Breit-Wigner formulas for the scattering phase and the total scattering cross-section in the semi-classical limit. (English) Zbl 0704.35114

Authors’ summary: In this paper we prove results in resonance scattering for the Schrödinger operator \(P_ V=-h^ 2\Delta +V\), V being a smooth, short range potential on \({\mathbb{R}}^ n\). More precisely, for energy \(\lambda\) near a trapping energy level \(\lambda_ 0\) for the classical system defined by the Hamiltonian \(p(x,\xi)=\xi^ 2+V(x)\), we prove that the scattering phase and the scattering cross sections associated to \((P_ V,P_ 0)\) have the Breit-Wigner form (“Lorentzian line shape”) in the limit \(h\to 0\).
Reviewer: X.-P.Wang


35P25 Scattering theory for PDEs
35Q40 PDEs in connection with quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
Full Text: DOI


[1] Aguilar, J., Combes, J.M.: A class of analytic perturbations for one body Schr?dinger Hamiltonians. Commun. Math. Phys.,22, 269-279 (1971) · Zbl 0219.47011
[2] Birman, M.S., Krein, M.G.: On the theory of wave operators and scattering operators. Dokl. Akad. Nauk SSSR144, 475-478 (1962), (in Russian)
[3] Colin de Verdi?re, Y.: Une formule de trace pour l’op?rateur de Schr?dinger dansR 3. Ann. Sci. Ec. Norm. Sup.14, 27-39 (1981)
[4] Combes, J.M., Duclos, P., Klein, M., Seiler, R.: The shape resonance. Commun. Math. Phys.110, 215-236 (1987) · Zbl 0629.47044
[5] Enss, V., Simon, B.: Total cross sections in non-relativistic scattering theory, quantum mechanics in mathematics, chemistry and physics. Gustafson K.E. (ed.). In: P. Reinhart. New York: Plenum Press 1981
[6] G?rard, C., Martinez, A.: Semiclassical asymptotics for the spectral function of long range Schr?dinger operators. J. Funct. Anal. (to appear)
[7] Principe d’absorption limite pour des op?rateurs de Schr?dinger ? longue port?e. C.R. Acad. Sci. Paris,306, S?r. I, 121-123 (1988) · Zbl 0672.35013
[8] Guillop?, L.: Assymptotique de la phase de diffusion pour l’op?rateur de Schr?dinger avec potentiel. C.R. Acad. Sci. Paris293, 601-603 (1981)
[9] Helffer, B., Martinez, A.: Comparaison entre les diverses notions de r?sonances. Helv. Phys. Acta,60, 992-1003 (1987)
[10] Helffer, B., Robert, D.: Calcul fonctionnel par la transformation de Mellin et op?rateurs admissibles. J. Funct. Anal.53, 246-268 (1983) · Zbl 0524.35103
[11] Helffer, B., Sj?strand, J.: R?sonances en limite semi-classique. Bull. S.M.F., m?moire no24/25, tome114 (1986)
[12] Hislop, P., Sigal, M.: Shape resonances in quantum mechanics, Proceedings of the International Conferences on Diff. Eq. In: Math. Phys., Birmingham (Alabama), March 1986 · Zbl 0653.46074
[13] Isozaki, H.: Differentiability of generalized Fourier Transforms associated with Schr?dinger Operators. J. Math. Kyoto Univ.25 789-806 (1985) · Zbl 0612.35004
[14] Ivrii, V.J., Shubin, M.A.: On the asymptotics of the spectral shift function. Dokl. Acad. Nauk. SSSR263 (1982); Sov. Math. Dokl.25 · Zbl 0541.58047
[15] Jensen, A., Kato, T.: Asymptotic behavior of the scattering phase for exterior domains. Commun. P.D.E.3, 1165-1195 (1978) · Zbl 0419.35067
[16] Landau, L., Lifchitz, E.: M?canique quantique, th?orie non relativiste. Editions Mir, Moscou (1967)
[17] Majda, A., Ralston, J.: An analogue of Weyl’s theorem for unbounded domains. I, II, and III. Duke Math. J.45, 183-196 (1978);45, 513-536 (1978);46, 725-731 (1979) · Zbl 0408.35069
[18] Nakamura, S.: Scattering theory for the shape resonance model, I-Non-resonant energies. II. Resonance scattering. Preprint Tokyo University · Zbl 0686.35090
[19] Newton, R.G.: Scattering theory of waves and particles, Texts and monographs in physics, 2nd ?d. Berlin, Heidelberg, New York: Springer
[20] Petkov, V., Popov, G.: Asymptotic behavior of the scattering phase for non-trapping obstacles. Ann. Inst. Fourier Grenoble32, 111-149 (1982) · Zbl 0476.35014
[21] Reed, M., Simon, B.: Methods of modern mathematical physics. New York: Academic Press 1978 · Zbl 0401.47001
[22] Robert, D., Tamura, H.: Semiclassical estimates for resolvents and asymptotics for total scattering cross-sections. Ann. I.H.P.46, 415-442 (1987) · Zbl 0648.35066
[23] Semiclassical asymptotics for local spectral densities and time delay problems in scattering processes. J. Funct Anal.80, 124-147 (1988) · Zbl 0663.47009
[24] Sobolev, A.V., Yafaev, D.R.: On the quasi-classical limit of the total scattering cross-section in non relativistic quantum mechanics. Ann. I.H.P.44, 195-210 (1986) · Zbl 0607.35070
[25] Yajima, K.: The quasi-classical limit of scattering amplitude ?L 2 approach for short range potential. Jpn. J. Math.13, 77-126 (1987) · Zbl 0648.35067
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.