Matrix Schrödinger Operators and Weighted Bergman Kernels

The aim of the present thesis is twofold: to study the problem of discreteness of the spectrum of Schrödinger 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 Schödinger operator with matrix-valued potential.

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