Abstract
We get new tests for the existence and completeness of wave operators under perturbation of a pseudodifferential operator with constant symbol P(ξ) by a bounded potential v(x). The term anisotropic is understood in the sense that the growth of P(ξ) as ξ→∞ and the decrease of v(x) as x→∞ can depend essentially on the direction of the vectors ξ and x respectively. This permits us to include in the sphere of applications of the abstract scattering theory of a nonelliptic unperturbed operator the D'Alembert operator, an ultrahyperbolic operator, nonstationary Schrödinger operator, etc. In view of the anisotropic character of the assumptions on the potential, the results obtained are new even in the elliptic case. As an example we consider a Schrödinger operator with potential close to the energy of a pair of interacting systems of many particles.
Similar content being viewed by others
Literature cited
M. Sh. Birman, “Scattering problems for differential operators with constant coefficients,” Funkts. Anal. Prilozhen.,3, No. 3, 1–16 (1969).
S. T. Kuroda, “Scattering theory for differential operators. II,” J. Math. Soc. Jpn.,25, 222–234 (1973).
S. T. Kuroda, “On the existence and the unitary property of the scattering operator,” Nuovo Cimento,12, 431–454 (1959).
L. P. Nizhnik, “Spectral properties of self-adjoint partial differential operators close to operators with constant coefficients,” Materials on the Soviet—American Symposium on Partial Differential Equations, Novosibirsk (1963).
J. Howland, “Stationary scattering theory for time-dependent Hamiltonians,” Math. Ann.,207, 315–335 (1974).
T. Kato and S. T. Kuroda, “An abstract theory of scattering,” Rocky Mountain Math. J.,1, 127–171 (1971).
V. G. Deich, “Stationary local method of scattering theory for a pair of spaces,” Dokl. Akad. Nauk SSSR,197, No. 6, 1247–1250 (1971).
R. Lavine, “Commutators and scattering theory. II,” Indiana Univ. Math. J.,27, 643–656 (1972).
T. Kato, “Wave operators and similarity for some non-self-adjoint operators,” Math. Ann.,162, 258–279 (1966).
E. L. Korotyaev and D. R. Yafaev, “Traces on surfaces in functional classes with dominating mixed derivatives,” J. Sov. Math.,10, No. 1 (1978).
V. G. Deich, E. L. Korotyaev, and D. R. Yafaev, “Potential scattering considering spatial anisotropy,” Dokl. Akad. Nauk SSSR,235, No. 4, 749–752 (1977).
E. M. Stein, Singular Integrals and Differential Properties of Functions [Russian translation], Mir, Moscow (1973).
M. Sh. Birman, “Spectrum of singular boundary problems,” Mat. Sb.,55 (97), 125–174 (1961).
L. P. Nizhnik, “Structure of the spectrum and self-adjointness of perturbed differential operators with constant coefficients,” Ukr. Mat. Zh.,25, No. 4, 385–399 (1963).
L. D. Faddeev, Mathematical Questions of Quantum Scattering Theory for a System of Three Particles [in Russian], Tr. Mat. Inst. Akad. Nauk SSSR, Vol. 69, Nauka, Moscow (1963).
Yu. M. Berezanskii, Decomposition of Self-adjoint Operators in Eigenfunctions [in Russian], Naukova Dumka, Kiev (1965).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 35–51, 1977.
Rights and permissions
About this article
Cite this article
Deich, V.G., Korotyaev, E.L. & Yafaev, D.R. Theory of potential scattering, taking into account spatial anisotropy. J Math Sci 34, 2040–2050 (1986). https://doi.org/10.1007/BF01741578
Issue Date:
DOI: https://doi.org/10.1007/BF01741578