INFINITEMATRICES · Spectra, Fredholm Properties and Stable Approximation of Infinite Matrices

Spectra, fredholm properties and stable approximation of infinite matrices

The research carried out under this project has produced significant contributions to both spectral theory and stable approximation schemes of general and concrete classes of infinite matrices (most notably in the field of random operators). The objectives of this project have been (i) to extend the Fredholm and spectral theory of large classes of infinite matrices in several directions; (ii) to develop new stable approximation schemes for the solution of operator equations; and (iii) to apply both to concrete operators from mathematical physics. Eigenvalue or spectral problems appear in countless fields of natural and engineering sciences. Often the matrix is too large for linear algebra to work. Then one is interested in asymptotic properties of the quantity that is to be studied (e.g. eigenvalues) as the matrix size goes to infinity and instead to study the infinite counterpart - an infinite matrix - by means of functional analysis. There are different ways to derive information about the spectrum of an infinite matrix. One of these ways is to study the asymptotic behaviour of the matrix entries. This leads to so-called limit operators and subsequently to a full description of the essential spectrum. To gather information about other parts of the spectrum, we have studied finite principal submatrices and were able to derive exciting new upper bounds on the spectrum that nicely complement the lower bounds derived before. This is especially useful since sharp upper bounds are rare in practice (typical candidates are Gershgorin's circles and the closed numerical range but these are in general far from being as sharp as ours) and since our results apply to the very general case of Jacobi matrices with operator entries and hence to arbitrary band matrices. Besides the study of spectra, we also focus on ways to solve infinite linear systems Ax = b by truncation techniques. The canonical way to do this is to cut finite square matrices out of the infinite matrix A and to solve the corresponding finite systems instead. This is the finite section method (FSM). However, it is very easy to give examples, where this method fails to approximate the solution x of our infinite system Ax = b. We have done a rigorous study of ways to guarantee convergence by putting the finite submatrices in other (well-defined) places, by avoiding certain sizes but also by using rectangular instead of square submatrices. All of the above has been done in a sufficiently general setting but has also been applied to particular operators A that are important in applications, such as scattering problems by acoustic or electromagnetic waves and equations from quantum mechanics (Schrödinger and related operators). Most notable are the results on spectra but also on the finite section method in the booming subject of random Jacobi (including Schrödinger) operators. The attached picture shows some features of the particularly intricate spectrum of such an operator.

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