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Vladimir S. Rabinovich

Publications and source records attributed to Vladimir S. Rabinovich.

4 recordsLinked to original sources
arXiv2011

Pseudodifferential operators on periodic graphs

The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph $Γ$ which is periodic with respect to the action of the group ${\mathbb Z}^n$. The operators under consideration are distinguished by their local behavior: they act as (Fourier) pseudodifferential operators in the class $OPS^0$ on every open edge of the graph, and they can be represented as a matrix Mellin pseudodifferential operator on a neighborhood of every vertex of $Γ$. We apply these results to study the Fredholm property of a class of singular integral operators and of certain locally compact operators on graphs.

math.FA↗
arXiv2006

The essential spectrum of Schrödinger operators on lattices

The paper is devoted to the study of the essential spectrum of discrete Schrödinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit operators method. This method has been applied by one of the authors to describe the essential spectrum of (continuous) electromagnetic Schrödinger operators, square-root Klein-Gordon operators, and Dirac operators under quite weak assumptions on the behavior of the magnetic and electric potential at infinity. The present paper is aimed to illustrate the applicability and efficiency of the limit operators method to discrete problems as well. We consider the following classes of the discrete Schrödinger operators: 1) operators with slowly oscillating at infinity potentials, 2) operators with periodic and semi-periodic potentials; 3) Schrödinger operators which are discrete quantum analogs of the acoustic propagators for waveguides; 4) operators with potentials having an infinite set of discontinuities; and 5) three-particle Schrödinger operators which describe the motion of two particles around a heavy nuclei on the lattice $\mathbb{Z}^3$.

math-ph↗
arXiv2006

The Fredholm index of locally compact band-dominated operators on $L^p (R)$

We establish a necessary and sufficient criterion for the Fredholmness of a general locally compact band-dominated operator $A$ on $L^p(R)$ and solve the long-standing problem of computing its Fredholm index in terms of the limit operators of $A$. The results are applied to operators of convolution type with almost periodic symbol.

math.FA↗