Resolvent bounds for Lipschitz potentials in dimension two and higher with singularities at the origin

TL;DR

This paper establishes resolvent bounds for semiclassical Schrödinger operators with potentials that have singularities at the origin in dimensions two and higher, showing exponential bounds in the general case and linear bounds outside a certain region.

Contribution

It provides new resolvent bounds for potentials with specific singularities at the origin, extending previous results to higher dimensions and more singular potentials.

Findings

Resolvent bound is exponential in h^{-1} for the full operator.

Exterior resolvent bound is linear in h^{-1}.

Results apply to potentials with controlled singularities at the origin.

Abstract

We consider, for $h,E>0$, the semiclassical Schr\"odinger operator $-h^2\Delta + V - E$ in dimension two and higher. The potential $V$, and its radial derivative $\dell_{r}V$ are bounded away from the origin, have long-range decay and $V$ is bounded by $r^{-\delta}$ near the origin while $\dell_{r}V$ is bounded by $r^{-1-\delta}$, where $0\leq\delta\leq 4(\sqrt{2}-1)$. In this setting, we show that the resolvent bound is exponential in $h^{-1}$, while the exterior resolvent bound is linear in $h^{-1}$.