Related papers: Criteria for embedded eigenvalues for discrete Sch…

Let $H_0$ be a periodic operator on $\R^+$(or periodic Jacobi operator on $\N$). It is known that the absolutely continuous spectrum of $H_0$ is consisted of spectral bands $\cup[\alpha_l,\beta_l]$. Under the assumption that $\limsup_{x\to…

We investigate the eigengenvalues problem for self-adjoint operators with the singular perturbations. The general results presented here includes weakly as well as strongly singular cases. We illustrate these results on two models which…

We study location of eigenvalues of one-dimensional discrete Schr\"odinger operators with complex $\ell^{p}$-potentials for $1\leq p\leq \infty$. In the case of $\ell^{1}$-potentials, the derived bound is shown to be optimal. For $p>1$, two…

We consider non-self-adjoint Schr\"{o}dinger operators $H_{{\rm c}}=-\Delta+V_{{\rm c}}$ (resp. $H_{{\rm r}}=-\Delta+V_{{\rm r}}$) acting in $L^2(\mathbb R^d)$, $d\ge 1$, with dilation analytic complex (resp. real) potentials. We were able…

For Schr\"odinger operator $H=-\Delta+ V({\mathbf x})\cdot$, acting in the space $L_2(\mathbb R^d)\,(d\ge 3)$, necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum.are obtained without assumption that…

We prove a dispersive estimate for the evolution of Schroedinger operators H = -\Delta + V(x) in three dimensions. The potential should belong to the closure of bounded compactly-supported functions with respect to the golbal Kato norm.…

It is known that the spectrum of Schr\"odinger operators with sparse potentials consists of singular continuous spectrum. We give a sufficient condition so that the edge of the singular continuous spectrum is not an eigenvalue and construct…

We obtain a representation formula for the derivative of the spectral shift function $\xi(\lambda; B, \epsilon)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$ and $H(B, \epsilon) = H_0(B, \epsilon) + V(x,y),…

The aim of this paper is to provide uniform estimates for the eigenvalue spacings of one-dimensional semiclassical Schr\"odinger operators with singular potentials on the half-line. We introduce a new development of semiclassical measures…

The goal of this paper is the spectral analysis of the Schr\"{o}dinger type operator $H=L+V$, the perturbation of the Taibleson-Vladimirov multiplier $L=\mathfrak{D}^{\alpha}$ by a potential $V$. Assuming that $V$ belongs to a certain class…

Following the proof given by Froese and Herbst in [FH82] with another conjugate operator, we show for a class of real potential that possible eigenfunction of the Schr\"odinger operator has to decay sub-exponentially. We also show that, for…

This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrodinger operator perturbed by a fast decreasing potential. We use a recently developed technique, the so called…

Commutator relations are used to investigate the spectra of Schr\"odinger Hamiltonians, $H = -\Delta + V({x}),$ acting on functions of a smooth, compact $d$-dimensional manifold $M$ immersed in $\bbr^{\nu}, \nu \geq d+1$. Here $\Delta$…

This work is a continuation of our previos paper, where for the Schr\"odinger operator $H=-\Delta+ V(\e)\cdot$ $(V(\e)\ge 0)$, acting in the space $L_2(\R^d)\,(d\ge 3)$, some sufficient conditions for discreteness of its spectrum have been…

The lowest eigenvalue of the Schr\"odinger operator $-\Delta+\mathcal{V}$ on a compact Riemannian manifold without boundary is studied. We focus on the particularly subtle case of a sign changing potential with positive average.

We show that whole-line Schr\"odinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential…

Eigenfunctions and eigenvalues of the free magnetic Schr\"odinger operator, describing a spinless particle confined to an infinite layer of fixed width, are discussed in detail. The eigenfunctions are realized as an orthonormal basis of a…

We review various attempts to localize the discrete spectra of semirelativistic Hamiltonians of the form H = \beta \sqrt{m^2 + p^2} + V(r) (w.l.o.g. in three spatial dimensions) as entering, for instance, in the spinless Salpeter equation.…

Given a selfadjoint magnetic Schr\"odinger operator \begin{equation*} H = ( i \partial + A(x) )^2 + V(x) \end{equation*} on $L^{2}(\mathbb{R}^n)$, with $V(x)$ strictly subquadratic and $A(x)$ strictly sublinear, we prove that the flow…

Let $H$ be any $\PT$ symmetric Schr\"odinger operator of the type $ -\hbar^2\Delta+(x_1^2+...+x_d^2)+igW(x_1,...,x_d)$ on $L^2(\R^d)$, where $W$ is any odd homogeneous polynomial and $g\in\R$. It is proved that $\P H$ is self-adjoint and…