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Scattering resonances of convex obstacles for general boundary conditions. (English) Zbl 1319.35142

Resonances for an obstacle problem are complex characteristic frequencies which are featured in decay of linear waves around the obstacle. They are typically defined as poles of the meromorphic continuation of the resolvent of the Laplacian in the exterior domain with some boundary conditions at the obstacle.
The present paper considers general smooth strictly convex obstacles and general Robin boundary conditions (these include the case of Neumann boundary conditions). The author shows that there is a cubic resonance-free region below the real axis and, under pinching conditions, deeper cubic resonance free strips; the resonances in the bands between these strips satisfy a Weyl law. The approach used is similar to the work on Dirichlet case by J. Sjöstrand and M. Zworski in [Acta Math. 183, No. 2, 191–253 (1999; Zbl 0989.35099)].

MSC:

35P25 Scattering theory for PDEs
35B34 Resonance in context of PDEs

Citations:

Zbl 0989.35099
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References:

[1] Aguilar J., Combes J.M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269-279 (1971) · Zbl 0219.47011 · doi:10.1007/BF01877510
[2] Balslev E., Combes J.M.: Spectral properties of many-body Schrödinger operators with dilation analytic interactions. Commun. Math. Phys. 22, 280-294 (1971) · Zbl 0219.47005 · doi:10.1007/BF01877511
[3] Babich V.M., Grigoreva N.S.: The analytic continuation of the resolvent of the exterior three-dimensional problem for the Laplace operator to the second sheet. Funktsional. Anal. i Prilozhen. 8, 71-74 (1974) · Zbl 0292.35064 · doi:10.1007/BF02028310
[4] Bardos C., Lebeau G., Rauch J.: Scattering frequencies and Gevrey 3 singularities. Invent. Math. 90, 77-114 (1987) · Zbl 0723.35058 · doi:10.1007/BF01389032
[5] Boutet de Monvel L., Kree P.: Pseudodifferential operators and Gevrey classes. Ann. Inst. Fourier 17, 295-323 (1967) · Zbl 0195.14403 · doi:10.5802/aif.258
[6] Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Note Series, vol. 268. Cambridge University Press, Cambridge (1999) · Zbl 0926.35002
[7] Filippov V.B., Zayaev A.B.: Rigorous justification of the asymptotic solutions of sliding wave type. J. Sov. Math. 30, 2395-2406 (1985) · Zbl 0567.73036 · doi:10.1007/BF02107400
[8] Hargé T., Lebeau G.: Diffraction par un convexe. Invent. Math. 118, 161-196 (1994) · Zbl 0831.35121 · doi:10.1007/BF01231531
[9] Heffler B., Sjöstrand J.: Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.), vol. 24/25. Bordas, Paris (1986) · Zbl 0631.35075
[10] Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. I-IV. Springer, Berlin (1983-1985) · Zbl 0521.35002
[11] Jin L.: Resonance-free region in scattering by a strictly convex obstacle. Ark. Mat. 52, 257-289 (2014) · Zbl 1317.35161 · doi:10.1007/s11512-013-0185-0
[12] Lax P., Phillips R.: Scattering Theory. Academic Press, New York (1967) · Zbl 0186.16301
[13] Lax P., Phillips R.: Decaying modes for the wave equation in the exterior of an obstacle. Commun. Pure Appl. Math. 22, 737-787 (1969) · Zbl 0181.38201 · doi:10.1002/cpa.3160220603
[14] Lax P., Phillips R.: A logarithmic bound on the location of the poles of the scattering. Mat. Arch. Ration. Mech. Anal. 40, 268-280 (1971) · Zbl 0216.13002
[15] Lebeau G.: Régularité Gevrey 3 pour la diffraction. Commun. Partial Differ. Equ. 9, 1437-1494 (1984) · Zbl 0559.35019 · doi:10.1080/03605308408820368
[16] Martinez A.: An Introduction to Semiclassical and Microlocal Analysis. Universitext. Springer, New York (2002) · Zbl 0994.35003 · doi:10.1007/978-1-4757-4495-8
[17] Melrose, R.B.: Polynomial bound on the distribution of poles in scattering by an obstacle. In: Journée “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1984), Exp. No. II. Soc. Math. France, Paris (1984) · Zbl 0621.35073
[18] Melrose R.B.: Geometric Scattering Theory, Stanford Lectures. Cambridge University Press, Cambridge (1995)
[19] Morawetz C.S., Ralston J.V., Strauss W.A.: Decay of solutions of the wave equation outside nontrapping obstacles. Commun. Pure Appl. Math. 30, 447-508 (1977) · Zbl 0372.35008 · doi:10.1002/cpa.3160300405
[20] Popov G.: Some estimates of Green’s functions in the shadow. Osaka J. Math. 24, 1-12 (1987) · Zbl 0656.35022
[21] Sjöstrand J.: Singularité analytiques microlocales, Astérisque, vol. 95. Soc. Math. France, Paris (1982)
[22] Sjöstrand J.: Density of resonances for strictly convex analytic obstacles. With an appendix by M. Zworski. Can. J. Math. 48, 397-447 (1996) · Zbl 0863.35072 · doi:10.4153/CJM-1996-022-9
[23] Stefanov P.: Sharp upper bounds on the number of the scattering poles. J. Funct. Anal. 231(1), 111-142 (2006) · Zbl 1099.35074 · doi:10.1016/j.jfa.2005.07.007
[24] Sjöstrand J., Zworski M.: Complex scaling and the distribution of scattering poles. J. Am. Math. Soc. 4, 729-769 (1991) · Zbl 0752.35046 · doi:10.2307/2939287
[25] Sjöstrand J., Zworski M.: Lower bounds on the number of scattering poles. Commun. Partial Differ. Equ. 18, 847-858 (1993) · Zbl 0784.35070 · doi:10.1080/03605309308820953
[26] Sjöstrand J., Zworski M.: Lower bounds on the number of scattering poles, II. J. Funct. Anal. 123, 336-367 (1994) · Zbl 0823.35137 · doi:10.1006/jfan.1994.1092
[27] Sjöstrand J., Zworski M.: Estimates on the number of scattering poles for strictly convex obstacles near the real axis. Ann. Inst. Fourier (Grenoble) 43, 769-790 (1993) · Zbl 0784.35073 · doi:10.5802/aif.1355
[28] Sjöstrand J., Zworski M.: The complex scaling method for scattering by strictly convex obstacles. Ark. Mat. 33, 135-172 (1995) · Zbl 0839.35095 · doi:10.1007/BF02559608
[29] Sjöstrand J., Zworski M.: Asymptotic distribution of resonances for convex obstacles. Acta Math. 183, 191-253 (1999) · Zbl 0989.35099 · doi:10.1007/BF02392828
[30] Sjöstrand J., Zworski M.: Elementary linear algebra for advanced spectral problems. Ann. Inst. Fourier Grenoble 57, 2095-2141 (2007) · Zbl 1140.15009 · doi:10.5802/aif.2328
[31] Vodev G.: Sharp bounds on the number of scattering poles in even-dimensional spaces. Duke Math. J. 74, 1-17 (1994) · Zbl 0813.35075 · doi:10.1215/S0012-7094-94-07401-2
[32] Watson G.N.: Diffraction of electric waves by the Earth. Proc. R. Soc. (Lond.) A95, 83-99 (1918) · JFM 46.0753.03 · doi:10.1098/rspa.1918.0050
[33] Zworski, M.: Couting scattering poles. In: Spectral and Scattering Theory (Sanda, 1992), pp. 301-331. Lecture Notes in Pure and Applied Mathematics, vol. 161. Dekker, New York (1994) · Zbl 0823.35139
[34] Zworski M.: Semiclassical Analysis. Graduate Studies in Mathematics, vol. 138. AMS, Providence (2012) · Zbl 1252.58001
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