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Geometric bounds on the density of resonances for semiclassical problems. (English) Zbl 0702.35188
The author gives upper bounds on the number of resonances in certain regions in the complex plane close to the real axis, for semi-classical operators \(like:-h^ 2\Delta +V(x)\) when h is small. A part of the results was announced in the Proc. of the VIII Latin American School of Mathematics (1986) [Lect. Notes Math. 1324, 286-292 (1988; Zbl 0674.35018)]. The resonances are defined using a microlocal version of the complex scaling due to the reviewer and the author [Mém. Soc. Math. Fr., Nouv. Sér. 24/25 (1986; Zbl 0631.35075)] and the results presented here are analogous to those by R. Melrose [Journ. “Equations Deriv. Partielles”, St. Jean-De Monts 1984, Conf. No.3, 8 p. (1984; Zbl 0621.35073)] in the context of the exterior problem.
Reviewer: B.Helffer

MSC:
35P15 Estimates of eigenvalues in context of PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
35J10 Schrödinger operator, Schrödinger equation
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[1] J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians , Comm. Math. Phys. 22 (1971), 269-279. · Zbl 0219.47011
[2] E. Balslev and J. M. Combes, Spectral properties of many-body Schrödinger operators with dilatation analytic interactions , Comm. Math. Phys. 22 (1971), 280-294. · Zbl 0219.47005
[3] C. Bardos, G. Lebeau, and J. Rauch, Scattering frequencies and Gevrey \(3\) singularities , Invent. Math. 90 (1987), no. 1, 77-114. · Zbl 0723.35058
[4] L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators , Ann. of Math. Studies, vol. 99, Princeton Univ. Press, Princeton, NJ, 1981. · Zbl 0469.47021
[5] P. Briet, J. M. Combes, and P. Duclos, On the location of resonances for Schrödinger operators in the semiclassical limit. I. Resonances free domains , J. Math. Anal. Appl. 126 (1987), no. 1, 90-99. · Zbl 0629.47043
[6] P. Briet, J. M. Combes, and P. Duclos, On the location of resonances for Schrödinger operators in the semiclassical limit. II. Barrier top resonances , Comm. Partial Differential Equations 12 (1987), no. 2, 201-222. · Zbl 0622.47047
[7] A. Cordoba and C. Fefferman, Wave packets and Fourier integral operators , Comm. Partial Differential Equations 3 (1978), no. 11, 979-1005. · Zbl 0389.35046
[8] N. Dunford and J. T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space , With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. · Zbl 0635.47002
[9] K. J. Falconer, The geometry of fractal sets , Cambridge tracts in mathematics, vol. 95, Cambridge University press, Cambridge, 1985. · Zbl 0587.28004
[10] C. Gérard, Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes , Mém. Soc. Math. France (N.S.) (1988), no. 31, 146. · Zbl 0654.35081
[11] C. Gérard and J. Sjöstrand, Semiclassical resonances generated by a closed trajectory of hyperbolic type , Comm. Math. Phys. 108 (1987), no. 3, 391-421. · Zbl 0637.35027
[12] C. Gérard and J. Sjöstrand, Resonances en limite semiclassique et exposants de Lyapunov , Comm. Math. Phys. 116 (1988), no. 2, 193-213. · Zbl 0698.35118
[13] W. Goodhue, Scattering theory for hyperbolic systems with coefficients of Gevrey type , Trans. Amer. Math. Soc. 180 (1973), 337-346. · Zbl 0266.47009
[14] B. Helffer and J. Sjöstrand, Résonances en limite semiclassique , Mém. Soc. Math. France (N.S.) (1986), no. 24-25, iv+228. · Zbl 0631.35075
[15] B. Helffer and A. Martinez, Comparaison entre les diverses notions de résonances , Helv. Phys. Acta 60 (1987), no. 8, 992-1003.
[16] L. Hörmander, The analysis of linear partial differential operators III , Grundlehren der math. wiss., vol. 274, Springer Verlag, Berlin, 1985. · Zbl 0601.35001
[17] 1 M. Ikawa, On the poles of the scattering matrix for two strictly convex obstacles , J. Math. Kyoto Univ. 23 (1983), no. 1, 127-194. · Zbl 0561.35060
[18] 2 M. Ikawa, On the poles of the scattering matrix for two strictly convex obstacles: An addendum , J. Math. Kyoto Univ. 23 (1983), no. 4, 795-802. · Zbl 0559.35061
[19] 3 M. Ikawa, On the distribution of the poles of the scattering matrix for two strictly convex obstacles , Hokkaido Math. J. 12 (1983), no. 3, 343-359. · Zbl 0542.35057
[20] 1 M. Ikawa, Trapping obstacles with a sequence of poles of the scattering matrix converging to the real axis , · Zbl 0617.35102
[21] 2 Mitsuru Ikawa, On the scattering matrix for two convex obstacles , Hyperbolic equations and related topics (Katata/Kyoto, 1984), Academic Press, Boston, MA, 1986, pp. 63-84. · Zbl 0695.35109
[22] M. Ikawa, Decay of solutions of the wave equations in the exterior of several convex bodies , · Zbl 0636.35045
[23] M. Ikawa, 1988, Proc. of Japan Academy. · Zbl 0636.35045
[24] A. Intissar, A polynomial bound on the number of the scattering poles for a potential in even-dimensional spaces \(\mathbf R^ n\) , Comm. Partial Differential Equations 11 (1986), no. 4, 367-396. · Zbl 0607.35069
[25] A. Kelley, The stable, center-stable, center, center-unstable and unstable manifolds , W. A. Benjamin Inc., New York, Amsterdam, 1967, Appendix C in Transversal mappings and flows by R. Abraham and J. Robbin. · Zbl 0173.11001
[26] M. Klein, On the absence of resonances for Schrödinger operators with nontrapping potentials in the classical limit , Comm. Math. Phys. 106 (1986), no. 3, 485-494. · Zbl 0651.47007
[27] P. Lax and R. Phillips, Scattering theory , Pure and Appl. Math., vol. 26, Academic Press, New York, 1967. · Zbl 0186.16301
[28] G. Lebeau, Régularité Gevrey \(3\) pour la diffraction , Comm. Partial Differential Equations 9 (1984), no. 15, 1437-1494. · Zbl 0559.35019
[29] R. Melrose, Polynomials bound on the distribution of poles in scattering by an obstacle , Proceedings of the Journées “Equations aux dérivées partielles” à St Jean de Montes, Société Mathématique de France, 4-8 Juin 1984. · Zbl 0621.35073
[30] S. Nakamura, A note on the absence of resonances for Schrödinger operators , Lett. Math. Phys. 16 (1988), no. 3, 217-223. · Zbl 0688.35073
[31] S. Nakamura, Shape resonances for distortion analytic Schrödinger operators , · Zbl 0699.35204
[32] V. Petkov Stojanov, Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles , Preprint, 1987.
[33] M. Shub, Global stability of dynamical systems , Springer Verlag, New York, 1987. · Zbl 0606.58003
[34] Ya. G. Sinai, Development of Krylov’s ideas , Princeton Univ. Press, 1979, Addendum to Works on the foundations of statistical physics, by N. S. Krylov.
[35] J. Sjöstrand, Singularités analytiques microlocales , Astérisque, 95, Astérisque, vol. 95, Soc. Math. France, Paris, 1982, pp. 1-166. · Zbl 0524.35007
[36] J. Sjöstrand, Propagation of singularities for operators with multiple involutive characteristics , Ann. Inst. Fourier (Grenoble) 26 (1976), no. 1, v, 141-155. · Zbl 0313.58021
[37] J. Sjöstrand, Semiclassical resonances generated by nondegenerate critical points , Pseudodifferential operators (Oberwolfach, 1986), Lecture Notes in Math., vol. 1256, Springer, Berlin, 1987, pp. 402-429. · Zbl 0627.35074
[38] J. Sjöstrand, Estimates on the number of resonances for semiclassical Schrödinger operators , Partial Differential Equations (Rio de Janeiro, 1986), Lecture Notes in Math., vol. 1324, Springer, Berlin, 1988, pp. 286-292. · Zbl 0674.35018
[39] C. Tricot, Two definitions of fractional dimension , Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 1, 57-74. · Zbl 0483.28010
[40] M. Zworski, Sharp polynomial bounds on the number of scattering poles , · Zbl 0705.35099
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