We study the inverse spectral problem of reconstructing energy-dependent
Sturm-Liouville equations from their Dirichlet spectra and sequences of the
norming constants. For the class of problems under consideration, we give a
complete description of the corresponding spectral data, suggest a
reconstruction algorithm, and establish uniqueness of reconstruction. The
approach is based on connection between spectral problems for energy-dependent
Sturm-Liouville equations and for Dirac operators of special form.

It is well known that the standard projection methods allow one to recover
the whole spectrum of a bounded self-adjoint operator but they often lead to
spectral pollution, i.e. to spurious eigenvalues lying in the gaps of the
essential spectrum. Methods using second order relative spectra are free from
this problem, but they have not been proven to approximate the whole spectrum.
L. Boulton (2006, 2007) has shown that second order relative spectra
approximate all isolated eigenvalues of finite multiplicity.

Let X be a compact Riemannian manifold of dimension two or three and let P be
a point of X. We derive comparison formulas relating the zeta-regularized
determinant of an arbitrary self-adjoint extension of (symmetric) Laplace
operator with domain, consisting of smooth functions with compact supports
which does not contain P, to the zeta-regularized determinant of the
self-adjoint Laplacian on X.

Let $\Lambda$ be a subgroup of an arithmetic lattice in SO(n+1,1). The
quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers
corresponding to primes in some ring of integers. We establish a super-strong
approximation result for Zariski-dense $\Lambda$ with some additional
regularity and thickness properties. Concretely, this asserts a quantitative
spectral gap for the Laplacian operators on the congruence covers. This
generalizes results of Sarnak and Xue (1991) and Gamburd (2002).

In this paper, we mainly focus on how to generalize some conclusions from
\emph{nonnegative irreducible tensors} to \emph{nonnegative weakly irreducible
tensors}. To do so, a basic lemma as Lemma 3.1 of \cite{s11} is proven using
new tools. First, we define the stochastic tensor. Then we show that every
nonnegative weakly irreducible tensor with spectral radius be 1 is diagonally
similar to a unique weakly irreducible stochastic tensor. Based on it, we prove
some important lemmas, which help us to generalize the results.

We obtain pointwise lower bounds for heat kernels of higher order
differential operators with Dirichlet boundary conditions on bounded domains in
$\R^N$. The bounds exhibit explicitly the nature of the spatial decay of the
heat kernel close to the boundary. We make no smoothness assumptions on our
operator coefficients which we assume only to be bounded and measurable.

The joint spectral radius of a bounded set of d times d real or complex
matrices is defined to be the maximum exponential rate of growth of products of
matrices drawn from that set. Under quite mild conditions such a set of
matrices admits an associated vector norm, called a Barabanov norm, which can
be used to characterise those sequences of matrices which achieve this maximum
rate of exponential growth. In this note we continue an earlier investigation
into the problem of determining when the Barabanov norm associated to such a
set of matrices is unique.

We study the direct and inverse spectral problems for semiclassical operators
of the form $S = S_0 +\h^2V$, where $S_0 = \frac 12 \Bigl(-\h^2\Delta_{\bbR^n}
+ |x|^2\Bigr)$ is the harmonic oscillator and $V:\bbR^n\to\bbR$ is a tempered
smooth function. We show that the spectrum of $S$ forms eigenvalue clusters as
$\h$ tends to zero, and compute the first two associated "band invariants". We
derive several inverse spectral results for $V$, under various assumptions.

We consider one-dimensional Schroedinger-type operators in a bounded interval
with non-self-adjoint Robin-type boundary conditions. It is well known that
such operators are generically conjugate to normal operators via a similarity
transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians
in quantum mechanics, we study properties of the transformations in detail. We
show that they can be expressed as the sum of the identity and an integral
Hilbert-Schmidt operator.

In this paper we study a semiclassical heat trace expansion for perturbations
of the harmonic oscillator, by adapting to the semiclassical setting techniques
developed by Hitrik and Polterovich in [HP]. We use the expansion to obtain
certain inverse spectral results.

We consider the Schr\"odinger operator $Hy=-y"+(p+q)y$ with a periodic
potential $p$ plus a compactly supported potential $q$ on the real line. The
spectrum of $H$ consists of an absolutely continuous part plus a finite number
of simple eigenvalues below the spectrum and in each spectral gap $\g_n\ne \es,
n\ge1$.

In this paper we develop and apply methods for the spectral analysis of
non-self-adjoint tridiagonal infinite and finite random matrices, and for the
spectral analysis of analogous deterministic matrices which are pseudo-ergodic
in the sense of E.B.Davies (Commun. Math. Phys. 216 (2001), 687-704). As a
major application to illustrate our methods we focus on the "hopping sign
model" introduced by J.Feinberg and A.Zee (Phys. Rev.

We consider a family of operators $-\Delta+ t V$ with a slowly decaying and
oscillating potential $V$. We prove that the absolutely continuous spectrum of
this operator is essentially supported by $[0,\infty)$ for almost every $t$.

This is a sequel of a recent article by Borichev-Golinskii-Kupin, where the
authors obtain Blaschke-type conditions for special classes of analytic
functions in the unit disk which satisfy certain growth hypotheses. These
results were applied to get Lieb-Thirring inequalities for complex compact
perturbations of a selfadjoint operator with a simply connected resolvent set.

We consider the fractional Laplacian on a domain and investigate the
asymptotic behavior of its eigenvalues. Extending methods from semi-classical
analysis we are able to prove a two-term formula for the sum of eigenvalues
with the leading (Weyl) term given by the volume and the subleading term by the
surface area. Our result is valid under very weak assumptions on the regularity
of the boundary.

We study Laplace operators on infinite networks $(G,c)$, and their
self-adjoint extensions. We consider the Laplacian $\Delta$ as on operator on
$\ell^2(G)$ and as an operator on the Hilbert space $\mathcal{H}_\mathcal{E}$
of finite energy functions on $G$, focusing on the case when $\Delta$ is
unbounded. It is known that $\Delta$ is essentially self-adjoint on its natural
domain in $\ell^2(G)$, but that it is \emph{not} essentially self-adjoint on
its natural domain in $\mathcal{H}_\mathcal{E}$.

We study spectral properties of the Laplace-Beltrami operator on
asymptotically hyperbolic manifolds and their applications to inverse
scattering. We deal with the general short-range perturbation of the metric.
The main part of the monograph deals with the direct problem, namely, (1)
Location of the essential spectrum. (2) Absence of eigenvalues embedded in the
continuous spectrum. (3) Discreteness of embedded eigenvalues in the continuous
spectrum when all the ends are cusps. (4) Limiting absorption principle for the
resolvent and the absolute continuity of the continuous spectrum.

We prove that the inverse spectral mapping reconstructing the square
integrable potentials on [0,1] of Dirac operators in the AKNS form from their
spectral data (two spectra or one spectrum and the corresponding norming
constants) is analytic and uniformly stable in a certain sense.

We prove that the potential of a Sturm--Liouville operator depends
analytically and Lipschitz continuously on the spectral data (two spectra or
one spectrum and the corresponding norming constants). We treat the class of
operators with real-valued distributional potentials in the Sobolev class
W^{s-1}_2(0,1), s\in[0,1].

Following the work of Marzuola & Simpson, we prove the absence of embedded
eigenvalues for a collection of nonlinear Schr\"odinger equations, including
some one and three dimensional supercritical equations, and the three
dimensional cubic-quintic equation.

The proof is computer assisted as it depends on the sign of certain inner
products which do not readily admit analytic representations. Our source code
is available for verification at
this http URL

The purpose of this paper is to give some refined results about the
distribution of resonances in potential scattering. We use techniques and
results from one and several complex variables, including properties of
functions of completely regular growth. This enables us to find asymptotics for
the distribution of resonances in sectors for certain potentials and for
certain families of potentials.

We explain an array of basic functional analysis puzzles on the way to
general spectral flow formulae and indicate a direction of future topological
research for dealing with these puzzles.

A Berezin-Li-Yau type inequality for $(-\Delta)^{\alpha/2}|_{\Omega},$ the
fractional Laplacian operators restriced to a bounded domain $\Omega\subset
\mathbb{R}^d$ for $\alpha\in(0,2],$ $d\ge 2,$ has not been known so far. First
we positively answer this question. Second, we provide an improvement to this
inequality consistent with the work in \cite{Melas,Selma} by using a pure
analytical approach.

We obtain two-sided bounds on kinetic and potential energies of a bound state
of a quantum particle in the semiclassical limit, as the Planck constant
$\hbar\ri 0$.

Proofs of these results rely on the generalized virial theorem obtained in
the paper as well as on a decay of eigenfunctions in the classically forbidden
region.

We prove spectral and dynamical localization for Anderson models on locally
finite graphs using the fractional moment method. Our theorems extend earlier
results on localization for the Anderson model on $\ZZ^d$. We establish
geometric assumptions for the underlying graph such that localization can be
proven in the case of sufficiently large disorder. All results are given with
full proofs.

In this paper we continue our study of the Laplacian on manifolds with axial
analytic asymptotically cylindrical ends initiated in~arXiv:1003.2538. By using
the complex scaling method and the Phragm\'{e}n-Lindel\"{o}f principle we prove
exponential decay of the eigenfunctions corresponding to the non-threshold
eigenvalues of the Laplacian on functions. In the case of a manifold with
(non-compact) boundary it is either the Dirichlet Laplacian or the Neumann
Laplacian.

We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly
singular potentials such as perturbed spherical Schroedinger operators (also
known as Bessel operators). It is known that in such situations one can still
define a corresponding singular Weyl m-function and it was recently shown that
there is also an associated spectral transformation. Here we will give a
general criterion when the singular Weyl function can be analytically extended
to the upper half plane.

We establish a sharp geometric constant for the upper bound on the resonance
counting function for surfaces with hyperbolic ends. An arbitrary metric is
allowed within some compact core, and the ends may be of hyperbolic planar,
funnel, or cusp type. The constant in the upper bound depends only on the
volume of the core and the length parameters associated to the funnel or
hyperbolic planar ends.

We report on recent results on the spectral statistics of the discrete
Anderson model in the localized phase. Our results show, in particular, that,
for the discrete Anderson Hamiltonian with smoothly distributed random
potential at sufficiently large coupling, the limit of the level spacing
distribution is that of i.i.d. random variables distributed according to the
density of states of the random Hamiltonian. This text is a contribution to the
proceedings of the conference "Spectra of Random Operators and Related Topics"
held at Kyoto University, 02-04/12/09 organized by N. Minami and N.

This paper studies eigenvalues of some Steklov problems. Among other things,
we show the following sharp estimtes. Let $\Omega$ be a bounded smooth domain
in an $n(\geq 2)$-dimensional Hadamard manifold an let $0=\lambda_0 <
\lambda_1\leq \lambda_2\leq ... $ denote the eigenvalues of the Steklov
problem: $\Delta u=0$ in $\Omega$ and $(\partial u)/(\partial \nu)=\lambda u$
on $\partial \Omega$. Then $\sum_{i=1}^{n} \lambda^{-1}_i \geq
(n^2|\Omega|)/(|\partial\Omega|) $ with equality holding if and only if
$\Omega$ is isometric to an $n$-dimensional Euclidean ball.

Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold
that is hyperbolic at infinity in the sense that outside of a compact set $K
\subset X$ the sectional curvatures of $g$ are identically equal to minus one.
We prove that the counting function for the resolvent resonances has maximal
order of growth $(n+1)$ generically for such manifolds.

We study the isoresonance problem on non-compact surfaces of finite area that
are hyperbolic outside a compact set. Inverse resonance problems correspond to
inverse spectral problems in the non-compact setting. We consider a conformal
class of surfaces with hyperbolic cusps where the deformation takes place
inside a fixed compact set. Inside this compactly supported conformal class we
consider isoresonant metrics, i.e. metrics for which the set of resonances is
the same, including multiplicities. We prove that sets of isoresonant metrics
inside the conformal class are sequentially compact.

We investigate the edge conductance of particles submitted to an Iwatsuka
magnetic field, playing the role of a purely magnetic barrier. We also consider
magnetic guides generated by generalized Iwatsuka potentials. In both cases we
prove quantization of the edge conductance. Next, we consider magnetic
perturbations of such magnetic barriers or guides, and prove stability of the
quantized value of the edge conductance. Further, we establish a sum rule for
edge conductances. Regularization within the context of disordered systems is
discussed as well.

We consider the "weighted" operator $P_k=-\partial_x a(x)\partial_x$ on the
line with a step-like coefficient which appears when propagation of waves
thorough a finite slab of a periodic medium is studied. The medium is
transparent at certain resonant frequencies which are related to the complex
resonance spectrum of $P_k.$ If the coefficient is periodic on a finite
interval (locally periodic) with $k$ identical cells then the resonance
spectrum of $P_k$ has band structure. In the present paper we study a
transition to semi-infinite medium by taking the limit $k\to \infty.$

We consider the periodic Jacobi operator $J$ with finitely supported
perturbations on $\ell^2(\N)$ subject to Dirichlet boundary condition at $n=0$.
We classify all states of $J$ and give their properties. We solve the inverse
resonance problem

(including characterization): we prove that mapping from real perturbations
to the associated regularized Jost functions is one-to-one and onto.

We discuss recent results on spectral properties of discrete alloy-type
random Schr\"odinger operators. They concern Wegner estimates and bounds on the
fractional moments of the Green's function.

We consider discrete Schroedinger operator J with Wigner-von Neumann
potential not belonging to l^2. We find asymptotics of orthonormal polynomials
associated to J. We prove the Weyl-Titchmarsh type formula, which relates the
spectral density of J to a coefficient in asymptotics of orthonormal
polynomials.

We develop relative oscillation theory for one-dimensional Dirac operators
which, rather than measuring the spectrum of one single operator, measures the
difference between the spectra of two different operators. This is done by
replacing zeros of solutions of one operator by weighted zeros of Wronskians of
solutions of two different operators.

The gap function on the space of compact Riemannian manifolds with boundary
is defined as the difference of the first two Dirichlet eigenvalues, where the
Riemannian metric is rescaled so that the diameter of the manifold is 1.
Estimating the gap function is known as the {\it gap problem}. Our main theorem
reduces the gap problem for domains in $\R^n$ to a certain Neumann problem in
$\R^{n+1}$. The infinitesimal version of this is related to Bakry-\'Emery
geometry; our second theorem embeds the Dirichlet gap problem into a certain
Bakry-\'Emery Neumann problem.

We consider the zigzag half-nanotubes

(tight-binding approximation) in a uniform magnetic field which is described
by the magnetic Schr\"odinger operator with a periodic potential plus a
finitely supported perturbation.

We describe all eigenvalues and resonances of this operator, and theirs
dependence on the magnetic field.

The proof is reduced to the analysis of the periodic Jacobi operators on the
half-line with finitely supported perturbations.

A $n\times n$ matrix $A$, which has a certain sign-symmetric structure
($J$--sign-symmetric), is studied in this paper. It is shown that such a matrix
is similar to a nonnegative matrix. The existence of the second in modulus
positive eigenvalue $\lambda_2$ of a $J$--sign-symmetric matrix $A$, or an odd
number $k$ of simple eigenvalues, which coincide with the $k$-th roots of
$\rho(A)^k$, is proved under the additional condition that its second compound
matrix is also $J$--sign-symmetric.

We study the connection of the existence of solutions with certain properties
and the spectrum of operators in the framework of regular Dirichlet forms on
infinite graphs.

Relations between half- and full-lattice CMV operators with scalar- and
matrix-valued Verblunsky coefficients are investigated. In particular, the
decoupling of full-lattice CMV operators into a direct sum of two half-lattice
CMV operators by a perturbation of minimal rank is studied. Contrary to the
Jacobi case, decoupling a full-lattice CMV matrix by changing one of the
Verblunsky coefficients results in a perturbation of twice the minimal rank.
The explicit form for the minimal rank perturbation and the resulting two
half-lattice CMV matrices are obtained.

We survey Barry Simon's principal contributions to the field of inverse
spectral theory in connection with one-dimensional Schrodinger and Jacobi
operators.

We consider Dirichlet-to-Neumann maps associated with (not necessarily
self-adjoint) Schrodinger operators describing nonlocal interactions in
$L^2(\Omega; d^n x)$, $n\geq 2$, where $\Omega$ is an open set with a compact,
nonempty boundary satisfying certain regularity conditions. As an application
we describe a reduction of a certain ratio of Fredholm perturbation
determinants associated with operators in $L^2(\Omega; d^n x)$ to Fredholm
perturbation determinants associated with operators in $L^2(\partial\Omega;
d^{n-1}\sigma)$.

We study a semiclassical inverse spectral problem based on the spectral
asymptotics of arXiv:math/0502032, which apply to small non-selfadjoint
perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2.
The eigenvalues in a suitable complex window have an expansion in terms of a
quantum Birkhoff normal form for the operator near several Lagrangian tori
which are invariant under the classical dynamics and satisfy a Diophantine
condition.

In this paper, we establish universal inequalities for eigenvalues of the
clamped plate problem on compact submanifolds of Euclidean spaces, of spheres
and of real, complex and quaternionic projective spaces. We also prove similar
results for the biharmonic operator on domains of Riemannian manifolds
admitting spherical eigenmaps (this includes the compact homogeneous Riemannian
spaces) and nally on domains of the hyperbolic space.

We consider a quasinilpotent operator whose resolvent is entire operator
function of exponential type. Let A be its one-dimensional perturbation. We
establish necessity of Muckenhoupt condition (A2) for two weights related to
operator A for unconditional basisness of its eigenfunctions. Note that no
apriori restrictions are imposed on the spectrum of A.

We consider the spectrum of the Fibonacci Hamiltonian for small values of the
coupling constant. It is known that this set is a Cantor set of zero Lebesgue
measure. Here we study the limit, as the value of the coupling constant
approaches zero, of its thickness and its Hausdorff dimension. We prove that
the thickness tends to infinity and, consequently, the Hausdorff dimension of
the spectrum tends to one. We also show that at small coupling, all gaps
allowed by the gap labeling theorem are open and the length of every gap tends
to zero linearly.

We classify the hulls of different limit-periodic potentials and show that
the hull of a limit-periodic potential is a procyclic group.

We also describe how limit-periodic potentials can be generated from a
procyclic group and answer some arising questions. As an expository paper, we
discuss the connection between limit-periodic potentials and profinite groups
as completely as possible and review recent results on Schroedinger operators
that were obtained in this context.

We consider a magnetic Schr\"odinger operator $H^h$, depending on the
semiclassical parameter $h>0$, on a two-dimensional Riemannian manifold. We
assume that there is no electric field. We suppose that the minimal value $b_0$
of the magnetic field $b$ is strictly positive, and there exists a unique
minimum point of $b$, which is non-degenerate. The main result of the paper is
a complete asymptotic expansion for the low-lying eigenvalues of the operator
$H^h$ in the semiclassical limit.

We prove the Spectral Mapping Theorem for the Helffer-Sj\"ostrand functional
calculus for linear operators on Banach spaces with real spectra and
consequently give a new proof for the Spectral Mapping Theorem for self-adjoint
operators on Hilbert spaces

The paper discusses the spectrum of Toeplitz operators in Bargmann spaces.
Our Toeplitz operators have real symbols with a variable sign and a compact
support. A class of examples is considered where the asymptotics of the
eigenvalues of such operators can be computed. These examples show that this
asymptotics depends on the geometry of the supports of the positive and
negative parts of the symbol. Applications to the perturbed Landau Hamiltonian
are given.

We prove the following. For any complex valued $L^2$-function $b(x)$ the
spectrum of a perturbed harmonic oscillator operator $L = -\frac{d^2}{dx^2} +
x^2 + b(x)$ in $L^2(\mathbb{R}^1)$ is discrete and eventually simple. Its SEAF
(system of eigen- and associated functions) is an unconditional basis in
$L^2(\mathbb{R}^1)$.

We study self-adjoint bounded Jacobi operators of the form:

(J \psi)(n) = a_n \psi(n + 1) + b_n \psi(n) +a_{n-1} \psi(n - 1) on
$\ell^2(\N)$. We assume that for some fixed q, the q-variation of $\{a_n\}$ and
$\{b_n\}$ is square-summable and $\{a_n\}$ and $\{b_n\}$ converge to q-periodic
sequences. Our main result is that under these assumptions the essential
support of the absolutely continuous part of the spectrum of J is equal to that
of the asymptotic periodic Jacobi operator. This work is an extension of a
recent result of S.A.Denisov.

The angular synchronization problem is to obtain an accurate estimation (up
to a constant additive phase) for a set of unknown angles
$\theta_1,...,\theta_n$ from $m$ noisy measurements of their offsets
$\theta_i-\theta_j \mod 2\pi$. Of particular interest is angle recovery in the
presence of many outlier measurements that are uniformly distributed in
$[0,2\pi)$ and carry no information on the true offsets. We introduce an
efficient recovery algorithm for the unknown angles from the top eigenvector of
a specially designed Hermitian matrix.

Let $\fre\subset\bbR$ be a finite union of disjoint closed intervals. We
study measures whose essential support is $\fre$ and whose discrete eigenvalues
obey a 1/2-power condition. We show that a Szeg\H{o} condition is equivalent to
\[ \limsup \f{a_1... a_n}{\ca(\fre)^n}>0 \] (this includes prior results of
Widom and Peherstorfer--Yuditskii). Using Remling's extension of the
Denisov--Rakhmanov theorem and an analysis of Jost functions, we provide a new
proof of Szeg\H{o} asymptotics, including $L^2$ asymptotics on the spectrum.

We study the inverse resonance problem for conformally compact manifolds
which are hyperbolic outside a compact set. Our results include compactness of
isoresonant metrics in dimension two and of isophasal negatively curved metrics
in dimension three. In dimensions four or higher we prove topological
finiteness theorems under the negative curvature assumption.

In this note we investigate the asymptotic behaviour of the $s$-numbers of
the resolvent difference of two generalized self-adjoint, maximal dissipative
or maximal accumulative Robin Laplacians on a bounded domain $\Omega$ with
smooth boundary $\partial\Omega$. For this we apply the recently introduced
abstract notion of quasi boundary triples and Weyl functions from extension
theory of symmetric operators together with Krein type resolvent formulae and
well-known eigenvalue asymptotics of the Laplace-Beltrami operator on
$\partial\Omega$.

We prove dynamical upper bounds for discrete one-dimensional Schroedinger
operators in terms of various spacing properties of the eigenvalues of finite
volume approximations. We demonstrate the applicability of our approach by a
study of the Fibonacci Hamiltonian.

We address the problem on the right definition of the Schroedinger operator
with potential $\delta'$, where $\delta$ is the Dirac delta-function. Namely,
we prove the uniform resolvent convergence of a family of Schroedinger
operators with regularized short-range potentials $\epsilon^{-2}V(x/\epsilon)$
tending to $\delta'$ in the distributional sense as $\epsilon\to 0$. In 1986,
P.

It is proved that small periodic singular perturbation of a cylindrical
waveguide surface may open a gap in the continuous spectrum of the Dirichlet
problem for the Laplace operator. If the perturbation period is long and the
caverns in the cylinder are small, the gap certainly opens.

We consider Sturm-Liouville operators $-y''+v(x)y$ on $[0,1]$ with Dirichlet
boundary conditions $y(0)=y(1)=0$. For any $1\le p<\infty$, we give a short
proof of the characterization theorem for the spectral data corresponding to
$v\in L^p(0,1)$.

A Szeg\"o-type theorem for Toeplitz operators was proved by Boutet de Monvel
and Guillemin for general Toeplitz structures. We give a local version of this
result in the setting of positive line bundles on compact symplectic manifolds.

Given a self-adjoint operator H, a self-adjoint trace class operator V and a
fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using
limiting absorption principle an explicit set of full Lebesgue measure is
defined such that for all points of this set the wave and the scattering
matrices can be defined and constructed unambiguously. Many well-known
properties of the wave and scattering matrices and operators are proved,
including the stationary formula for the scattering matrix.

We propose a simple proof of characterization of the eigenspaces
corresponding to the eigenvalues $\pm m$ of a supersymmetric Dirac operator
$H=Q + m\tau$, where $Q$ is a supercharge, $m$ a positive constant, and $\tau$
the unitary involution. The proof is abstract, but not relevant to the abstract
Foldy-Wouthuysen transformation.

The Bethe-Sommerfeld conjecture states that the spectrum of the stationary
Schrodinger operator with a periodic potential in dimensions higher than 1 has
only finitely many gaps. After work done by many authors, it has been proven by
now in full generality. The similar conjecture in presence of a periodic
magnetic potential has been proven in dimension 2 only. Another case of a
significant interest, due to its importance for the photonic crystal theory, is
of a periodic Maxwell operator, where apparently no results of such kind are
known.

For certain compactly supported metric and/or potential perturbations of the
Laplacian on $\mathbb{H}^{n+1}$, we establish an upper bound on the resonance
counting function with an explicit constant that depends only on the dimension,
the radius of the unperturbed region in $\mathbb{H}^{n+1}$, and the volume of
the metric perturbation. This constant is shown to be sharp in the case of
scattering by a spherical obstacle.

Let $\mu$ be a matrix-valued measure with the essential spectrum a single
interval and countably many point masses outside of it. Under the assumption
that the absolutely continuous part of $\mu$ satisfies Szego's condition and
the point masses satisfy a Blaschke-type condition, we obtain the asymptotic
behavior of the orthonormal polynomials on and off the support of the measure.

The result generalizes the scalar analogue of Peherstorfer-Yuditskii and the
matrix-valued result of Aptekarev-Nikishin, which handles only a finite number
of mass points.

A simple sufficient condition on curved end of a straight cylinder is found
that provides a localization of the principal eigenfunction of the mixed
boundary value for the Laplace operator with the Dirichlet conditions on the
lateral side. Namely, the eigenfunction concentrates in the vicinity of the
ends and decays exponentially in the interior. Similar effects are observed in
the Dirichlet and Neumann problems, too.

This is the second in a series of papers on scattering theory for
one-dimensional Schr\"odinger operators with Miura potentials admitting a
Riccati representation of the form $q=u'+u^2$ for some $u\in L^2(R)$. We
consider potentials for which there exist `left' and `right' Riccati
representatives with prescribed integrability on half-lines. This class
includes all Faddeev--Marchenko potentials in $L^1(R,(1+|x|)dx)$ generating
positive Schr\"odinger operators as well as many distributional potentials with
Dirac delta-functions and Coulomb-like singularities.

Selfadjoint Sturm-Liouville operators $H_\omega$ on $L_2(a,b)$ with random
potentials are considered and it is proven, using positivity conditions, that
for almost every $\omega$ the operator $H_\omega$ does not share eigenvalues
with a broad family of random operators and in particular with operators
generated in the same way as $H_\omega$ but in $L_2(\tilde a,\tilde b)$ where
$(\tilde a,\tilde b)\subset(a,b)$.

In this article we improve a lower bound for $\sum_{j=1}^k\beta_j$ (a
Berezin-Li-Yau type inequality) in [E. M. Harrell II and S. Yildirim Yolcu,
Eigenvalue inequalities for Klein-Gordon Operators, J. Funct. Analysis, 256(12)
(2009) 3977-3995]. Here $\beta_j$ denotes the $j$th eigenvalue of the Klein
Gordon Hamiltonian $H_{0,\Omega}=|p|$ when restricted to a bounded set
$\Omega\subset {\mathbb R}^n$. $H_{0,\Omega}$ can also be described as the
generator of the Cauchy stochastic process with a killing condition on
$\partial \Omega$. (cf. [R. Banuelos, T.

In this paper such Riemann metrics are established whose Laplace-Beltrami
operators are identical to familiar Hamilton operators of elementary particle
systems. Such metrics are the natural positive definite invariant metrics
defined on two-step nilpotent Lie groups. The corresponding wave and
Schroedinger operators emerge in the Laplacians of the static resp. solvable
extensions of these nilpotent groups. The latter manifolds are endowed with
natural invariant indefinite metric of Lorentz signature.

In this paper a tight lower bound for algebraic connectivity of graphs
(second smallest eigenvalue of the Laplacian matrix of the graph) based on
connection-graph-stability method is introduced. The connection-graph-stability
score for each edge is defined as the sum of the length of all the shortest
paths making use of that edge. We prove that the algebraic connectivity of the
graph is lower bounded by the size of the graph divided by the maximum
connection graph stability of the edges.

For the Toeplitz quantization of complex-valued functions on a
$2n$-dimensional torus we prove that the expected number of eigenvalues of
small random perturbations of a quantized observable satisfies a natural Weyl
law. In numerical experiments the same Weyl law also holds for ``false''
eigenvalues created by pseudospectral effects.

A refined notion of curvature for a linear system of Hermitian vector spaces,
in the sense of Grothendieck, leads to the unitary classification of a large
class of analytic Hilbert modules. Specifically, we study Hilbert sub-modules,
for which the localizations are of finite (but not constant) dimension, of an
analytic function space with a reproducing kernel.

Given a self-adjoint involution J on a Hilbert space H, we consider a
J-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjoint
operator commuting with J and V a bounded J-self-adjoint operator
anti-commuting with J. We establish optimal estimates on the position of the
spectrum of L with respect to the spectrum of A and we obtain norm bounds on
the operator angles between maximal uniformly definite reducing subspaces of
the unperturbed operator A and the perturbed operator L.

We turn back to the well known problem of interpretation of the Schrodinger
operator with the pseudopotential being the first derivative of the Dirac
function. We show that the problem in its conventional formulation contains
hidden parameters and the choice of the proper selfadjoint operator is
ambiguously determined. We study the asymptotic behavior of spectra and
eigenvectors of the Hamiltonians with increasing smooth potentials perturbed by
short-range potentials. Appropriate solvable models are constructed and the
corresponding approximation theorems are proved.

We answer Mark Kacs famous question - can one hear the shape of a drum - in
the negative for orbifolds that are spherical space forms. This is done by
extending the techniques developed by A. Ikeda on Lens Spaces to the orbifold
setting. Several results are proved to show that with certain restrictions on
the dimensionalities of orbifold Lens spaces we can obtain infinitely many
pairs of isospectral non-isometric Lens spaces. These results are then
generalized to show that for any dimension greater than 8 we can have pairs of
isospectral non-isometric orbifold Lens spaces.

We consider the 3D Pauli operator with nonconstant magnetic field B of
constant direction, perturbed by a symmetric matrix-valued electric potential V
whose coefficients decay fast enough at infinity. We investigate the low-energy
asymptotics of the corresponding spectral shift function. As a corollary, for
generic negative V, we obtain a generalized Levinson formula, relating the
low-energy asymptotics of the eigenvalue counting function and of the
scattering phase of the perturbed operator.

Spectral properties of 1-D Schr\"odinger operators
$\mathrm{H}_{X,\alpha}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2} + \sum_{x_{n}\in
X}\alpha_n\delta(x-x_n)$ with local point interactions on a discrete set
$X=\{x_n\}_{n=1}^\infty$ are well studied when
$d_*:=\inf_{n,k\in\N}|x_n-x_k|>0$. Our paper is devoted to the case $d_*=0$. We
consider $\mathrm{H}_{X,\alpha}$ in the framework of extension theory of
symmetric operators by applying the technique of boundary triplets and the
corresponding Weyl functions.

We examine the adjacency matrices of three-regular graphs representing
one-face maps. Numerical studies reveal that the limiting eigenvalue statistics
of these matrices are the same as those of much larger, and more widely studied
classes from Random Matrix Theory.