Degenerate Schr\"odinger equations with irregular potentials

TL;DR

This paper studies degenerate Schr"odinger equations with irregular potentials, establishing well-posedness, spectral properties, and $L^p$ bounds using a H"ormander metric and weighted symbol classes.

Contribution

It introduces a framework with a H"ormander metric and weights to analyze degenerate Schr"odinger equations, deriving well-posedness, spectral, and $L^p$ estimates for these operators.

Findings

Established well-posedness for degenerate Schr"odinger and parabolic equations.

Derived spectral properties for degenerate Hamiltonians under subellipticity.

Obtained sharp $L^p$ estimates and Schatten class properties for Schr"odinger operators.

Abstract

In this work we investigate a class of degenerate Schr\"odinger equations associated to degenerate elliptic operators with irregular potentials on $\Ran$ by introducing a suitable H\"ormander metric $g$ and a $g$-weight $m$. We establish the well-posedness for the corresponding degenerate Schr\"odinger 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 Schr\"odinger operators for H\"ormander sums of squares are also investigated.