Semiclassical Resolvent Bounds for Long-Range Lipschitz Potentials

Abstract

Abstract We give an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 \Delta + V(x) - E$ in dimension $n \neq 2$, where $h, \, E> 0$. The potential is real valued and $V$ 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.