Degenerate Schrödinger equations with irregular potentials

Abstract

In this work we investigate a class of degenerate Schrödinger equations associated to degenerate elliptic operators with irregular potentials on $\Ran$ by introducing a suitable Hörmander metric $g$ and a $g$-weight $m$. We establish the well-posedness for the corresponding degenerate Schrödinger and degenerate parabolic equations. When the subelliticity is available on the degenerate elliptic operator we deduce spectral properties for a class of degenerate Hamiltonians. We also study the $L^p$ mapping properties for operators with symbols in the $S(m^{-\beta },g)$ classes in the spirit of classical Fefferman's $L^p$-bounds for the $(\rho ,\delta )$ calculus. Finally, within our $S(m,g)$-classes, sharp $L^p$-estimates and Schatten properties for Schrödinger operators for Hörmander sums of squares are also investigated.

45 Pages