Semiclassical resolvent bounds for weakly decaying potentials

In this note, we prove weighted resolvent estimates for the semi-classical Schrödinger operator $-h^2 \Delta+V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n), n \neq 2$. The potential $V$ is real-valued, and assumed to either decay at infinity or to obey a radial $\alpha$-Hölder continuity condition, $0 \leq \alpha \leq 1$, with sufficient decay of the local radial $C^\alpha$ norm toward infinity. Note, however, that in the Hölder case, the potential need not decay. If the dimension $n \geq 3$, the resolvent bound is of the form $\exp \left( Ch^{-1-\frac{1-\alpha}{3+\alpha}} [(1-\alpha) \log(h^{-1} + c] \right)$, 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.

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1fundingThis material is based upon work supported by the National Science Foundation under Grant No. 1440140, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall semester of 2019. J. Shapiro was also supported in part by the Australian Research Council through grant DP180100589.

Received 21 March 2020

Accepted 23 November 2020

Published 29 September 2022