Special Analysis Seminar: Semiclassical resolvent estimates and wave decay in low regularity

Seminar | June 14 | 4:10-5 p.m. | 740 Evans Hall

 Jacob Shapiro, Purdue University

 Department of Mathematics

We study weighted resolvent bounds for semiclassical Schrödinger operators. When the potential function is Lipschitz with long range decay, the resolvent norm grows exponentially in the inverse semiclassical parameter $h$. When the potential belongs to $L^\infty $ and has compact support, the resolvent norm grows exponentially in $h^{-4/3}\log (h^{-1})$. This extends the works of Burq, Cardoso-Vodev, and Datchev. Our main tool is a global Carleman estimate. Applying the resolvent estimates along with the resonance theory for blackbox perturbations, we show local energy decay for the wave equation with wavespeeds that are an $L^\infty $ perturbation of unity.