Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits. (English) Zbl 0659.35026

Robert, D.; Tamura, H.

Ann. Inst. Fourier 39, No. 1, 155-192 (1989).

We study the semi-classical asymptotic behavior as \(h\to 0\) of scattering amplitudes for Schrödinger operators \(-()h^ 2\Delta +V.\) The asymptotic formula is obtained for energies fixed in a non-trapping energy range and also is applied to study the low energy behavior of scattering amplitudes for a certain class of slowly decreasing repulsive potentials without spherical symmetry.

Reviewer: D.Robert

MSC:

35J10	Schrödinger operator, Schrödinger equation
35P25	Scattering theory for PDEs
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

semi-classical asymptotic behavior; scattering amplitudes; Schrödinger operators; non-trapping energy range; low energy behavior

References:

[1]	[1] , Spectral properties of Schrödinger operators and scattering theory, Ann. Norm. Sup. Pisa, 2 (1975), 151-218. · Zbl 0315.47007
[2]	[2] , Some new results in spectral and scattering theory of differential operators on Rn, Séminaire Goulaouic-Schwartz, École Polytechnique, 1978. · Zbl 0406.35052
[3]	[3] , and , The low energy expansion in nonrelativistic scattering theory, Ann. Inst. Henri Poincaré, 37 (1982), 1-28. · Zbl 0528.35076
[4]	[4] , and , Low-energy parameters in nonrelativistic scattering theory, Ann. Phys., 148 (1983), 308-326. · Zbl 0542.35056
[5]	[5] and , Finite total cross sections in nonrelativistic quantum mechanics, Comm. Math. Phys., 76 (1980), 177-209. · Zbl 0471.35065
[6]	[6] and , Total cross sections in nonrelativistic scattering theory, Quantum Mechanics in Mathematics, Chemistry and Physics, edited by K.E. Gustafson and W. P. Reinhart, Plenum Press, 1981. · Zbl 0471.35065
[7]	[7] and , Principe d’absorption limite pour des opérateurs de Schrödinger à longue portée, Université de Paris-Sud, preprint, 1987. · Zbl 0672.35013
[8]	[8] and Modified wave operators with time-independent modifiers, J. Fac. Sci. Univ. Tokyo Sect. IA, 32 (1985), 77-104. · Zbl 0582.35036
[9]	[9] and , Scattering matrices for two-body Schrödinger operators, Sci. Papers College Arts Sci. Univ. Tokyo, 35 (1985), 81-107. · Zbl 0615.35065
[10]	[10] and , A remark on the microlocal resolvent estimates for two-body Schrödinger operators, Publ. RIMS Kyoto Univ., 21 (1985), 889-910. · Zbl 0611.35090
[11]	[11] , Scattering by long-range potentials at low energies, Theoretical and Mathematical Physics, 59 (1984), 629-633.
[12]	[12] and , Semi-classical Approximation in Quantum Mechanics, Reidel, 1981. · Zbl 0458.58001
[13]	[13] , Quasiclassical asymptotics of the scattering amplitude for the scattering of a plane wave by inhomogeneities of the medium, Math. USSR Sbornik, 45 (1983), 487-506. · Zbl 0549.35101
[14]	[14] , Autour de l’approximation Semi-classique, Birkhaüser, 1987. · Zbl 0621.35001
[15]	[15] and , Semi-classical estimates for resolvents and asymptotics for total scattering cross-sections, Ann. Inst. Henri Poincaré, 46 (1987), 415-442. · Zbl 0648.35066
[16]	B. R. VAINGERG, Quasi-classical approximation in stationary scattering problems, Func. Anal. Appl., 11 (1977), 247-257.0413.35025 · Zbl 0413.35025
[17]	[3]
[18]	[18] , The quasi-classical limit of scattering amplitude — L2 — approach for short range potentials — Japan J. Math., 13 (1987), 77-126. · Zbl 0648.35067
[19]	[19] and , Methods of Modern Mathematical Physics, III: Scattering Theory, Academic Press, 1979. · Zbl 0405.47007
[20]	[20] , Scattering Theory of Waves and Particles, 2nd édition, Springer, 1982. · Zbl 0496.47011