Semiclassical resolvent bounds for short range $L^{\infty }$ potentials with singularities at the origin

Abstract

We consider, for $h,E>0$, resolvent estimates for the semiclassical Schrödinger operator $-h^2\Delta +V-E$. Near infinity, the potential takes the form $V=V_L+V_S$, where $V_L$ is a long range potential which is Lipschitz with respect to the radial variable, while $V_S=O(|x{|}^{-1}(\log |x|{)}^{-\rho })$ for some $\rho >1$. Near the origin, $|V|$ may behave like $|x{|}^{-\beta }$, provided $0\le \beta <2(\sqrt{3}-1)$. We find that, for any $\tilde{\rho }>1$, there are $C,h_0>0$ such that we have a resolvent bound of the form $\exp (C{h}^{-2}(\log (h^{-1}){)}^{1+\tilde{\rho }})$ for all $h\in (0,h_0]$. The $h$-dependence of the bound improves if $V_S$ decays at a faster rate toward infinity.

arXiv comment19 pages