Spectral Theory
We study the trace class perturbations of the whole-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we refine the Lieb--Thirring…
Spectral inclusion and spectral pollution results are proved for sequences of linear operators of the form $T_0 + i \gamma s_n$ on a Hilbert space, where $s_n$ is strongly convergent to the identity operator and $\gamma > 0$. We work in…
We consider boundary conditions of self-adjoint banded Toeplitz matrices. We ask if boundary conditions exist for banded self-adjoint Toeplitz matrices which satisfy operator inequalities of Dirichlet-Neumann bracketing type. For a special…
Upper and lower estimates of eigenvalues of the Laplacian on a metric graph have been established in 2017 by G. Berkolaiko, J.B. Kennedy, P. Kurasov and D. Mugnolo. Both these estimates can be achieved at the same time only by highly…
We consider the inverse Sturm-Liouville problem with one discontinuous point on three-star graph, we deduced the distribution of the eigenvalues, and proved that one spectrum could uniquely determine the unknown potential and jump…
In 1977, Victor Guillemin published a paper discussing geometric scattering theory, in which he related the Lax-Phillips Scattering matrices (associated to a noncompact hyperbolic surface with cusps) and the sojourn times associated to a…
We study the Schr\"odinger operator in the plane with a step magnetic field function. The bottom of its spectrum is described by the infimum of the lowest eigenvalue band function, for which we establish the existence and uniqueness of the…
Unfortunately the proof of the main result of [1], Theorem 1, has a flaw. Namely, Lemma 13 used in the proof of Proposition 11 is correct only under an additional assumption that the operator $A$ is normal (adjoint for the one-sided shift…
We develop a theory of regularity for Dirac operators with uniformly locally square-integrable operator data. This is motivated by Stahl--Totik regularity for orthogonal polynomials and by recent developments for continuum Schr\"odinger…
The spectrum of the normalized complex Laplacian for electrical networks is analyzed. We show that eigenvalues lie in a larger region compared to the case of the real Laplacian. We show the existence of eigenvalues with negative real part…
The paper is concerned with the stability property under perturbation $Q\to\widetilde Q$ of different spectral characteristics of a BVP associated in $L^2([0,1];\Bbb C^2)$ with the following $2\times2$ Dirac type equation…
The Krein-von Neumann extension is studied for Schr\"odinger operators on metric graphs. Among other things, its vertex conditions are expressed explicitly, and its relation to other self-adjoint vertex conditions (e.g.…
We study orthogonal polynomials with periodically modulated Jacobi parameters in the case when $0$ lies on the soft edge of the spectrum of the corresponding periodic Jacobi matrix. We determine when the orthogonality measure is absolutely…
Our studies concern some aspects of scattering theory of the singular differential systems $ y'-x^{-1}Ay-q(x)y=\rho By, \ x>0 $ with $n\times n$ matrices $A,B, q(x), x\in(0,\infty)$, where $A,B$ are constant and $\rho$ is a spectral…
We obtain upper bounds for the Steklov eigenvalues $\sigma_k(M)$ of a smooth, compact, connected, $n$-dimensional submanifold $M$ of Euclidean space with boundary $\Sigma$ that involve the intersection indices of $M$ and of $\Sigma$. One of…
This article is a brief presentation of results surrounding the fundamental gap. We begin by recalling Bakry-Emery geometry and demonstrate connections between eigenvalues of the Laplacian with the Dirichlet and Neumann boundary conditions.…
We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the…
We demonstrate lower bounds for the eigenvalues of compact Bakry-Emery manifolds with and without boundary. The lower bounds for the first eigenvalue rely on a generalised maximum principle which allows gradient estimates in the Riemannian…
We prove that the presence or absence of corners is spectrally determined in the following sense: any simply connected domain with piecewise smooth Lipschitz boundary cannot be isospectral to any connected domain, of any genus, which has…
In this paper we define a new convergence called "asymptotically conic convergence" in which a smooth family of Riemannian metrics on a fixed compact manifold degenerate to a metric with isolated conic singularity. Our results are:…