon>0$
there exists $C(\varepsilon)>0$ such that

(A.3) $\int_{\Gamma}|\upsilon_{1}||\tau
u|^{2}\,ds\leq\varepsilon\|\Pi(A)u\|^{2}_{{L}^{2}({\mathbb{R}}^{2})}+C(\varepsilon)\|u\|^{2}_{{L}^{2}({\mathbb{R}}^{2})}.$

By [19, Lemma 5.3] the operator of multiplication
$\mathcal{M}_{|\upsilon_{2}|}$ with $|\upsilon_{2}|$ is bounded from ${\rm
H}^{1/2}(\Gamma)$ into ${\rm H}^{-1/2}(\Gamma)$ and, moreover, its norm
between these two spaces is estimated from above by

$\|\mathcal{M}_{|\upsilon_{2}|}\|_{{\rm H}^{1/2}(\Gamma)\rightarrow{\rm
H}^{-1/2}(\Gamma)}\leq c\|\upsilon_{2}\|_{{L}^{p}(\Gamma)}=c\delta,$

where $c=c(\Gamma,p)>0$. Using the above estimate of the norm of
$\mathcal{M}_{|\upsilon_{2}|}$ and that the mapping $\tau$ is bounded from
${\rm H}_{A}^{1}({\mathbb{R}}^{2})$ into ${\rm H}^{1/2}(\Gamma)$ we get

(A.4) $\displaystyle\int_{\Gamma}|\upsilon_{2}||\tau u|^{2}\,ds$
$\displaystyle\leq\|{\mathcal{M}}_{|\upsilon_{2}|}\tau u\|_{\rm
H^{-1/2}(\Gamma)}\|\tau u\|_{\rm H^{1/2}(\Gamma)}$ $\displaystyle\leq
c\delta\|\tau\|_{{\rm H}_{A}^{1}({\mathbb{R}}^{2})\rightarrow{\rm
H}^{1/2}(\Gamma)}^{2}\|u\|^{2}_{{\rm H}_{A}^{1}({\mathbb{R}}^{2})}.$

Combining the estimates (A.2), (A.3), (A.4), and taking into account that the
decomposition of $\upsilon$ can be chosen such that the parameter $\delta$ is
arbitrarily small, we conclude that for any $\varepsilon^{\prime}>0$ there
exists $C^{\prime}(\varepsilon^{\prime})>0$ so that

$\left|\int_{\Gamma}\upsilon|\tau
u|^{2}\,ds\right|\leq\varepsilon^{\prime}\|\Pi(A)u\|^{2}_{{L}^{2}({\mathbb{R}}^{2})}+C^{\prime}(\varepsilon^{\prime})\|u\|^{2}_{{L}^{2}({\mathbb{R}}^{2})}.$

Hence, it follows from the perturbation result [27, Theorem VI.1.33] that the
quadratic form in (A.1) is closed and semibounded.

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