Matrix Schr\"odinger Operators and Weighted Bergman Kernels

TL;DR

This thesis investigates the spectral properties of matrix-valued Schr"odinger operators in real space and derives new bounds for weighted Bergman kernels in complex space, linking the two through the analysis of the weighted Kohn Laplacian.

Contribution

It provides new insights into the discreteness of the spectrum for matrix Schr"odinger operators and establishes novel pointwise bounds for weighted Bergman kernels, connecting these areas via the weighted Kohn Laplacian.

Findings

Characterization of spectrum discreteness for matrix Schr"odinger operators.

New pointwise bounds for weighted Bergman kernels in ${f C}^n$.

Connection between Bergman kernel analysis and matrix-valued Schr"odinger operators.

Abstract

The aim of the present thesis is twofold: to study the problem of discreteness of the spectrum of Schr\"odinger operators with matrix-valued potentials in ${\mathbb R}^d$ (Chapter 1), and to prove new pointwise bounds for weighted Bergman kernels in ${\mathbb C}^n$ (chapters 2 and 3). These two themes are connected by the observation that the weighted Bergman kernel may be effectively studied by means of the analysis of the weighted Kohn Laplacian, which in turn is unitarily equivalent to a Sch\"odinger operator with matrix-valued potential.