Skip to main content

Some aspects of the theory of Schrödinger operators

  • Conference paper
  • First Online:
Schrödinger Operators

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1159))

  • 766 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Reed and B. Simon, Methods of modern mathematical physics, Vol. II. Fourier analysis, self adjointness, Academic Press, 1975.

    Google Scholar 

  2. M. Reed and B. Simon, Methods of modern mathematical physics, Vol. III. Scattering theory, Academic Press, 1978.

    Google Scholar 

  3. M. Reed and B. Simon, Methods of modern mathematical physics, Vol. IV. Analysis of operators, Academic Press, 1977.

    Google Scholar 

  4. H. Cycon, R. Froese, W. Kirsch and B. Simon, Lectures on Schrödinger operators, in preparation.

    Google Scholar 

  5. M. Reed and B. Simon, Methods of modern mathematical physics, Vol. I. Functional analysis, Academic Press, 1972.

    Google Scholar 

  6. T. Kato, Perturbation theory for linear operators, Springer, 1966.

    Google Scholar 

  7. M. Schechter, Spectra of partial differential operators, North Holland, 1971.

    Google Scholar 

  8. T. Kato, Schrödinger operators with singular potentials, Is. J. Math. 13 (1973), 135–148.

    Article  Google Scholar 

  9. B. Simon, Maximal and minimal Schrödinger forms, J. Op. Th. 1 (1979), 37–47.

    MATH  Google Scholar 

  10. H. Leinfelder and C. Simader, Schrödinger operators with singular magnetic vector potentials, Math. Z. 176 (1981), 1–19.

    Article  MATH  MathSciNet  Google Scholar 

  11. F. Rellich, Störungstheorie der spectralzerlegung, II, Math. Ann. 116 (1939), 555–570.

    Article  MathSciNet  Google Scholar 

  12. T. Kato, Fundamental properties of Hamiltonian operators of Schrödinger type, Trans. Am. Math. Soc. 70 (1951), 195–211.

    MATH  Google Scholar 

  13. F. Stummel, Singulare elliptische differentialoperatoren in Hilbertschen Räumen, Math. Ann. 132 (1956), 150–176.

    Article  MATH  MathSciNet  Google Scholar 

  14. A.G. Sigalov and I.M. Sigal, Description of the spectrum of the energy operator of quantum-mechanical systems…, Theor. and Math. Phys. 5 (1970), 990.

    Article  ADS  MathSciNet  Google Scholar 

  15. P. Deift, W. Hunziker, B. Simon and E. Vock, Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, IV, Commun. Math. Phys. 64 (1978), 1–34.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. B. Simon, Schrödinger semigroups, Bull. AMS 7 (1982), 447–526.

    Article  MATH  Google Scholar 

  17. M. Aizenman and B. Simon, Brownian motion and Harnack's inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), 209–273.

    Article  MATH  MathSciNet  Google Scholar 

  18. Z. Zhao, Uniform boundedness of conditional gauge and Schrödinger equations, Commun. Math. Phys., to appear, and Continuity of conditioned Feynman-Kac functional and integral kernel for Schrödinger equation, Stanford preprint.

    Google Scholar 

  19. J. Brossard, The problem of Dirichlet for the Schrödinger operator, Institut Fourier preprint, 1984.

    Google Scholar 

  20. B. Simon, On the number of bound states of two-body Schrödinger operators — A review, from Studies in mathematical physics, essays in honor of Valentine Bargmann, Princeton Press, 1976, pp. 305–326.

    Google Scholar 

  21. B. Simon, Functional integration and quantum physics, Academic Press, 1979.

    Google Scholar 

  22. E.H. Lieb and W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, in Studies in mathematical physics, essays in honor of Valentine Bargmann, Princeton Press, 1976, pp. 269–304.

    Google Scholar 

  23. G. Rosenbljum, The distribution of the discrete spectrum for singular differential operators, Dokl. Akad. Nauk SSSR 202 (1972); transl. Soviet Math. Dokl. 13 (1972), 245–249.

    Google Scholar 

  24. M. Cwikel, Weak type estimates and the number of bound states of Schrödinger operators, Ann. Math. 106 (1977), 93–102.

    Article  MATH  MathSciNet  Google Scholar 

  25. E. Lieb, Bounds on the eigenvalues of the Laplace and Schrödinger operators, Bull. AMS 82 (1976), 751–753, and Proc. 1979 AMS Honolulu Conference.

    Article  MATH  MathSciNet  Google Scholar 

  26. B. Simon, Trace ideals and their applications, Cambridge Univ. Press, 1979.

    Google Scholar 

  27. P. Li and S.T. Yau, On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys. 88 (1983), 309.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. C. Fefferman, The uncertainty principle, Bull. Am. Math. Soc. 9 (1983), 129.

    Article  MATH  MathSciNet  Google Scholar 

  29. M. Birman and V.V. Borzov, On the asymptotics of the discrete spectrum of some singular differential operators, Topics in Math. Phys. 5 (1972), 19–30.

    MathSciNet  Google Scholar 

  30. A. Martin, Bound states in the strong coupling limit, Helv. Phys. Acta. 45 (1972), 140–148.

    Google Scholar 

  31. H. Tamura, The asymptotic eigenvalue distribution for nonsmooth elliptic operators, Proc. Japan Acad. 50 (1974), 19–22.

    Article  MATH  MathSciNet  Google Scholar 

  32. M. Klaus and B. Simon, Coupling constant threshold in non-relativistic quantum mechanics, II. Two body thresholds in N-body systems, Commun. Math. Phys. 78 (1980), 153–168.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. A. Persson, Bounds for the discrete part of the spectrum of a semibounded Schrödinger operator, Math. Semd. 8 (1960), 143–153.

    MATH  MathSciNet  Google Scholar 

  34. S. Agmon, Lectures on exponential decay of solutions of second order elliptic equations. Bounds on eigenfunctions of N-body Schrödinger operators, Mathematical Notes, Princeton Univ, Press, Princeton, NJ 1982.

    MATH  Google Scholar 

  35. S. Agmon, On exponential decay of solutions of second order elliptic equations in unbounded domains, Proc. A. Pleijel Conf.

    Google Scholar 

  36. L. Garding, On the essential spectrum of Schrödinger operators, J. Func. Anal. 52 (1983), 1–10.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. W. Hunziker, On the spectra of Schrödinger multiparticle Hamiltonians, Helv. Phys. Acta 39 (1966), 451–462.

    MATH  MathSciNet  Google Scholar 

  38. C. van Winter, Theory of finite systems of particles, I. Mat. Fys. Skr. Danske Vid. Selsk 1 (1964), 1–60.

    Google Scholar 

  39. G. Zhislin, Discussion of the spectrum of the Schrödinger operator for systems of many particles, Trudy. Mosk. Mat. Obs. 9 (1960), 81–128.

    Google Scholar 

  40. V. Enss, A note on Hunziker's theorem, Commun. Math. Phys. 52 (1977), 233.

    Article  ADS  MathSciNet  Google Scholar 

  41. B. Simon, Geometric methods in multiparticle quantum systems, Commun. Math. Phys. 55 (1977), 259–274.

    Article  ADS  MATH  Google Scholar 

  42. I.M. Sigal, Geometric methods in the quantum many-body problem. Nonexistence of very negative ions, Commun. Math. Phys. 85 (1982), 309–324.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  43. G.M. Zhislin, On the finiteness of the discrete spectrum of the energy operator of negative atomic and molecular ions, Theor. Math. Phys. 7 (1971), 571–578.

    Article  Google Scholar 

  44. M.A. Antonets, G.M. Zhislin and J.A. Sheresherskii, On the discrete spectrum of the Hamiltonian of an n-particle quantum system, Theor. Math. Phys. 16 (1973), 800.

    Article  Google Scholar 

  45. M.B. Ruskai, Absence of discrete spectrum in highly negative ions, I, II. Commun. Math. Phys. 82 (1982), 457; 85 (1982), 325.

    Article  ADS  MathSciNet  Google Scholar 

  46. E.H. Lieb, Bound on the maximum negative ionization of atoms and molecules, Phys. Rev. A, to appear.

    Google Scholar 

  47. E.H. Lieb, I.M. Sigal, B. Simon and W. Thirring, Asymptotic bulk neutrality of large Z ions, Phys. Rev. Lett., to appear.

    Google Scholar 

  48. P. Deift and B. Simon, A time-dependent approach to the completeness of multiparticle quantum systems, Commun. Pure Appl. Math. 30 (1977), 573–583.

    Article  MATH  MathSciNet  Google Scholar 

  49. E.B. Davies, On Enss' approach to scattering theory, Duke Math. J. 47 (1980), 171–185.

    Article  MATH  MathSciNet  Google Scholar 

  50. P. Perry, Scattering theory by the Enss method, Mathematical Reports, Vol. I, 1983.

    Google Scholar 

  51. P. Perry, I.M. Sigal and B. Simon, Spectral analysis of multiparticle Schrödinger operators, Ann. Math. 114 (1981), 519–567.

    Article  MATH  MathSciNet  Google Scholar 

  52. L. Faddeev, Mathematical aspects of the three body problem in quantum scattering theory, Steklov Institute, 1963.

    Google Scholar 

  53. V. Enss, Topics in scattering theory for multiparticle quantum mechanics, to appear in Boulder IAMP Proceedings.

    Google Scholar 

  54. E. Mourre, Operateurs conjugués et propriétés de propagation, II, preprint.

    Google Scholar 

  55. G. Hagedorn, Asymptotic completeness for classes of two, three, and four particle Schrödinger operators, Trans. AMS 258 (1980), 1–75.

    MATH  MathSciNet  Google Scholar 

  56. I.M. Sigal, On quantum mechanics of many-body systems with dilation-analytic potentials, Bull. AMS 84 (1978), 152–154.

    Article  MATH  MathSciNet  Google Scholar 

  57. E. Mourre and I.M. Sigal, Phase space analysis and scattering theory for N-particle systems, preprint.

    Google Scholar 

  58. I.M. Sigal, Mathematical foundations of quantum scattering theory for multi-particle systems, Mem. Am. Math. Soc. 209 (1978).

    Google Scholar 

  59. E. Mourre, Absence of singular spectrum for certain self-adjoint operators, Commun. Math. Phys. 78 (1981), 391.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  60. R. Froese and I. Herbst, A new proof of the Mourre estimate, Duke Math. J. 49 (1982), 1075.

    Article  MATH  MathSciNet  Google Scholar 

  61. E. Mourre, Operateurs conjugués et propriétés de propagation, I, Commun. Math. Phys. 91 (1983), 279.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  62. A. Jensen, E. Mourre and P. Perry, Multiple commutator estimates and resolvent smoothness in quantum scattering theory, preprint, 1983.

    Google Scholar 

  63. R. Froese and I. Herbst, Exponential bounds and absence of positive eigenvalues for N-body Schrödinger operators, Commun. Math. Phys. 87 (1982), 429.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  64. R. Froese, I. Herbst, M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof, L2-exponential lower bounds to solutions of the Schrodinger equation, Commun. Math. Phys. 87 (1982), 265–286.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  65. B. Simon and B. Souillard, Franco-American meeting on the mathematics of random and almost periodic potentials, J. Stat. Phys., to appear.

    Google Scholar 

  66. R. Prange, D. Grempel and S. Fishman, A solvable model of quantum motion in an incommensurate potential, Phys. Rev., in press; Localization in an incommensurate potential: An exactly solvable model, Phys. Rev. Lett. 49 (1982), 833.

    Google Scholar 

  67. B. Simon, Almost periodic Schrödinger operators, IV. The Maryland model, Ann. Phys., to appear.

    Google Scholar 

  68. L. Pastur and A. Figotin, Localization in an incommensurate potential: Exactly solvable multidimensional model, JETP Lett. 37 (1983), 686; paper to appear in Commun. Math. Phys.

    ADS  Google Scholar 

  69. H. Kunz and B. Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Commun. Math. Phys. 78 (1980), 201–246.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  70. W. Craig and B. Simon, Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices, Commun. Math. Phys. 90 (1983), 207–218.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  71. F. Delyon and B. Souillard, Remark on the continuity of the density of states of ergodic finite difference operators, Commun. Math. Phys., to appear.

    Google Scholar 

  72. F. Delyon, H. Kunz and B. Souillard, One dimensional wave equations in disordered media, J. Phys. A16 (1983), 25.

    ADS  MathSciNet  Google Scholar 

  73. I. Goldsheid, S. Molchanov and L. Pastur, A pure point spectrum of the stochastic and one dimensional Schrödinger equation, Funct. Anal. Appl. 11 (1977), 1–10.

    Article  Google Scholar 

  74. R. Carmona, Exponential localization in one dimensional disordered systems, Duke Math. J. 49 (1982), 191.

    Article  MATH  MathSciNet  Google Scholar 

  75. B. Simon, Some Jacobi matrices with decaying potential and dense point spectrum, Commun. Math. Phys. 87 (1982), 253–258.

    Article  ADS  MATH  Google Scholar 

  76. F. Delyon, B. Simon and B. Souillard, From power law localized to extended states in a disordered system, preprint.

    Google Scholar 

  77. J. Moser, An example of a Schrödinger equation with an almost periodic potential and nowhere dense spectrum, Comm. Math. Helv. 56 (1981), 198.

    Article  MATH  Google Scholar 

  78. V. Chulaevsky, On perturbations of a Schrödinger operator with periodic potential, Russian Math. Surveys 36, No. 5 (1981), 143.

    Article  ADS  Google Scholar 

  79. J. Avron and B. Simon, Almost periodic Schrödinger operators, I. Limit periodic potentials, Commun. Math. Phys. 82 (1982), 101–120.

    Article  ADS  MathSciNet  Google Scholar 

  80. J. Avron and B. Simon, Transient and recurrent spectrum, J. Func. Anal. 43 (1981), 1–31.

    Article  MATH  MathSciNet  Google Scholar 

  81. W. Craig, Pure point spectrum for discrete almost periodic Schrödinger operators, Commun. Math. Phys. 88 (1983), 113–131.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  82. J. Pöschel, Examples of discrete Schrödinger operators with pure point spectrum, Commun. Math. Phys. 88 (1983), 447–463.

    Article  ADS  MATH  Google Scholar 

  83. J. Bellissard, R. Lima and E. Scoppola, Localization in v-dimensional incommensurate structures, Commun. Math. Phys. 88 (1983), 465–477.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  84. P. Sarnak, Spectral behavior of quasi periodic potentials, Commun. Math. Phys. 84 (1982), 377–401.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  85. J. Avron and B. Simon, Almost periodic Schrödinger operators, II. The integrated density of states, Duke Math. J. 50 (1983), 369–391.

    Article  MATH  MathSciNet  Google Scholar 

  86. S. Aubry and G. Andre, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Israel Phys. Soc. 3 (1980), 133.

    MathSciNet  Google Scholar 

  87. A. Gordon, Usp. Math. Nauk. 31 (1976), 257.

    MATH  Google Scholar 

  88. S. Kotani, Proceedings of the Conference on Stochastic Processes, Kyoto, 1982.

    Google Scholar 

  89. B. Simon, Kotani theory for one dimensional stochastic Jacobi matrices, Commun. Math. Phys. 89 (1983), 227.

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Simon, B. (1985). Some aspects of the theory of Schrödinger operators. In: Graffi, S. (eds) Schrödinger Operators. Lecture Notes in Mathematics, vol 1159. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0080333

Download citation

  • DOI: https://doi.org/10.1007/BFb0080333

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16035-9

  • Online ISBN: 978-3-540-39706-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics