  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.
