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The topics of this paper are Fredholm properties and the applicability of the finite section method for band operators on/P-spaces as well as for their norm limits which we call band-dominated operators. The derived criteria will be established in terms of the limit operators of the given band-dominated operator. After presenting the generM theory, we present its specifications to concrete classes of band-dominated operators. IntroductionThe topic of the present paper is band operators and norm limits of band operators -the latter we call band-dominated operators. Let 12(2Z) denote the ttilbert space of the two-sided infinite and squared-summable sequences provided with its standard basis It is easy to check that every band operator can be uniquely written as a finite sum ~tr a~V; where the a~ are bounded multiplication operators (i.e. they are given by a diagonal matrix with respect to the standard basis), and where the V; are the shift operators ej ~-+ ei+j. Conversely, every finite sum ~ aiVi is a band operator. This equivalence allows to think of band operators as being composed of two kinds of 'generators' -multiplication and shift operators -and we shall adopt this point of view in what follows.Band and band-dominated operators appear in numerous branches of mathematics. Archetypical appearances are that as discretizations of partial differential or pseudo-differential operators, and that in signal processing. We shall examine these operators with *Supported by the DFG grant 436 RUS/17/148/95 ?Supported by a DFG Heisenberg grant

The central theme of the present paper are band and band-dominated operators, i.e. norm limits of band operators. In the first part, we generalize the results from [24] and [25] concerning the Fredholm properties of band-dominated operators and the applicability of the finite section method to the case of operators with operator-valued coefficients. We characterize these properties in terms of the limit operators of the given band-dominated operator. The main objective of the second part is to apply these results to pseudodifferential operators on cones in IR n which is possible after a suitable discretization.

We derive a dispersion equation for determining eigenvalues of inhomogeneous quantum wells in terms of spectral parameter power series and apply it for the numerical treatment of eigenvalue problems. The method is algorithmically simple and can be easily implemented using available routines of such environments for scientific computing as MATLAB.

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We prove a statement on the boundedness of a certain class of singular type operators in the weighted spaces L p(·) (R n , w) with variable exponent p(x) and a power type weight w, from which we derive the boundedness of pseudodifferential operators of Hörmander class S 0 1,0 in such spaces. This gives us a possibility to obtain a necessary and sufficient condition for pseudodifferential operators of the class OP S m 1,0 with symbols slowly oscillating at infinity, to be Fredholm within the frameworks of weighted Sobolev spaces H s,p(·) w (R n ) with constant smoothness s, variable p(·)-exponent, and exponential weights w. (2000). Primary 47G30. Mathematics Subject Classification

The paper is devoted to the study of the essential spectrum of discrete Schrödinger operators on the lattice 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 Z 3 .

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