Semiclassical Resolvent Estimates and Wave Decay in Low Regularity
In this thesis, we prove weighted resolvent upper bounds for semiclassical Schrödinger operators. These upper bounds hold in the semiclassical limit. First, we consider operators in dimension two when the potential is Lipschitz with long range decay. We prove that the resolvent norm grows at most exponentially in the inverse semiclassical parameter, while near infinity it grows at most linearly. Both of these bounds are optimal. Second, we work in any dimension and require that the potential be bounded and have compact support. Again, we find that the weighted resolvent norm grows at most exponentially, but this time with an additional loss in the exponent. Finally, we apply the resolvent bounds to prove two logarithmic local energy decay rates for the wave equation, one when the wavespeed is a compactly supported Lipschitz perturbation of unity, and the other when the wavespeed is a compactly supported, bounded perturbation of unity.
Datchev, Purdue University.
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