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arXiv · 2302.02413

Degenerate Schr\"odinger equations with irregular potentials

Duván Cardona·Marianna Chatzakou·Julio Delgado·Michael Ruzhansky
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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.

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Duván Cardona, Marianna Chatzakou, Julio Delgado, Michael Ruzhansky. 2023-02-05. Degenerate Schr\"odinger equations with irregular potentials. https://arxiv.org/abs/2302.02413

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