Vladimir S. Rabinovich

The essential spectrum of Schrödinger operators on lattices

The paper is devoted to the study of the essential spectrum of discrete Schr?dinger operators on the lattice 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 Schr?dinger operators, square-root Klein?Gordon operators and Dirac operators under quite weak assumptions on the behaviour of the magnetic and electric potential at infinity. The present paper aims at illustrating the applicability and efficiency of the limit operators method to discrete problems as well. We consider the following classes of the discrete Schr?dinger operators: (1) operators with slowly oscillating at infinity potentials, (2) operators with periodic and semi-periodic potentials, (3) Schr?dinger operators which are discrete quantum analogues of the acoustic propagators for waveguides, (4) operators with potentials having an infinite set of discontinuities and (5) three-particle Schr?dinger operators which describe the motion of two particles around a heavy nuclei on the lattice .

Essential spectra of difference operators on -periodic graphs

Let (X, ρ) be a discrete metric space. We suppose that the group acts freely on X and that the number of orbits of X with respect to this action is finite. Then we call X a -periodic discrete metric space. We examine the Fredholm property and essential spectra of band-dominated operators on lp(X) when 1 < p < ∞. Our approach is based on the theory of band-dominated operators on and their limit operators. In the case where X is the set of vertices of a combinatorial graph, the graph structure defines a Schrodinger operator on lp(X) in a natural way. We illustrate our approach by determining the essential spectra of Schrodinger operators with slowly oscillating potential both on zig-zag and on hexagonal graphs, the latter being related to nano-structures.

Algebras of Approximation Sequences: Finite Sections of Band-Dominated Operators

We develop the stability theory for the finite section method for general band-dominated operators on lp spaces over Zk. The main result says that this method is stable if and only if each member of a whole family of operators – the so-called limit operators of the method – is invertible and if the norms of these inverses are uniformly bounded.

Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics

This paper is devoted to estimates of the exponential decay of eigenfunctions of difference operators on the lattice which are discrete analogs of the Schrodinger, Dirac and square-root Klein–Gordon operators. Our investigation of the essential spectra and the exponential decay of eigenfunctions of the discrete spectra is based on the calculus of pseudodifference operators (i.e., pseudodifferential operators on the group with analytic symbols), and the limit operators method. We obtain a description of the location of the essential spectra and estimates of the eigenfunctions of the discrete spectra of the main lattice operators of quantum mechanics, namely: matrix Schrodinger operators on , Dirac operators on and square root Klein–Gordon operators on .

Spectral parameter power series analysis of isotropic planarly layered waveguides

This work addresses the analysis of an isotropic planarly layered waveguide consisting of an inhomogeneous core that is enclosed between two homogeneous layers forming the cladding. The analysis relies on an auxiliary one-dimensional spectral problem that is intimately linked with the scalar wave equation for planarly layered media. We construct the Green function of the waveguide as an expansion involving the eigenfunctions of the continuous and the discrete spectrum of the auxiliary problem. From the eigenvalues of the discrete spectrum, we calculate the allowed propagation constants of the guided modes. The Spectral Parameter Power Series (SPPS) method [Math. Method Appl. Sci. 2010;33: 459–468] leads us to analytic expressions for the eigenfunctions of the auxiliary problem in the form of power series of the spectral parameter. In addition, we obtain an SPPS representation for the dispersion relation without making any kind of approximation or discretisation to the core of the waveguide. The SPPS analysis here presented is well suited for its numerical implementation, since all these series can be truncated due to their uniform convergence.

The Finite Sections Approach to the Index Formula for Band-dominated Operators

Recently J. Roe and two of the authors derived a formula which expresses the Fredholm index of a band-dominated operator on l p (ℤ) in terms of local indices of its limit operators. The proof makes thoroughly use of K-theory for C*-algebras (which, of course, appears as a natural approach to index problems). The purpose of this short note is to develop a completely different approach to the index formula for band-dominated operators which is exclusively based on ideas and results from asymptotic numerical analysis.

Quaternionic Fundamental Solutions for Electromagnetic Scattering Problems and Application

We propose a new class of fundamental solutions for the numerical analysis of boundary value problems for the Maxwell equations. We prove completeness of systems of such fundamental solutions in appropriate Sobolev spaces on a smooth boundary and support the relevancy of our approach by numerical results.

Finite sections of band-dominated operators with almost periodic coefficients

We consider the sequence of the finite sections R n AR n of a band-dominated operator A on l 2(ℤ) with almost periodic coefficients. Our main result says that if the compressions of A onto ℤ+ and ℤ− are invertible, then there is a distinguished subsequence of (R n AR n) which is stable. Moreover, this subsequence proves to be fractal, which allows us to establish the convergence in the Hausdorff metric of the singular values and pseudoeigenvalues of the finite section matrices.

Time-Frequency Integrals and the Stationary Phase Method in Problems of Waves Propagation from Moving Sources ?

The time-frequency integrals and the two-dimensional stationary phase method are applied to study the electromagnetic waves radiated by moving modulated sources in dispersive media. We show that such unified approach leads to explicit expressions for the field amplitudes and simple relations for the field eigenfrequencies and the retardation time that become the coupled variables. The main features of the technique are illustrated by examples of the moving source fields in the plasma and the Cherenkov radiation. It is emphasized that the deeper insight to the wave effects in dispersive case already requires the explicit formulation of the dispersive material model. As the advanced application we have considered the Doppler frequency shift in a complex single-resonant dispersive metamaterial (Lorenz) model where in some frequency ranges the negativity of the real part of the refraction index can be reached. We have demonstrated that in dispersive case the Doppler frequency shift acquires a nonlinear dependence on the modulating frequency of the radiated particle. The detailed frequency dependence of such a shift and spectral behavior of phase and group velocities (that have the opposite directions) are studied numerically.

Effective methods of estimates of acoustic fields in the ocean generated by moving sources

We consider underwater acoustic fields generated by non-uniformly moving modulated sources. We introduce a large parameter which characterizing simultaneously a slowness of variations of source amplitudes, and their horizontal velocities, and a large distances between sources and receivers with respect to the characteristic length of the acoustic wave. We present the asymptotic formulas for the acoustic fields as a sum of propagating modes. Applying the spectral parameter power series method (SPPS method), we obtain an analytic form of the dispersion equation and the propagating modes. We show that the SPPS method is an effective tool for the numerical implementation of the underwater acoustic fields generated by non uniformly moving sources.