×

Resonance-free region in scattering by a strictly convex obstacle. (English) Zbl 1317.35161

The author establishes the existence of a cubic resonance-free region in scattering by a strictly convex obstacle under a Robin boundary condition. Previous results were known in the case of Dirichlet and Neumann boundary conditions. Lower bounds for an associated scaled operator are also obtained.

MSC:

35P25 Scattering theory for PDEs
47A40 Scattering theory of linear operators
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aguilar, J. and Combes, J. M., A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys.22 (1971), 269-279. · Zbl 0219.47011 · doi:10.1007/BF01877510
[2] Balslev, E. and Combes, J. M., Spectral properties of many-body Schrödinger operators with dilation analytic interactions, Comm. Math. Phys.22 (1971), 280-294. · Zbl 0219.47005 · doi:10.1007/BF01877511
[3] Bardos, C., Lebeau, G. and Rauch, J., Scattering frequencies and Gevrey 3 singularities, Invent. Math.90 (1987), 77-114. · Zbl 0723.35058 · doi:10.1007/BF01389032
[4] Cordoba, A. and Fefferman, C., Wave packets and Fourier integral operators, Comm. Partial Differential Equations3 (1978), 979-1006. · Zbl 0389.35046 · doi:10.1080/03605307808820083
[5] Delort, J.-M., F.B.I. Transformation. Second Microlocalization and Semilinear Caustics, Lecture Notes in Math. 1522, Springer, Berlin-Heidelberg, 1992. · Zbl 0760.35004
[6] Folland, G., Harmonic Analysis in Phase Space, Ann. of Math. Stud. 122, Princeton Univ. Press, Princeton, NJ, 1989. · Zbl 0682.43001
[7] Hargé, T. and Lebeau, G., Diffraction par un convexe, Invent. Math.118 (1994), 161-196. · Zbl 0831.35121 · doi:10.1007/BF01231531
[8] Hörmander, L., The Analysis of Linear Partial Differential Operators, I-IV, Springer, Berlin-Heidelberg, 1983, 1985. · Zbl 0521.35002
[9] Lascar, B. and Lascar, R., FBI transforms in Gevrey classes, J. Anal. Math.72 (1997), 105-125. · Zbl 0898.35069 · doi:10.1007/BF02843155
[10] Lax, P. and Phillips, R., A logarithmic bound on the location of the poles of the scattering matrix, Arch. Ration. Mech. Anal.40 (1971), 268-280. · Zbl 0216.13002 · doi:10.1007/BF00252678
[11] Lebeau, G., Régularité Gevrey 3 pour la diffraction, Comm. Partial Differential Equations9 (1984), 1437-1494. · Zbl 0559.35019 · doi:10.1080/03605308408820368
[12] Martinez, A., An Introduction to Semiclassical and Microlocal Analysis, Springer, New York, 2002. · Zbl 0994.35003 · doi:10.1007/978-1-4757-4495-8
[13] Melrose, R. B., Singularities and energy decay in acoustical scattering, Duke Math. J.46 (1979), 43-59. · Zbl 0415.35050 · doi:10.1215/S0012-7094-79-04604-0
[14] Morawetz, C. S., Ralston, J. V. and Strauss, W. A., Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math.30 (1977), 447-508. · Zbl 0372.35008 · doi:10.1002/cpa.3160300405
[15] Popov, G., Some estimates of Green’s functions in the shadow, Osaka J. Math.24 (1987), 1-12. · Zbl 0656.35022
[16] Sjöstrand, J., Singularités analytiques microlocales, Astérisque 95, Soc. Math. France, Paris, 1982. · Zbl 0524.35007
[17] Sjöstrand, J., Density of resonances for strictly convex analytic obstacles, Canad. J. Math.48 (1996), 397-447. · Zbl 0863.35072 · doi:10.4153/CJM-1996-022-9
[18] Sjöstrand, J. and Zworski, M., Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc.4 (1991), 729-769. · Zbl 0752.35046 · doi:10.2307/2939287
[19] Sjöstrand, J. and Zworski, M., Lower bounds on the number of scattering poles, Comm. Partial Differential Equations18 (1993), 847-858. · Zbl 0784.35070 · doi:10.1080/03605309308820953
[20] Sjöstrand, J. and Zworski, M., Estimates on the number of scattering poles for strictly convex obstacles near the real axis, Ann. Inst. Fourier (Grenoble)43 (1993), 769-790. · Zbl 0784.35073 · doi:10.5802/aif.1355
[21] Sjöstrand, J. and Zworski, M., The complex scaling method for scattering by strictly convex obstacles, Ark. Mat.33 (1995), 135-172. · Zbl 0839.35095 · doi:10.1007/BF02559608
[22] Sjöstrand, J. and Zworski, M., Asymptotic distribution of resonances for convex obstacles, Acta Math.183 (1999), 191-253. · Zbl 0989.35099 · doi:10.1007/BF02392828
[23] Stefanov, P., Sharp upper bounds on the number of the scattering poles, J. Funct. Anal.231 (2006), 111-142. · Zbl 1099.35074 · doi:10.1016/j.jfa.2005.07.007
[24] Tang, S.-H. and Zworski, M., Resonance expansions of scattered waves, Comm. Pure Appl. Math.53 (2000), 1305-1334. · Zbl 1032.35148 · doi:10.1002/1097-0312(200010)53:10<1305::AID-CPA4>3.0.CO;2-#
[25] Vainberg, B. R., On exterior elliptic problems depending on a spectral parameter, and the asymptotic behavior for large time of solutions of nonstationary problems, Mat. Sb.92 (1973), 224-241 (Russian). English transl.: Math. USSR-Sb.21 (1973), 221-239. · Zbl 0294.35031
[26] Watson, G. N., The diffraction of electric waves by the Earth, Proc. R. Soc. Lond. Ser. A95 (1918), 83-99. · doi:10.1098/rspa.1918.0050
[27] Wunsch, J. and Zworski, M., The FBI transform on compact C∞ manifolds, Trans. Amer. Math. Soc.353 (2000), 1151-1167. · Zbl 0974.35005 · doi:10.1090/S0002-9947-00-02751-3
[28] Zworski, M., Semiclassical Analysis, Graduate Studies in Mathematics 138, Amer. Math. Soc., Providence, RI, 2012. · Zbl 1252.58001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.