Semiclassical resolvent bounds for weakly decaying potentials

Abstract

In this note, we prove weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 Δ+ V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $n \neq 2$. The potential $V$ is real-valued, and assumed to either decay at infinity or to obey a radial $α$-Hölder continuity condition, $0\leq α\leq 1$, with sufficient decay of the local radial $C^α$ norm toward infinity. Note, however, that in the Hölder case, the potential need \emph{not} decay. If the dimension $n \ge 3$, the resolvent bound is of the form $\exp \left(C h^{-1 - \frac{1 - α}{3 + α}} [(1-α) \log(h^{-1})+c]\right)$, while for $n = 1$ it is of the form $\exp(Ch^{-1})$. A new type of weight and phase function construction allows us to reduce the necessary decay even in the pure $L^\infty$ case.