Marko Lindner

Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators.

Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrodinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

Approximating the Inverse of Banded Matrices by Banded Matrices with Applications to Probability and Statistics

In the first part of this paper we give an elementary proof of the fact that if an infinite matrix

Convergence and Numerics of a Multisection Method for Scattering by Three-Dimensional Rough Surfaces

A

The Finite Section Method in the Space of Essentially Bounded Functions: An Approach Using Limit Operators

, which is invertible as a bounded operator on

Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non–self–adjoint random operator

\ell^2

Boundary Integral Equations on Uubounded Rough Surfaces: Fredholmness and the Finite Section Method

, can be uniformly approximated by banded matrices then so can the inverse of

An affirmative answer to a core issue on limit operators

A

Finite Sections of Random Jacobi Operators

. We give explicit formulas for the banded approximations of

Invertibility at Infinity of Band-Dominated Operators on the Space of Essentially Bounded Functions

A^{-1}