Harmonic Analysis and Differential Equations Student Seminar: Semiclassical resolvent bound for compactly supported Hölder continuous potentials

Seminar | November 5 | 3:40-5 p.m. | 740 Evans Hall

 Jacob Shapiro, ANU

 Department of Mathematics

We prove a weighted resolvent estimate for the semiclassical Schrödinger operator $-h^2 \Delta + V : L^2(\mathbb { R }^n) \to L^2(\mathbb { R }^n)$, $n \ge 3$. We assume the potential $V$ is compactly supported and α-Hölder continuous, $0< \alpha < 1$. The logarithm of the resolvent norm grows like $h^{-1-\frac {1-\alpha }{3 + \alpha }}\log (h^{-1})$ as the semiclassical parameter $h \to 0^+$. This bound interpolates between the previously known $h$-dependent resolvent bounds for Lipschitz and $L^\infty $ potentials. To key step is to prove a suitable global Carleman estimate, which we establish via a spherical energy method. This is joint work with Jeffrey Galkowski.