Semiclassical resolvent estimates and wave decay in low regularity


We study weighted resolvent bounds for semiclassical Schrodinger 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 \(h^{-4/3}\log(h^{-1})\). Vodev recently showed this bound also holds for short range \(L^\infty\) potentials. 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.


  •  Jacob Shapiro
    Jacob Shapiro, Australian National University