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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Schrödinger operators with rapidly oscillating central potentials
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by Denis A. W. White PDF
Trans. Amer. Math. Soc. 275 (1983), 641-677 Request permission

Abstract:

Spectral and scattering theory is discussed for the Schrödinger operators $H = - \Delta + V$ and ${H_0} = - \Delta$ when the potential $V$ is central and may be rapidly oscillating and unbounded. A spectral representation for $H$ is obtained along with the spectral properties of $H$. The existence and completeness of the modified wave operators is also demonstrated. Then a condition on $V$ is derived which is both necessary and sufficient for the Møller wave operators to exist and be complete. This last result disproves a recent conjecture of Mochizuki and Uchiyama.
References
  • Shmuel Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151–218. MR 397194
  • V. de Alfaro and T. Regge, Potential scattering, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1965. MR 0191316
  • W. O. Amrein and D. B. Pearson, The scattering matrix for rapidly oscillating potentials, J. Phys. A 13 (1980), no. 4, 1259–1264. MR 565769, DOI 10.1088/0305-4470/13/4/019
  • Masaharu Arai, Eigenfunction expansions associated with the Schrödinger operators with long-range potentials, Publ. Res. Inst. Math. Sci. 16 (1980), no. 1, 35–59. MR 574029, DOI 10.2977/prims/1195187499
  • F. V. Atkinson, The asymptotic solution of second-order differential equations, Ann. Mat. Pura Appl. (4) 37 (1954), 347–378. MR 67289, DOI 10.1007/BF02415105
  • M. L. Baeteman and K. Chadan, Scattering theory with highly singular oscillating potentials, Ann. Inst. H. Poincaré Sect. A (N.S.) 24 (1976), no. 1, 1–16. MR 400975
  • Matania Ben-Artzi, Eigenfunction expansions for a class of differential operators, J. Math. Anal. Appl. 69 (1979), no. 2, 304–314. MR 538219, DOI 10.1016/0022-247X(79)90144-6
  • Matania Ben-Artzi, On the absolute continuity of Schrödinger operators with spherically symmetric, long-range potentials. I, II, J. Differential Equations 38 (1980), no. 1, 41–50, 51–60. MR 592867, DOI 10.1016/0022-0396(80)90023-6
    • —, On the absolute continuity of Schrödinger operators with spherically symmetric long range potentials. II, J. Differential Equations 38 (1980), 51-60.
  • Matania Ben-Artzi, Spectral properties of linear ordinary differential operators with slowly decreasing coefficients, J. Math. Anal. Appl. 68 (1979), no. 1, 68–91. MR 531423, DOI 10.1016/0022-247X(79)90100-8
  • Matania Ben-Artzi and Allen Devinatz, Spectral and scattering theory for the adiabatic oscillator and related potentials, J. Math. Phys. 20 (1979), no. 4, 594–607. MR 529723, DOI 10.1063/1.524128
  • Brian Bourgeois, On scattering theory for oscillatory potentials of slow decay, Ann. Physics 121 (1979), no. 1-2, 415–431. MR 548178, DOI 10.1016/0003-4916(79)90103-9
  • Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
  • Monique Combescure, Spectral and scattering theory for a class of strongly oscillating potentials, Comm. Math. Phys. 73 (1980), no. 1, 43–62. MR 573612, DOI 10.1007/BF01942693
  • M. Combescure and J. Ginibre, Spectral and scattering theory for the Schrödinger operator with strongly oscillating potentials, Ann. Inst. H. Poincaré Sect. A (N.S.) 24 (1976), no. 1, 17–30. MR 400976
  • Allen Devinatz, The existence of wave operators for oscillating potentials, J. Math. Phys. 21 (1980), no. 9, 2406–2411. MR 585593, DOI 10.1063/1.524678
  • John D. Dollard, Asymptotic convergence and the Coulomb interaction, J. Mathematical Phys. 5 (1964), 729–738. MR 163620, DOI 10.1063/1.1704171
  • John D. Dollard, Quantum-mechanical scattering theory for short-range and Coulomb interactions, Rocky Mountain J. Math. 1 (1971), no. 1, 5–88. MR 270673, DOI 10.1216/RMJ-1971-1-1-5
  • John D. Dollard and Charles N. Friedman, Existence of the Møller wave operators for $V(r)=(\lambda \textrm {sin}(\mu r^{\alpha })/r^{\beta })$, Ann. Physics 111 (1978), no. 1, 251–266. MR 489510, DOI 10.1016/0003-4916(78)90230-0
  • Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
  • Volker Enss, Asymptotic completeness for quantum mechanical potential scattering. I. Short range potentials, Comm. Math. Phys. 61 (1978), no. 3, 285–291. MR 523013, DOI 10.1007/BF01940771
  • Volker Enss, Asymptotic completeness for quantum-mechanical potential scattering. II. Singular and long-range potentials, Ann. Physics 119 (1979), no. 1, 117–132. MR 535624, DOI 10.1016/0003-4916(79)90252-5
    • J. Ginibre, La méthode ’dépendant du temps’ dans le problème de la complétude asymptotique, preprint.
  • T. A. Green and O. E. Lanford III, Rigorous derivation of the phase shift formula for the Hilbert space scattering operator of a single particle, J. Mathematical Phys. 1 (1960), 139–148. MR 128356, DOI 10.1063/1.1703644
  • W. A. Harris Jr. and D. A. Lutz, Asymptotic integration of adiabatic oscillators, J. Math. Anal. Appl. 51 (1975), 76–93. MR 369840, DOI 10.1016/0022-247X(75)90142-0
  • Lars Hörmander, The existence of wave operators in scattering theory, Math. Z. 146 (1976), no. 1, 69–91. MR 393884, DOI 10.1007/BF01213717
  • Teruo Ikebe and Hiroshi Isozaki, Completeness of modified wave operators for long-range potentials, Publ. Res. Inst. Math. Sci. 15 (1979), no. 3, 679–718. MR 566076, DOI 10.2977/prims/1195187871
  • Hiroshi Isozaki, Eikonal equations and spectral representations for long-range Schrödinger Hamiltonians, J. Math. Kyoto Univ. 20 (1980), no. 2, 243–261. MR 582166, DOI 10.1215/kjm/1250522277
    • A. R. Its and V. B. Matveev, Co-ordinate asymptotics for the Schrödinger equation with a rapidly oscillating potential, J. Soviet Math. 11 (1979), 442-444.
  • Konrad Jörgens and Joachim Weidmann, Spectral properties of Hamiltonian operators, Lecture Notes in Mathematics, Vol. 313, Springer-Verlag, Berlin-New York, 1973. MR 0492941, DOI 10.1007/BFb0060821
  • Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
  • Hitoshi Kitada, Scattering theory for Schrödinger operators with long-range potentials. I. Abstract theory, J. Math. Soc. Japan 29 (1977), no. 4, 665–691. MR 634802, DOI 10.2969/jmsj/02940665
  • Hitoshi Kitada, Scattering theory for Schrödinger operators with long-range potentials. I. Abstract theory, J. Math. Soc. Japan 29 (1977), no. 4, 665–691. MR 634802, DOI 10.2969/jmsj/02940665
  • Hitoshi Kitada and Kenji Yajima, A scattering theory for time-dependent long-range potentials, Duke Math. J. 49 (1982), no. 2, 341–376. MR 659945
    • V. B. Matveev, Wave operators and positive eigenvalues for a Schrödinger equation with oscillating potential, Theoret. and Math. Phys. 15 (1973), 574-583.
  • V. B. Matveev and M. M. Skriganov, Wave operators for a Schrödinger equation with rapidly oscillating potential, Dokl. Akad. Nauk SSSR 202 (1972), 755–757 (Russian). MR 0300135
  • Kiyoshi Mochizuki and Jun Uchiyama, Radiation conditions and spectral theory for $2$-body Schrödinger operators with “oscillating” long-range potentials. I. The principle of limiting absorption, J. Math. Kyoto Univ. 18 (1978), no. 2, 377–408. MR 492943, DOI 10.1215/kjm/1250522579
  • Kiyoshi Mochizuki and Jun Uchiyama, Radiation conditions and spectral theory for $2$-body Schrödinger operators with “oscillating” long-range potentials. II, J. Math. Kyoto Univ. 19 (1979), no. 1, 47–70. MR 527395, DOI 10.1215/kjm/1250522468
    • —, Time dependent representations of the stationary wave operators for ’oscillating’ long-range potentials, preprint. J. von Neumann and E. Wigner, Über merkwürdige diskrete Eigenwerte, Z. Phys. 30 (1929), 465-467.
  • D. B. Pearson, Scattering theory for a class of oscillating potentials, Helv. Phys. Acta 52 (1979), no. 4, 541–554 (1980). MR 566255
  • Peter A. Perry, Propagation of states in dilation analytic potentials and asymptotic completeness, Comm. Math. Phys. 81 (1981), no. 2, 243–259. MR 632760, DOI 10.1007/BF01208898
  • Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
    • —, Methods of modern mathematical physics, vol. III, Academic Press, New York, 1979.
  • Yoshimi Sait\B{o}, Spectral representations for Schrödinger operators with long-range potentials, Lecture Notes in Mathematics, vol. 727, Springer, Berlin, 1979. MR 540891
  • Martin Schechter, Spectra of partial differential operators, 2nd ed., North-Holland Series in Applied Mathematics and Mechanics, vol. 14, North-Holland Publishing Co., Amsterdam, 1986. MR 869254
  • Martin Schechter, Scattering theory for elliptic operators of arbitrary order, Comment. Math. Helv. 49 (1974), 84–113. MR 367484, DOI 10.1007/BF02566721
  • Barry Simon, Phase space analysis of simple scattering systems: extensions of some work of Enss, Duke Math. J. 46 (1979), no. 1, 119–168. MR 523604
  • M. M. Skriganov, The spectrum of a Schrödinger operator with rapidly oscillating potential, Trudy Mat. Inst. Steklov. 125 (1973), 187–195, 235 (Russian). Boundary value problems of mathematical physics, 8. MR 0344705
    • —, The eigenvalues of the Schrödinger operator situated on the continuous spectrum, J. Soviet Math. 8 (1977), 464-467.
  • Hideo Tamura, The principle of limiting absorption for uniformly propagative systems with perturbations of long-range class, Nagoya Math. J. 82 (1981), 141–174. MR 618813, DOI 10.1017/S0027763000019334
    • M. Wolfe, Asymptotic behavior of solutions of the radial Schroedinger equation and applications to long-range potential scattering, Dissertation, University of Texas at Austin, 1978.
  • D. R. Yafaev, On the asymptotics of scattering phases for the Schrödinger equation, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), no. 3, 283–299 (English, with French summary). MR 1084881
    • B. Bourgeois, Quantum mechanical scattering for potentials of the form $\sin \,r/{r^\beta },\frac {1} {3} < \beta \leqslant \frac {1} {2}$, J. Math. Phys. 23 (1982), 790-797.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 275 (1983), 641-677
  • MSC: Primary 35P25; Secondary 34B25, 35P10, 81F05
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0682723-7
  • MathSciNet review: 682723