Semiclassical resolvent bounds for long range Lipschitz potentials

TL;DR

This paper provides an elementary proof of weighted resolvent estimates for semiclassical Schrödinger operators with long-range potentials, revealing sharp energy dependence and growth behavior in the resolvent norm.

Contribution

It introduces a simplified proof method for resolvent bounds in long-range Lipschitz potentials, clarifying energy dependence and growth rates, and addresses a previously open question.

Findings

Resolvent norm grows exponentially in 1/h, linearly near infinity.

When potential is compactly supported, resolvent growth is linear with appropriate weights.

The energy dependence of the bounds is shown to be sharp.

Abstract

We give an elementary proof of weighted resolvent estimates for the semiclassical Schr\"odinger operator $-h^2 \Delta + V(x) - E$ in dimension $n \neq 2$, where $h, \, E > 0$. The potential is real-valued, $V$ and $\partial_r V$ exhibit long range decay at infinity, and may grow like a sufficiently small negative power of $r$ as $r \to 0$. The resolvent norm grows exponentially in $h^{-1}$, but near infinity it grows linearly. When $V$ is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{-1/2}$ for some $C > 0$. This $E$-dependence is sharp and answers a question of Datchev and Jin.