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Semiclassical resolvent bounds for long range Lipschitz potentials by Galkowski, Jeffrey; Shapiro, Jacob is a scholarly article available to read on EtoBox.

What is Semiclassical resolvent bounds for long range Lipschitz potentials about?

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.

Author

Galkowski, Jeffrey; Shapiro, Jacob

Published

2020

Language

EN