Monday, October 30:
Jacob Shapiro (Purdue)
``Resolvent estimates and local energy decay in low regularity via Carleman estimates and distribution of resonances''
We study weighted resolvent bounds for semiclassical Schrödinger operators in dimension two. We require the potential function to be Lipschitz with long range decay. The resolvent norm grows exponentially in the inverse semiclassical parameter. 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 logarithmic local energy decay for the wave equation with a wavespeed that is a Lipschitz perturbation of unity. The decay rate is the same as that proved by Burq for a smooth wavespeed outside an obstacle.
We also present what appears to be the first explicit resolvent bound for compactly supported $L^\infty$ potentials, which holds in the semiclassical limit. This is a step toward showing the optimal bound as conjectured by Vodev.