Distribution of Resonances in Scattering by Thin Barriers

Abstract: We study high energy resonances for the operators $-\Delta +\delta_{\partial\Omega}\otimes V$ and $-\Delta+\delta_{\partial\Omega}'\otimes V\partial_\nu$ where $\Omega$ is strictly convex with smooth boundary, $V:L^2(\partial\Omega)\to L^2(\partial\Omega)$ may depend on frequency, and $\delta_{\partial\Omega}$ is the surface measure on $\partial\Omega$. These operators are model Hamiltonians for quantum corrals and leaky quantum graphs. We give a quantum version of the Sabine Law from the study of acoustics for both the $\delta$ and $\delta'$ interactions. It characterizes the decay rates (imaginary parts of resonances) in terms of the system's ray dynamics. In particular, the decay rates are controlled by the average reflectivity and chord length of the barrier. For the $\delta$ interaction we show that generically there are infinitely many resonances arbitrarily close to the resonance free region found by our theorem. In the case of the $\delta'$ interaction, the quantum Sabine law gives the existence of a resonance free region that converges to the real axis at a fixed polynomial rate and is optimal in the case of the unit disk in the plane. As far as the author is aware, this is the only class of examples that is known to have resonances converging to the real axis at a fixed polynomial rate but no faster. The proof of our theorem requires several new technical tools. We adapt intersecting Lagrangian distributions to the semiclassical setting and give a description of the kernel of the free resolvent as such a distribution. We also construct a semiclassical version of the Melrose--Taylor parametrix for complex energies. We use these constructions to give a complete microlocal description of boundary layer operators and to prove sharp high energy estimates on the boundary layer operators in the case that $\partial\Omega$ is smooth and strictly convex.