Research in Schroedinger Operators

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

DMS-0400940 Title: Research in Schroedinger operators PI: Dirk Hundertmark (University of Illinois) ABSTRACT The PI proposes research in three areas in the spectral theory of Schr"odinger operators. Firstly, the PI will continue work on bounds for eigenvalue moments of single-particle Schr"odinger operators, the Lieb-Thirring inequalities. Lieb-Thirring inequalities are known to be an indispensable tool in the spectral theory of Schr"odinger operators, especially for the study of macroscopic Coulomb systems. Besides their original application, they have proven to be useful in other fields, for example, in non-linear PDE. The objective of the proposal is twofold: to achieve a good understanding of the scalar case for small moments and low dimensions, and to develop a general method to lift bounds from the scalar case to operator-valued potentials. In this way, estimates in low dimension would have immediate consequences for bounds in higher dimensions. Secondly, the author plans to study Wegner estimates for the integrated density of states for singular random potentials. These are important a-priori estimates in the spectral theory of random Schr"odinger operators. The third problem focuses on spectral properties of a non self-adjoint matrix Schr"odinger operator, given by the linearization of NLS around solitons. In particular, the question of embedded eigenvalues will be studied. These results should lead to a better understanding of asymptotic completeness for focusing NLS. The proposed projects aim at some basic questions in mathematical physics, more precisely, the rigorous study of non-relativistic quantum mechanics. Recent results on Lieb-Thirring inequalities have revived the general interest in this field and have proven useful in other areas of mathematical physics. It is expected that they will continue to do so. One aim of this proposal is to find new tools to establish these kind of inequalities. Random Schr"odinger operators model the behaviour of electrons in disordered matter. It is a fact from every day life that highly disordered materials are bad conductors. In the last two decades, our mathematical understanding of this and related effects has grown rapidly. But basically all models still have some unrealistic assumptions. An important a priori estimate for the spectral theory of random Schr"odinger operators are regularity results (i.e., smoothness) for the so-called density of states. Despite a lot of attention from several groups working in theory of random Schr"odinger operators, the regularity proofs are still mainly based on the old ideas of Wegner. They require strong nd mainly technical assumptions, excluding most physically relevant rough random potentials. Except for some very special cases, singular random potentials are still out of reach of the existing methods. So new ideas and, in particular, a better understand of the underlying physical mechanism will be required, in order to be able to establish the regularity results for rough random potentials. The third problem, embedded eigenvalues for a certain class of non self-adjoint matrix Schr"odinger operators, is motivated by the study of the non-linear Schr"odinger equation. By itself it is still a problem in linear functional analysis. However, with the twist that very a fine and detailed knowledge of the properties of this linearized operator is needed in order to improve the understanding of the long-time properties of solutions of the non-linear Schr"odinger equation.