The resonance expansion for the Green's function of the Schrödinger and wave equations

  • S. Albeverio
  • R. Høegh-Krohn
II Mathematical Framework
Part of the Lecture Notes in Physics book series (LNP, volume 211)


We give a survey of some recent mathematical work on resonances, in particular on perturbation series, low energy expansions and on resonances for point interactions. Expansions of the kernels of \(e^{ - it\sqrt {H_ + } } \) and e−itH in terms of resonances are also given (where H+ is the positive part of the Hamiltonian).


Compact Support Imaginary Axis Point Interaction Resonant State SchrSdinger Equation 
These keywords were added by machine and not by the authors.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Jost, Über die falschen Nullstellen der Eigenwerte der S-Matrix, Helv. Phys. Acta 20, 250–266 (1947)Google Scholar
  2. [2]
    R. Newton, Scattering theory of waves and particles, McGraw Hill, 2nd ed. (1982)Google Scholar
  3. [3]
    W.O. Amrein, J.M. Jauch, H.B. Sinha, Scattering theory in quantum mechanics, Reading, Benjamin (1977)Google Scholar
  4. [4]
    A. Grossmann, Nested Hilbert spaces in quantum mechanics I, J. Math. Phys. 5, 1025–1037 (1964)CrossRefGoogle Scholar
  5. [5]
    L.P. Horwitz, E. Katznelson, A partial inner product space of analytic functions for resonances, J. Math. Phys. 24, 848–859 (1983)CrossRefGoogle Scholar
  6. [6]
    L.P. Horwitz, J.P. Marchard, The decay scattering system, Rocky Mount. J. Math. 1, 225–253 (1971)Google Scholar
  7. [7]
    L.P. Horwitz, I.M. Sigal, On a mathematical model for non stationary physical systems, Helv. Phys. Acta 51, 685–715 (1978)Google Scholar
  8. [8]
    J. Rauch, Perturbation theory for eigenvalues and resonances for Schrödinger Hamiltonians, J. Funct. Anal. 35, 304–315 (1980)Google Scholar
  9. [9]
    G. Parravicini, V. Gorini, E.C.G. Sudarshan, Resonances, scattering theory, and rigged Hilbert spaces, J. Math. Phys. 21, 2208–2226 (1980)Google Scholar
  10. [10]
    S. Albeverio, On bound states in the continuum of N-body systems and the virial theorem, Ann. Phys. 71, 167–276 (1972)CrossRefGoogle Scholar
  11. [11]
    M. Reed, B. Simon, Methods of Modern Mathematical Physics, III. Scattering Theory; IV, Analysis Operators, Academic Press, New York (1978).Google Scholar
  12. [12]
    J.S. Howland, The Livsic matrix in perturbation theory, J. Math. Anal. Appl. 50, 415–437 (1975)CrossRefGoogle Scholar
  13. [13]
    H. Baumgärtel, M. Wollenberg, Mathematical Scattering Theory, Birkhäuser, Basel (1983)Google Scholar
  14. [14]
    M. Demuth, On the perturbation theory of instable isolated eigenvalues, Math. Nachr. 64, 345–356 (1974)Google Scholar
  15. [15]
    P.D. Lax, R.S. Phillips, Scattering theory, Acad. Press, New York (1967)Google Scholar
  16. [16]
    P.D. Lax, R.S. Phillips, Scattering theory for automorphic forms, Princeton Univ. Press, Princeton (1976);Bull. Am. Math. Soc. 2, 265–295 (1980)Google Scholar
  17. [17]
    C. Bardos, J.C. Guillot, J. Ralston, La relation de Poisson pour 1'équation des ondes dans un ouvert non borne. Application a la théoree de la diffusion, Commun. Part. Diff. Eqts. 7, 905–958 (1982)Google Scholar
  18. [18]
    E. Balslev, these Proc.Google Scholar
  19. [19]
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn, The low energy expansion in non relativistic scattering theory, Ann. Inst. H. Poincaré A37, 1–28 (1982)Google Scholar
  20. [20]
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable models of mathematical physics, book in preparation.Google Scholar
  21. [21]
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Some exactly solvable models in quantum mechanics and the low energy expansion, to appear Proc. Leipzig Conf. Operator algebras '83, TeubnerGoogle Scholar
  22. [22]
    A. Jensen, T. Kato, Spectral properties of the Schrödinger operators and time-decay of the wave functions, Duke Math. J. 46, 583–611 (1979)CrossRefGoogle Scholar
  23. [23]
    R. Newton, Non central potentials: The generalized Levinson theorem and the structure of the spectrums, J. Math. Phys. 18, 1348–1357 (1977)CrossRefGoogle Scholar
  24. [24]
    S. Albeverio, R. Høegh-Krohn, Perturbation of resonances in quantum mechanics, J. Math. Anal. Appl. 101, 491–513 (1984)CrossRefGoogle Scholar
  25. [25]
    B. Simon, On the absorption of eigenvalues by the continuum spectrum in regular perturbation problems, J. Funct. Anal. 25, 338–344 (1977)CrossRefGoogle Scholar
  26. [26]
    A.G. Ramm, Perturbation of resonances, J. Math. Anal. Appl. 88, 1–7 (1982)CrossRefGoogle Scholar
  27. [27]
    M. Klaus, B. Simon, Coupling constant thresholds in non relativistic quantum mechanics, Ann. Phys. 130, 251–281 (1980)CrossRefGoogle Scholar
  28. [28]
    A. Baz, Ya. Zeldovich, A. Perelomov, Scattering, Reactions and Decay in Nonrelativistic Quantum Mechanics, Isr. Progr. Scient. Transl. Jerusalem 1969Google Scholar
  29. [29]
    J.S. Bell, Electromagnetic properties of unstable particles, Nuovo Cim 24, 452–460 (1962)Google Scholar
  30. [30]
    N. Hokkyo, A remark on the norm of the unstable state, Prop. Theor. Phys. 33, 1116–1128 (1965)Google Scholar
  31. [31]
    A. Bohm, Resonance poles and Gamov vectors in the rigged Hilbert space formulation of quantum mechanics, J. Math. Phys. 22, 2813–2823 (1981)Google Scholar
  32. [32]
    T. Berggren, A note on function space spanned by complex energy eigenfunctions, Phys. Letts. B 38, 61–63 (1972)CrossRefGoogle Scholar
  33. [33]
    W.J. Romo, A study of the completeness properties of resonant states, J. Math. Phys. 21, 311–326 (1980)CrossRefGoogle Scholar
  34. [34]
    B. Gyarmati, T. Vertse, On the normalization of Gamov functions, Nucl. Phys. A 160, 523–528 (1971)CrossRefGoogle Scholar
  35. [35]
    R. Moore, E. Gerjuoy, Properties of resonance wave functions, Phys. Rev. A7, 1288–1303 (1973)Google Scholar
  36. [36]
    G. Garcia-Calderon, R. Peierls, Resonant states and their uses, Nucl. Phys. A 265, 443–460 (1976)CrossRefGoogle Scholar
  37. [37]
    A. Bohm, Quantum mechanics, Springer, New York (1979)Google Scholar
  38. [38]
    F. Gesztesy, these ProceedingsGoogle Scholar
  39. [39]
    S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, Singular perturbations and non standard analysis, Trans. Ann. Math. Soc. 252, 275–295 (1979)Google Scholar
  40. [40]
    S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrom, Non standard methods in stochastic analysis and mathematical physics, Acad. PressGoogle Scholar
  41. [41]
    S. Albeverio, R. Høegh-Krohn, Schrödinger operators with point interactions and short range expansions, Physica 124A, 11–28 (1984)MathSciNetGoogle Scholar
  42. [42]
    T. Berggren, Inner product for resonant states and shell-model application, Nucl. Phys. A116, 618–636 (1968)Google Scholar
  43. [43]
    T. Berggren, On the case of resonant states in eigenfunction expansions of scattering and reaction amplitudes, Nucl. Phys. A109, 265–287 (1968)Google Scholar
  44. [44]
    T. Berggren, On resonance contributions to sum rules in nuclear physics, Phys. Lett. B44, 23–25 (1973)Google Scholar
  45. [45]
    B. Berrondo, G. Garcia-Calderon, An eigenfunction expansion involving resonant states, Lett. Nuovo Cim 20, 34–38 (1977)Google Scholar
  46. [46]
    Y. Colin de Verdiére, Pseudo-Laplaciens I, II, Ann. Inst. Fourier 32,3,275–286 (1982); 33,2, 87–113 (1983)Google Scholar
  47. [47]
    H.M. Nussenzveio, The poles of the S-matrix of a rectangular potential well or barrier, Nucl. Phys. 11, 499–521 (1959)CrossRefGoogle Scholar
  48. [48]
    R.G. Newton Non central potentials, J. Math. Phys. 18, 1348–1357 (1977); Czech. J. Phys. B24, 1195–1204 (1974)CrossRefGoogle Scholar
  49. [49]
    S. Albeverio, J.F. Fenstad, R. Høegh-Krohn, W. Karwowski, T. Lindstrom, Perturbation of the Laplacian supported by zero measure sets, ZiF, Preprint 1984, to appear in Phys. Letts. A.Google Scholar
  50. [50]
    S. Albeverio, R. Høegh-Krohn, L. Streit, Energy forms, Hamiltonians and distorted Brownian paths, J. Math. Phys. 18, 907–917 (1977)CrossRefGoogle Scholar
  51. [51]
    L. Dabrovsky, H. Grosse, On non local point interactions in one, two and three dimensions, J. Math. Phys.Google Scholar
  52. [52]
    R. Høegh-Krohn, M. Mebkhout, The 1/r expansion for the critical multiple well problem, Commun. Math. Phys. 91, 65–73 (1983)Google Scholar
  53. [53]
    H. Holden, R. Høegh-Krohn, T. Wahl, Some explicit results on point interactions, Oslo Preprint (1983)Google Scholar
  54. [54]
    A. Grossman, T.T. Wu, A class of potentials with extremely narrow resonances I. Case with discrete rotational symmetry, CNRS-CPT Marseille Preprint, 1981.Google Scholar
  55. [55]
    A. Grossman, R. Høegh-Krohn, M. Mebkhout, The one particle theory of periodic point interactions, Commun. Math. Phys. 77, 87–110 (1980)CrossRefGoogle Scholar
  56. [56]
    S. Albeverio, R. Høegh-Krohn, Point interactions as limits of short range interactions, J. Operator Th. 6, 313–339 (1981)Google Scholar
  57. [57]
    H. Holden, R. Høegh-Krohn, S. Johannesen, The short range expansion, Adv. Appl. Math. 4, 402–421 (1983)Google Scholar
  58. [58]
    H. Holden, R. Høegh-Krohn, S. Johannesen, The short range expansion in solid state, Ann. I. H. Poincare (1984)Google Scholar
  59. [59]
    S. Albeverio, R. Høegh-Krohn, M. Mebkhout, Scattering by impurities in a solvable model of a 3-dimensional crystal, J. Math. Phys. 25, 1327–1334 (1984)CrossRefGoogle Scholar
  60. [60]
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn, L. Streit, Charged particles with short range interactions, Ann. Inst. H. Poincaré A38, 303–333 (1983)Google Scholar
  61. [61]
    S. Albeverio, D. Bollé, F. Gesztesy, R. Høegh-Krohn, L. Streit, Low energy parameters in non relativistic scattering theory, Ann. Phys. 140, 308–326 (1983)Google Scholar
  62. [62]
    S. Albeverio, L.S. Ferreira, F. Gesztesy, R. Høegh-Krohn, L. Streit, Model dependence of Coulomb-corrected scattering lengths, Phys. Rev. C29, 680–683 (1984)Google Scholar
  63. [63]
    S. Lang,SL2(ℜ), Addison-Wesley(1975)Google Scholar
  64. [64]
    B. Simon, Schrödinger semi groups, Bull. Am. Math. Soc. 7, 447–526 (1982)Google Scholar
  65. [65]
    A.G. Ramm. Theory and Applications of Some New Classes of Integral Equations, Springer, New York (1980)Google Scholar
  66. [66]
    O. Zohni, Generalized completeness relations in the theory of resonant scattering, J. Math. Phys. 23, 798–802 (1982)CrossRefGoogle Scholar
  67. [67]
    S. Fassari, On the Schrodinger equation with periodic point interactions in the 3-dimensional case, J. Math. Phys. 1984Google Scholar
  68. [68]
    J. Persson, The wave equation with measures as potentials and related topics, to appear in Rend. Sem. Mat. Univ. Politec. TorinoGoogle Scholar
  69. [69]
    T.A. Osborn, R. Wong, Time Decay and Spectral Kernel Asymptotics, University of Maryland Preprint April 1984Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • S. Albeverio
    • 1
    • 2
    • 3
  • R. Høegh-Krohn
    • 1
    • 4
    • 5
  1. 1.Zentrum für interdisziplinare ForschungUniversität BielefeldBielefeld
  2. 2.Mathematisches InstitutRuhr-UniversitätBochum 1Germany
  3. 3.Centre de Physique Theorique CNRSUniversité d'Aix-Marseille IIMarseille
  4. 4.CNRSUniversite de Provence and Centre de Physique TheoriqueMarseille
  5. 5.Matematisk InstituttUniversitetet i OsloOslo

Personalised recommendations