## Pure and Applied Analysis

### The quantum Sabine law for resonances in transmission problems

Jeffrey Galkowski

#### Abstract

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.

#### Article information

Source
Pure Appl. Anal., Volume 1, Number 1 (2019), 27-100.

Dates
Revised: 25 June 2018
Accepted: 8 August 2018
First available in Project Euclid: 4 February 2019

https://projecteuclid.org/euclid.paa/1549297978

Digital Object Identifier
doi:10.2140/paa.2019.1.27

Mathematical Reviews number (MathSciNet)
MR3900029

Zentralblatt MATH identifier
07027485

#### Citation

Galkowski, Jeffrey. The quantum Sabine law for resonances in transmission problems. Pure Appl. Anal. 1 (2019), no. 1, 27--100. doi:10.2140/paa.2019.1.27. https://projecteuclid.org/euclid.paa/1549297978

#### References

• I. Alexandrova, “Semi-classical wavefront set and Fourier integral operators”, Canad. J. Math. 60:2 (2008), 241–263.
• C. Bardos, G. Lebeau, and J. Rauch, “Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary”, SIAM J. Control Optim. 30:5 (1992), 1024–1065.
• M. C. Barr, M. P. Zalatel, and E. J. Heller, “Quantum corral resonance widths: lossy scattering as acoustics”, Nano Lett. 10:9 (2010), 3253–3260.
• M. Bellassoued, “Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization”, Asymptot. Anal. 35:3-4 (2003), 257–279.
• H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics”, Rev. Modern Phys. 87:1 (2015), 61–111.
• F. Cardoso and G. Vodev, “Boundary stabilization of transmission problems”, J. Math. Phys. 51:2 (2010), art. id. 023512.
• F. Cardoso, G. Popov, and G. Vodev, “Distribution of resonances and local energy decay in the transmission problem, II”, Math. Res. Lett. 6:3-4 (1999), 377–396.
• F. Cardoso, G. Popov, and G. Vodev, “Asymptotics of the number of resonances in the transmission problem”, Comm. Partial Differential Equations 26:9-10 (2001), 1811–1859.
• M. F. Crommie, C. P. Lutz, D. M. Eigler, and E. J. Heller, “Quantum corrals”, Phys. D 83:1-3 (1995), 98–108.
• M. Dimassi and J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Note Series 268, Cambridge Univ. Press, 1999.
• S. Dyatlov, “Asymptotic distribution of quasi-normal modes for Kerr–de Sitter black holes”, Ann. Henri Poincaré 13:5 (2012), 1101–1166.
• S. Dyatlov and C. Guillarmou, “Microlocal limits of plane waves and Eisenstein functions”, Ann. Sci. Éc. Norm. Supér. $(4)$ 47:2 (2014), 371–448.
• S. Dyatlov and M. Zworski, “Quantum ergodicity for restrictions to hypersurfaces”, Nonlinearity 26:1 (2013), 35–52.
• S. Dyatlov and M. Zworski, “Mathematical theory of scattering resonances”, preprint, 2018, http://math.mit.edu/~dyatlov/res/res_20181026.pdf.
• B. Elliott and M. Gilmore, Fiber optic cabling, 2nd ed., Newnes, Oxford, 2002.
• C. L. Epstein, “Pseudodifferential methods for boundary value problems”, pp. 171–200 in Pseudo-differential operators: partial differential equations and time-frequency analysis (Toronto, 2006), edited by L. Rodino et al., Fields Inst. Commun. 52, Amer. Math. Soc., Providence, RI, 2007.
• P. Exner, “Leaky quantum graphs: a review”, pp. 523–564 in Analysis on graphs and its applications (Cambridge, 2007), edited by P. Exner et al., Proc. Sympos. Pure Math. 77, Amer. Math. Soc., Providence, RI, 2008.
• J. Galkowski, “Distribution of resonances in scattering by thin barriers”, 2014. To appear in Mem. Amer. Math. Soc.
• J. Galkowski, “Resonances for thin barriers on the circle”, J. Phys. A 49:12 (2016), art. id. 125205.
• J. Galkowski and H. F. Smith, “Restriction bounds for the free resolvent and resonances in lossy scattering”, Int. Math. Res. Not. 2015:16 (2015), 7473–7509.
• A. Grigis and J. Sjöstrand, Microlocal analysis for differential operators, London Math. Soc. Lecture Note Series 196, Cambridge Univ. Press, 1994.
• V. Guillemin and S. Sternberg, Geometric asymptotics, Mathematical Surveys 14, Amer. Math. Soc., Providence, RI, 1977.
• V. Guillemin and G. Uhlmann, “Oscillatory integrals with singular symbols”, Duke Math. J. 48:1 (1981), 251–267.
• X. Han and M. Tacy, “Sharp norm estimates of layer potentials and operators at high frequency”, J. Funct. Anal. 269:9 (2015), 2890–2926.
• A. Hassell and S. Zelditch, “Quantum ergodicity of boundary values of eigenfunctions”, Comm. Math. Phys. 248:1 (2004), 119–168.
• L. Hörmander, The analysis of linear partial differential operators, III: Pseudodifferential operators, Grundlehren der Math. Wissenschaften 274, Springer, 1985.
• L. Hörmander, The analysis of linear partial differential operators, IV: Fourier integral operators, Grundlehren der Math. Wissenschaften 275, Springer, 1985.
• N. Ida, Engineering electromagnetics, Springer, 2000.
• D. Jakobson, Y. Safarov, and A. Strohmaier, “The semiclassical theory of discontinuous systems and ray-splitting billiards”, Amer. J. Math. 137:4 (2015), 859–906.
• H. Koch and D. Tataru, “On the spectrum of hyperbolic semigroups”, Comm. Partial Differential Equations 20:5-6 (1995), 901–937.
• V. Kovachev and G. Popov, “Invariant tori for the billiard ball map”, Trans. Amer. Math. Soc. 317:1 (1990), 45–81.
• S. Marvizi and R. Melrose, “Spectral invariants of convex planar regions”, J. Differential Geom. 17:3 (1982), 475–502.
• R. B. Melrose, “Equivalence of glancing hypersurfaces”, Invent. Math. 37:3 (1976), 165–191.
• R. Melrose and M. Taylor, “Boundary problems for wave equations with grazing and gliding rays”, preprint, 2010, http://www.unc.edu/math/Faculty/met/glide.pdf.
• L. Miller, “Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary”, J. Math. Pures Appl. $(9)$ 79:3 (2000), 227–269.
• F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (editors), NIST handbook of mathematical functions, U.S. Dept. Commerce, Washington, DC, 2010.
• V. M. Petkov and L. N. Stoyanov, Geometry of reflecting rays and inverse spectral problems, Wiley, Chichester, 1992.
• G. Popov and G. Vodev, “Distribution of the resonances and local energy decay in the transmission problem”, Asymptot. Anal. 19:3-4 (1999), 253–265.
• G. Popov and G. Vodev, “Resonances near the real axis for transparent obstacles”, Comm. Math. Phys. 207:2 (1999), 411–438.
• W. C. Sabine, Collected papers on acoustics, Harvard Univ. Press, Cambridge, 1922.
• Y. G. Safarov, “On the second term of the spectral asymptotics of the transmission problem”, Acta Appl. Math. 10:2 (1987), 101–130.
• J. Sjöstrand and M. Zworski, “Asymptotic distribution of resonances for convex obstacles”, Acta Math. 183:2 (1999), 191–253.
• M. E. Taylor, Partial differential equations, II: Qualitative studies of linear equations, 2nd ed., Applied Mathematical Sciences 116, Springer, 2011.
• S. Vũ Ng\doc, Systèmes intégrables semi-classiques: du local au global, Panoramas et Synthèses 22, Soc. Math. France, Paris, 2006.
• W. R. E. Weiss and G. A. Hagedorn, “Reflection and transmission of high frequency pulses at an interface”, Transport Theory Statist. Phys. 14:5 (1985), 539–565.
• M. Zaletel, “The Sabine law and a trace formula for lossy billiards”, unpublished note, 2010.
• M. Zworski, Semiclassical analysis, Graduate Studies in Math. 138, Amer. Math. Soc., Providence, RI, 2012.