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We study the cut-off resolvent of semiclassical Schrödinger operators on with bounded compactly supported potentials . We prove that for real energies in a compact interval in and for any smooth cut-off function supported in a ball near the support of the potential , for some constant , one has
This bound shows in particular an upper bound on the imaginary parts of the resonances , defined as a pole of the meromorphic continuation of the resolvent as an operator : any resonance with real part in a compact interval away from has imaginary part at most
This is related to a conjecture by Landis: The principal Carleman estimate in our proof provides as well a lower bound on the decay rate of solutions to with . We show that there exists a constant such that for any such , for sufficiently large, one has
We prove a quantum version of the Sabine law from acoustics describing the location of resonances in transmission problems. This work extends the work of the author to a broader class of systems. Our main applications are to scattering by transparent obstacles, scattering by highly frequency-dependent delta potentials, and boundary stabilized wave equations. We give a sharp characterization of the resonance-free regions in terms of dynamical quantities. In particular, we relate the imaginary part of resonances, or generalized eigenvalues, to the chord lengths and reflectivity coefficients for the ray dynamics, thus proving a quantum version of the Sabine law.
We prove space-time dispersive estimates for solutions to the wave equation on compact Riemannian manifolds with bounded curvature tensor, where we assume that the metric tensor is of regularity for some , which ensures that the curvature tensor is well-defined in the weak sense. The estimates are established for the same range of Lebesgue and Sobolev exponents that hold in the case of smooth metrics. Our results are for bounded time intervals, so by finite propagation velocity they hold also on noncompact manifolds under appropriate uniform geometry conditions.
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