Related papers: On the Singular Weyl-Titchmarsh Function of Pertur…

In this paper the asymmetric generalization of the Glazman-Povzner-Wienholtz theorem is proved for one-dimensional Schr\"{o}dinger operators with strongly singular matrix potentials from the space $H_{loc}^{-1}(\mathbb{R},…

Using the machinery of unitary spherical harmonics due to Koornwinder, Folland and other authors, we~obtain expansions for the Szeg\"o and the weighted Bergman kernels of $M$-harmonic functions, i.e.~functions annihilated by the invariant…

We study perturbation theory in certain quantum mechanics problems in which the perturbing potential diverges at some points, even though the energy eigenvalues are smooth functions of the coefficient of the potential. We discuss some of…

We give an exposition on the $L^2$ theory of the perturbed Fourier transform associated with a Schr\"odinger operator $H=-d^2/dx^2 +V$ on the real line, where $V$ is a real-valued \mbox{finite} measure. In the case $V\in L^1\cap L^2$, we…

In this paper, we obtain conditions of Noetherian solvability and the Index formula for a singular integral equation with a Cauchy kernel and a Carleman shift in Besov space, which is embedded into the space of continuous functions on a…

Schr\"odinger operator on half-line with complex potential and the corresponding evolution are studied within perturbation theoretic approach. The total number of eigenvalues and spectral singularities is effectively evaluated. Wave…

We construct a class of matrix-valued Schr\"odinger operators with prescribed finite-band spectra of maximum spectral multiplicity. The corresponding matrix potentials are shown to be stationary solutions of the KdV hierarchy. The methods…

This work focuses on the analysis of the spectral $\zeta$-function associated with a Schr\"{o}dinger operator endowed with a P\"oschl--Teller potential. We construct the spectral $\zeta$-function using a contour integral representation and,…

The transmutation (transformation) operator associated with the perturbed Bessel equation is considered. It is shown that its integral kernel can be uniformly approximated by linear combinations of constructed here generalized wave…

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the…

We study discrete Schroedinger operators with compactly supported potentials on the square lattice. Constructing spectral representations and representing S-matrices by the generalized eigenfunctions, we show that the potential is uniquely…

Exact solvability (ES) of one-dimensional quantum potentials $V(x)$ is a vague concept. We propose that beyond its most conventional range the ES status should be attributed also to many less common interaction models for which the wave…

We established a generalized version of the Christ-Kiselev's multi-linear operator technique to deal with the spectral theory of Schr\"odinger operators. As applications, several spectral results of perturbed periodic Schr\"odinger…

In this paper, we provide the boundedness property of the Riesz transforms associated to the Schr\"odinger operator $\mathcal{L}=-\Delta + \mathbf{V}$ in a new weighted Morrey space which is the generalized version of many previous Morrey…

We consider Sturm-Liouville operators on geometrical graphs without cycles (trees) with singular potentials from the class $W_2^{-1}$. We suppose that the potentials are known on a part of the graph, and study the so-called partial inverse…

We derive an asymptotic expansion for the Weyl function of a one-dimensional Schr\"odinger operator which generalizes the classical formula by Atkinson. Moreover, we show that the asymptotic formula can also be interpreted in the sense of…

We introduce a notion of spectral singularity that applies for a general class of nonlinear Schreodinger operators involving a confined nonlinearity. The presence of the nonlinearity does not break the parity-reflection symmetry of spectral…

We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the $A-$amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl $m-$function associated with the…

Let ${\bf R}$ denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear…

We give three formulas for meromorphic eigenfunctions (scattering states) of Sutherland's integrable N-body Schroedinger operators and their generalizations. The first is an explicit computation of the Etingof-Kirillov traces of…