Steffen Roch

C[*]-algebras and numerical analysis

The language of numerical analysis regularization approximation of spectra stability analysis representation theory Fredholm sequences self-adjoint approximation sequences.

Band-dominated operators with operator-valued coefficients , their Fredholm properties and finite sections

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 ℝn which is possible after a suitable discretization.

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 .

Algebras generated by idempotents and the symbol calculus for singular integral operators

It is proved that in Banach algebras generated by two idempotents and, perhaps, by a certain flip operator the standard identity F4 is fulfilled. The maximal ideal space of such algebras is determined and the corresponding symbol is given. By means of local techniques these results are applied to obtain a symbol calculus for singular integral operators with Carleman shift (changing the orientation) in weighted Banach spaces.

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.

Approximation methods for singular integral equations with conjugation on curves with corners

Approximation methods for singular integral operators with continuous coefficients and conjugation as well as for the double-layer potential operator on curves with corners are investigated with respect to their stability. The methods under consideration include several kinds of quadrature rules and collocation and qualocation methods. The approach is based on the thorough use of Banach algebra techniques and local principles. Complete necessary and sufficient stability conditions are derived.

Finite Section Method in some Algebras of Multiplication and Convolution Operators and a Flip

Abstract. This paper is concerned with the applicability of the finite section method to oper-ators belonging to the closed subalgebra of L(L 2 ( R )) generated by operators of multiplicationby piecewise continuous functions in R˙ , convolution operators – also with piecewise continuousgenerating functions – and the flip operator (Ju)(x) = u(−x). For this, a larger algebra ofsequences is introduced, which contains the special sequences we are interested in. There is adirect relationship between the applicability of the finite section method for a given operatorand the invertibility of the corresponding sequence in this algebra. Exploring this relationship,the methods of essentialization, localization and identification of the local algebras throughconstruction of locally equivalent representations are used and so useful invertibility criteriaare derived. Finally, examples are presented, including explicit conditions for the applicabilityof the finite section method to a Wiener-Hopf plus Hankel operator with piecewise continuoussymbols, and some relations between the approximation operators and the limit operator arediscussed.Keywords: Finite section method, Wiener-Hopf operators, Hankel operators.AMS subject classification: 65R20, 47C15.

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 .

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.