Related papers: Spectrally determined singularities in a potential…

In this work, we study the inverse spectral problems for the Sturm-Liouville operators on [0,1] with complex coefficients and a discontinuity at $x=a\in(0,1)$. Assume that the potential on (a,1) and some parameters in the discontinuity and…

This note deals with the direct and inverse spectral analysis for a class of infinite band symmetric matrices. This class corresponds to operators arising from difference quations with usual and inner boundary conditions. We give a…

In this paper, we investigate the $W^{s,p}$-boundedness for stationary wave operators of the Schr\"odinger operator with inverse-square potential $$\mathcal L_a=-\Delta+\tfrac{a}{|x|^2}, \quad a\geq -\tfrac{(d-2)^2}{4},$$ in dimension…

We consider eigenvalues of the Dirichlet-to-Neumann operator for Laplacian in the domain (or manifold) with edges and establish the asymptotics of the eigenvalue counting function \begin{equation*} \mathsf{N}(\lambda)= \kappa_0\lambda^d…

It has been recently conjectured that the spectral determinants of operators associated to mirror curves can be expressed in terms of a generalization of theta functions, called quantum theta functions. In this paper we study the symplectic…

Consider the operator $H\p=-\p''+q\p=\l\p$, $\p(0)=0$, $\p'(1)+b\p(1)=0$ acting in $L^2(0,1)$, where $q\in L^2(0,1)$ is a real potential. Let $\l_n(q,b)$, $n\ge 0$, be the eigenvalues of $H$ and $\n_n(q,b)$ be the so-called norming…

We study the phenomenon of an eigenvalue emerging from essential spectrum of a Schroedinger operator perturbed by a fast oscillating compactly supported potential. We prove the sufficient conditions for the existence and absence of such…

We prove sufficient conditions for Hausdorff convergence of the spectra of sequences of closed operators defined on varying Hilbert spaces and converging in norm-resolvent sense, i.e. $\|J_\varepsilon(1+A_\varepsilon)^{-1} -…

We consider self-adjoint extensions of differential operators of the type $ (-\frac{d^2}{dr^2} + \frac{l(l+1)}{r^2})^3 $ on the real semi-axis for l=1,2 with two kinds of boundary conditions: first that nullify the value of a function and…

First we study the spectral singularity at infinity and investigate the connections of the spectral singularities and the spectrality of the Hill operator. Then we consider the spectral expansion when there is not the spectral singularity…

We obtain a solution to the Bessis-Moussa-Villani conjecture for a trace-class perturbation of a semi-bounded operator and answer affirmatively the question on positivity of higher order spectral shift functions in the setting of…

We determine the $L^p$-spectrum of the Schr\"odinger operator with the inverted harmonic oscillator potential $V(x)=-x^2$ for $1 \leq p \leq \infty$.

A Borg-Marchenko type uniqueness theorem (in terms of the Weyl function) is obtained here for the system auxiliary to the N-wave equation. A procedure to solve inverse problem is used for this purpose. The asymptotic condition on the Weyl…

We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{\nu,p}(a)=\sum_{n\geq 1} (an^p/2)^{-\nu} K_\nu(an^p)\qquad (a>0,\ 0\leq\nu<1)\] as the parameter $a\to 0+$, where $p$…

This paper provides a method to study the non-negativity of certain linear operators, from other operators with similar spectral properties. If these new operators are formally self-adjoint and non-negative, we can study the complex powers…

In this survey we review positive inverse spectral and inverse resonant results for the following kinds of problems: Laplacians on bounded domains, Laplace-Beltrami operators on compact manifolds, Schr\"odinger operators, Laplacians on…

We consider an inverse problem for Schr\"odinger operators on connected equilateral graphs with standard matching conditions. We calculate the spectral determinant and prove that the asymptotic distribution of a subset of its zeros can be…

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependance on the spectral parameter. The corresponding hierarchy of integrable equations…

A Hankel operator $\Gamma$ in $L^2(\mathbb{R}_+)$ is an integral operator with the integral kernel of the form $h(t+s)$, where $h$ is known as the kernel function. It is known that $\Gamma$ is positive semi-definite if and only if $h$ is…

In this paper we discuss the spectral properties of one-term symmetric differential operators of even order with interior singularity, namely, we determine the deficiency numbers, describe its self-adjoint extensions and their spectrum. It…