We consider, for $h, E > 0$, resolvent estimates for the semiclassical Schr\"odinger 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 $\exp(Ch^{ -2} (\log(h^{ -1}))^{ 1 + \tilde{ \rho}})$ for all $h \in (0, h_0]$. The $h$-dependence of the bound improves if $V_S$ decays at a faster rate toward infinity.
Categories: math.AP