The HADES seminar on Tuesday, November 5th will be given by Jacob Shapiro in Evans 740 from 3:40 to 5 pm.

Speaker: Jacob Shapiro, ANU

Abstract: 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 $\alpha$-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.