Domain characterization for Schr\"odinger operators with sub-quadratic singularity

TL;DR

This paper provides a comprehensive characterization of the domain of Schr"odinger operators with sub-quadratic singularities across all parameter ranges, extending known results to previously unaddressed cases.

Contribution

It offers an explicit description of the domain of Schr"odinger operators with singular potentials for all parameter ranges, including cases not covered by prior methods.

Findings

Explicit domain descriptions for all parameter ranges

Extension of known results to new parameter regimes

Unified framework for Schr"odinger operators with singularities

Abstract

We characterize the domain of the Schr\"odinger operators $S=-\Delta+c|x|^{-\alpha}$ in $L^p(\mathbb{R}^N)$, with $0<\alpha<2$ and $c\in\mathbb{R}$. When $\alpha p< N$, the domain characterization is essentially known and can be proved using different tools, for instance kernel estimates and potentials in the Kato class or in the reverse H\"older class. However,the other cases seem not to be known, so far.In this paper, we give the explicit description of the domain of $S$ for all range of parameters $p,\alpha$ and $c$.