Galkowski, J; (2019) Distribution of Resonances in Scattering by Thin Barriers. Memoirs of the American Mathematical Society , 259 , Article 1248. 10.1090/memo/1248.

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Abstract

We study high energy resonances for the operators −∆∂Ω,δ := −∆ + δ∂Ω ⊗ V and − ∆∂Ω,δ0 := −∆ + δ 0 ∂Ω ⊗ V ∂ν where Ω ⊂ R d is strictly convex with smooth boundary, V : L 2 (∂Ω) → L 2 (∂Ω) may depend on frequency, and δ∂Ω is the surface measure on ∂Ω. These operators are model Hamiltonians for the quantum corrals studied in [AL05, BZH10, CLEH95a] and for leaky quantum graphs [Exn08]. We give a quantum version of the Sabine Law [Sab64] from the study of acoustics for both −∆∂Ω,δ and −∆∂Ω,δ0. 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 −∆∂Ω,δ with Ω smooth and strictly convex, our results improve those given for general ∂Ω in [GS14] and are generically optimal. Indeed, we show that for generic domains and potentials there are infinitely many resonances arbitrarily close to the resonance free region found by our theorem. In the case of −∆∂Ω,δ0, the quantum Sabine law gives the existence of a resonance free region that converges to the real axis at a fixed polynomial rate. The size of this resonance free region is optimal in the case of the unit disk in R 2 . 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 from [MU79] 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 [MT] for complex energies. We use these constructions to give a complete microlocal description of the single, double, and derivative double layer operators in the case that ∂Ω is smooth and strictly convex. These operators are given respectively for x ∈ ∂Ω by G(λ)f(x) := Z ∂Ω R0(λ)(x, y)f(y)dS(y), N˜(λ)f(x) := Z ∂Ω ∂νyR0(λ)(x, y)f(y)dS(y) ∂νD`(λ)f(x) := Z ∂Ω ∂νx∂νyR0(λ)(x, y)f(y)dS(y). This microlocal description allows us to prove sharp high energy estimates on G, N˜, and ∂νD` when Ω is smooth and strictly convex, removing the log losses from the estimates for G in [GS14, HT14] and proving a conjecture from [HT14, Appendix A].

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