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Distribution of Resonances in Scattering by Thin Barriers

About this Title

Jeffrey Galkowski, Department of Mathematics, Northeastern University, 567 Lake Hall, 360 Huntington Ave., Boston, MA, 02115

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 259, Number 1248
ISBNs: 978-1-4704-3572-1 (print); 978-1-4704-5251-3 (online)
DOI: https://doi.org/10.1090/memo/1248
Published electronically: April 18, 2019
Keywords: Transmission problems, layer potentials, layer operators, high frequency, resonances, potential theory, hypersingular operator
MSC: Primary 35P20, 35P25, 31B10, 31B20, 31B35; Secondary 45C05, 47F05

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Table of Contents

Chapters

• Acknowledgements
• 1. Introduction
• 2. Preliminaries
• 3. Meromorphic Continuation of the Resolvent
• 4. Boundary Layer Operators
• 5. Dynamical Resonance Free Regions
• 6. Existence of Resonances for the Delta Potential
• A. Model Cases
• B. Semiclassical Intersecting Lagrangian Distributions
• C. The Semiclassical Melrose–Taylor Parametrix

Abstract

We study high energy resonances for the operators \begin{equation*} -\Delta _{\partial \Omega ,\delta }:=-\Delta +\delta _{\partial \Omega }\otimes V\quad \text {and}\quad -\Delta _{\partial \Omega ,\delta ’}:=-\Delta +\delta _{\partial \Omega }’\otimes V\partial _\nu \end{equation*} where $\Omega \subset \mathbb {R}^d$ 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 the quantum corrals studied in Aligia and Lobos (2005), Barr, Zaletel, and Heller (2010), and Crommie et al. (1995) and for leaky quantum graphs in Exner (2008). We give a quantum version of the Sabine Law (Sabine, 1964) from the study of acoustics for both $-\Delta _{\partial \Omega ,\delta }$ and $-\Delta _{\partial \Omega ,\delta ’}$. 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 $-\Delta _{\partial \Omega ,\delta }$ with $\Omega$ smooth and strictly convex, our results improve those given for general $\partial \Omega$ in Galkowski and Smith (2015) 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 $-\Delta _{\partial \Omega ,\delta ’}$, 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 $\mathbb {R}^2$. 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 Melrose and Uhlman (1979) 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 (Melrose and Taylor, unpublished manuscript) 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 $\partial \Omega$ is smooth and strictly convex. These operators are given respectively for $x\in \partial \Omega$ by \begin{align*} G(\lambda )f(x)&:=\int _{\partial \Omega }R_0(\lambda )(x,y)f(y)dS(y)\,,\\ \tilde {N}(\lambda )f(x)&:=\int _{\partial \Omega }\partial _{\nu _y}R_0(\lambda )(x,y)f(y)dS(y)\\ \partial _{\nu }\mathcal {D}\ell (\lambda )f(x)&:=\int _{\partial \Omega }\partial _{\nu _x}\partial _{\nu _y}R_0(\lambda )(x,y)f(y)dS(y)\,. \end{align*} This microlocal description allows us to prove sharp high energy estimates on $G$, $\tilde {N}$, and $\partial _{\nu }\mathcal {D}\ell$ when $\Omega$ is smooth and strictly convex, removing the log losses from the estimates for $G$ in Galkowski and Smith (2015) and Han and Tacy (2015) and proving a conjecture from Appendix A in Han and Tacy (2015).