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On the spectral theory of schrödinger and dirac operators with strongly singular potentials

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Spectral Theory and Differential Equations

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References

  1. Barut, A.O.: Some unusual applications of Lie algebra representations in quantum theory. SIAM J. Appl. Math. 25, 247–259 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. Behncke, H.: Some remarks on singular attractive potentials. Nuovo Cimento 55, 780–785 (1968)

    Article  Google Scholar 

  3. Carleman, T.: Sur les équations intégrales singulières à noyau réel et symétrique. Uppsala universitets årsskrift 1923.

    Google Scholar 

  4. Carleman, T.: Sur la théorie mathématique de Schrödinger. Arkiv för mat., astr. och fysik 24 B, N:o 11 (1934) (= Edition complète des articles de Torsten Carleman: Malmö 1960)

    Google Scholar 

  5. Case, K.M.: Singular potentials. Phys. Rev. 80, 797–806 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  6. Courant, R., und Hilbert, D.: Methoden der Mathematischen Physik I. 2. Auflage. Berlin: Springer 1931

    Book  MATH  Google Scholar 

  7. Evans, W.D.: On the unique self-adjoint extension of the Dirac operator and the existence of the Green matrix. Proc. London Math. Soc. (3) 20, 537–557 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  8. Everitt, W.N., Giertz, M., and McLeod, J.B.: On the strong and weak limit-point classification of second-order differential expressions. The University of Wisconsin Technical Summary Report # 1338, September 1973

    Google Scholar 

  9. Frank, W.M., Land, D.J., and Spector, R.M.: Singular potentials. Rev. Mod. Physics 43, 36–98 (1971)

    Article  MathSciNet  Google Scholar 

  10. Frank, Ph., und v.Mises, R.: Die Differential-und Integralgleichungen der Mechanik und Physik II. 2. Auflage. Braunschweig: Vieweg 1934

    Google Scholar 

  11. Freudenthal, H.: Über die Friedrichssche Fortsetzung halbbeschränkter Hermitescher Operatoren. Nederl. Akad. Wetensch. Proc. 39, 832–833 (1936)

    MATH  Google Scholar 

  12. Friedrichs, K.: Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren I,II. Math. Ann. 109, 465–487, 685–713 (1933/34) (Berichtigung, Math. Ann. 110, 777–779 (1934/35))

    Article  MathSciNet  MATH  Google Scholar 

  13. Friedrichs, K.: Über die ausgezeichnete Randbedingung in der Spektraltheorie der halbbeschränkten gewöhnlichen Differentialoperatoren zweiter Ordnung. Math. Ann. 112, 1–23 (1935/36)

    Article  MathSciNet  MATH  Google Scholar 

  14. Glazman, I.M.: Direct methods of qualitative spectral analysis of singular differential operators. Jerusalem: Israel program for scientific translations 1965

    Google Scholar 

  15. Glimm, J., and Jaffe, A.: Singular perturbations of self-adjoint operators. Comm. Pure Appl. Math. 22, 401–414 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gordon, W.: Die Energieniveaus des Wasserstoffatoms nach der Diracschen Theorie des Elektrons. Z. Physik 48, 11–14 (1928)

    Article  MATH  Google Scholar 

  17. Grütter, A.: Wesentliche Selbstadjungiertheit eines Schrödinger-Operators. Math. Z. 135, 289–291 (1974)

    Article  MATH  Google Scholar 

  18. Gustafson, K.E., and Rejto, P.A.: Some essentially self-adjoint Dirac operators with spherically symmetric potentials. Israel J. Math. 14, 63–75 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hartman, Ph.: Ordinary Differential Equations. New York: John Wiley 1964

    MATH  Google Scholar 

  20. Hellwig, B.: Ein Kriterium für die Selbstadjungiertheit elliptischer Differentialoperatoren im Rn. Math. Z. 86, 255–262 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hellwig, B.: Ein Kriterium für die Selbstadjungiertheit singulärer elliptischer Differentialoperatoren im Gebiet G. Math. Z. 89, 333–344 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hellwig, B.: A criterion for self-adjointness of singular elliptic differential operators. J. Math. Anal. Appl. 26, 279–291 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hellwig, G.: Differential operators of mathematical physics. Reading: Addison-Wesley 1967

    MATH  Google Scholar 

  24. Ikebe, T., and Kato, T.: Uniqueness of the self-adjoint extensions of singular elliptic differential operators. Arch. Rational Mech. Anal. 9, 77–92 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jansen, K.-H.: Neue Kriterien für das Fehlen von L2-Lösungen für − Δ v = f(x,v) im Rn unter besonderer Berücksichtigung des linearen Falles. Jber.Deutsch.Math.Verein. 72, 190–204 (1971).

    MATH  Google Scholar 

  26. Jauch, J.M.: Foundations of quantum mechanics. Reading: Addison-Wesley 1968

    MATH  Google Scholar 

  27. Jörgens, K.: Wesentliche Selbstadjungiertheit singulärer elliptischer Differentialoperatoren zweiter Ordnung in C ∞o (G). Math. Scand. 15, 5–17 (1964)

    MathSciNet  MATH  Google Scholar 

  28. Kalf, H.: On the characterization of the Friedrichs extension of ordinary or elliptic differential operators with a strongly singular potential. J. Functional Anal. 10, 230–250 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kalf, H.: A limit-point criterion for separated Dirac operators and a little known result on Riccati’s equation. Math. Z. 129, 75–82 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kalf, H.: Self-adjointness for strongly singular potentials with a − |x|2 fall-off at infinity. Math. Z. 133, 249–255 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kalf, H.: The quantum mechanical virial theorem and the absence of positive energy bound states of Schrödinger operators. Submitted to J. Analyse Math.

    Google Scholar 

  32. Kalf, H., and Walter, J.: Strongly singular potentials and essential self-adjointness of singular elliptic operators in C ∞o (Rn / {0}). J. Functional Anal. 10, 114–130 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kalf, H., and Walter, J.: Note on a paper of Simon on essentially self-adjoint Schrödinger operators with singular potentials. Arch. Rational Mech. Anal. 52, 258–260 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  35. Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966

    Book  MATH  Google Scholar 

  36. Kato, T.: Schrödinger operators with singular potentials. Israel J. Math. 13, 135–148 (1972)

    Article  MathSciNet  Google Scholar 

  37. Kemble, E.C.: The fundamental principles of quantum mechanics. New York: Dover 1958

    MATH  Google Scholar 

  38. Landau, L.D., and Lifshitz, E.M.: Quantum Mechanics. Non-relativistic theory. London: Pergamon 1959

    MATH  Google Scholar 

  39. Landau, L.D., und Lifschitz, E.M.: Relativistische Quantentheorie. Berlin: Akademie-Verlag 1971

    Google Scholar 

  40. Meetz, K.: Singular potentials in nonrelativistic quantum mechanics. Nuovo Cimento 34, 690–708 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  41. Morse, Ph.M., and Feshbach, H.: Methods of theoretical physics II. New York: Mc Graw-Hill 1953

    MATH  Google Scholar 

  42. v. Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Berlin: Springer 1932

    MATH  Google Scholar 

  43. Nilsson, N.: Essential self-adjointness and the spectral resolution of Hamiltonian operators. Kungl. Fysiogr. Sällsk. i Lund Förh. Bd. 29, Nr. 1 (1959)

    Google Scholar 

  44. Oppenheimer, J.R.: Three notes on the quantum theory of aperiodic effects. Phys. Rev. 31, 66–81 (1928)

    Article  MATH  Google Scholar 

  45. Oseen, C.W.: Über die Fundamentalintegrale einiger wellenmechanischen Differentialgleichungen. Akiv för mat., astr. och fysik. 22 A, N:o 2 (1930)

    Google Scholar 

  46. Oseen, C.W.: Deux remarques sur la méthode des perturbation dans la mécanique ondulatoire Arkiv för math. astr. och fysik 25 A, N:o 2 (1934)

    Google Scholar 

  47. Popov, V.S.: "Collapse to the center" at Z > 137 and critical nuclear charge. Soviet J. Nuclear Phys. 12, 235–243 (1971) (english translation of Yad. Fiz. 12, 429–447 (1970))

    Google Scholar 

  48. Popov, V.S.: On the properties of the discrete spectrum for Z close to 137. Soviet Phys. JETP 33, 665–673 (1971) (english translation of Ž. Eksp. Teor. Fiz. 60, 1228–1244 (1971))

    Google Scholar 

  49. Povzner, A.Ya.: The expansion of arbitrary functions in terms of eigenfunctions of the operator − Δ u + cu. Amer. Math. Soc. Transl. (2) 60, 1–49 (1967) (english translation of Mat. Sb. 32, 109–156 (1953))

    Article  MATH  Google Scholar 

  50. Reed, M., and Simon, B.: Methods of modern mathematical physics I: Functional analysis. New York: Academic Press 1972

    MATH  Google Scholar 

  51. Rein, D.: Über den Grundzustand überschwerer Atome. Z. Physik 221, 423–430 (1969)

    Article  Google Scholar 

  52. Rejto, P.A.: Some essentially self-adjoint one-electron Dirac operators. Israel J. Math. 9, 144–171 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  53. Rellich, F.: Die zulässigen Randbedingungen bei den singulären Eigenwertproblemen der mathematischen Physik. Math. Z. 49, 702–723 (1943/44)

    Article  MathSciNet  MATH  Google Scholar 

  54. Rellich, F.: Eigenwerttheorie partieller Differentialgleichungen II. Vervielfältigtes Vorlesungsmanuskript: Göttingen 1953

    Google Scholar 

  55. Rellich, F.: Halbbeschränkte Differentialoperatoren höherer Ordnung. Proc. Internat. Congress Math. Amsterdam 1954. Vol. 3, 243–250

    MATH  Google Scholar 

  56. Rellich, F.: Perturbation theory of eigenvalue problems. New York: Gordon and Breach 1969

    MATH  Google Scholar 

  57. Rohde, H.-W.: Über die Symmetrie elliptischer Differentialoperatoren. Math. Z. 86, 21–33 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  58. Rohde, H.-W.: Regularitätsaussagen mit Anwendungen auf die Spektraltheorie elliptischer Differentialoperatoren. Math.Z. 91, 30–49 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  59. Rohde, H.-W.: Kriterien zur Selbstadjungiertheit elliptischer Differentialoperatoren I,II. Arch. Rational Mech. Anal. 34, 188–201, 202–217 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  60. Rohde, H.-W.: Ein Kriterium für das Fehlen von Eigenwerten elliptischer Differentialoperatoren. Math. Z. 112, 375–388 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  61. Rohde, H.-W.: Die Weyl-Stonesche Theorie für Systeme gewöhnlicher Differentialoperatoren. RWTH Aachen 1969 (unveröffentlicht)

    Google Scholar 

  62. Rohde, H.-W., und Wienholtz, E.: Ein Regularitätssatz zur schwachen 2. Randbedingung mit Anwendungen auf elliptische Differentialoperatoren. Math. Z. 98, 9–26 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  63. Schmincke, U.-W.: Über das Verhalten der Eigenfunktionen eines singulären elliptischen Differentialoperators. Math. Z. 111, 267–288 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  64. Schmincke, U.-W.: Über die Potenzräume eines Schrödinger-Operators. Math. Z. 114, 349–360 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  65. Schmincke, U.-W.: Essential selfadjointness of a Schrödinger operator with strongly singular potential. Math. Z. 124, 47–50 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  66. Schmincke, U.-W.: Essential selfadjointness of Dirac operators with a strongly singular potential. Math. Z. 126, 71–81 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  67. Schmincke, U.-W.: Distinguished selfadjoint extensions of Dirac operators. Math. Z. 129, 335–349 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  68. Schmincke, U.-W.: A spectral gap theorem for Dirac operators with central field. Math. Z. 131, 351–356 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  69. Shortley, G.H.: The inverse-cube central force field in quantum mechanics. Phys. Rev. 38, 120–127 (1931)

    Article  MATH  Google Scholar 

  70. Simader, C.G.: Bemerkungen über Schrödinger-Operatoren mit stark singulären Potentialen. Math. Z. (erscheint demnächst)

    Google Scholar 

  71. Simon, B.: Quantum mechanics for Hamiltonians defined as quadratic forms. Princeton University Press 1971

    Google Scholar 

  72. Simon, B.: Essential self-adjointness of Schrödinger operators with positive potentials. Math. Ann. 201, 211–220 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  73. Simon, B.: Essential self-adjointness of Schrödinger operators with singular potentials. Arch. Rational Mech. Anal. 52, 44–48 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  74. Sommerfeld, A.: Atombau und Spektrallinien. Wellenmechanischer Ergänzungsband zur 4. Aufl. Braunschweig: Vieweg 1929

    MATH  Google Scholar 

  75. Stetkær-Hansen, H.: A generalization of a theorem of Wienholtz concerning essential self-adjointness of singular elliptic operators. Math. Scand. 19, 108–112 (1966)

    MathSciNet  MATH  Google Scholar 

  76. Stone, M.H.: Linear transformations in Hilbert space and their applications to analysis. New York: Amer. Math. Soc. Colloq. Publ. 1932

    Book  MATH  Google Scholar 

  77. Stummel, F.: Singuläre elliptische Differentialoperatoren in Hilbertschen Räumen. Math. Ann. 132, 150–176 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  78. Triebel, H.: Erzeugung nuklearer lokalkonvexer Räume durch singuläre Differentialoperatoren zweiter Ordnung. Math. Ann. 174, 163–176 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  79. Walter, J.: Symmetrie elliptischer Differentialoperatoren I, II. Math. Z. 98, 401–406 (1967); 106, 149–152 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  80. Walter, J.: Note on a paper by Stetkær-Hansen concerning essential self-adjointness of Schrödinger operators. Math. Scand. 25, 94–96 (1969)

    MathSciNet  MATH  Google Scholar 

  81. Watanabe, S.: Knowing and guessing. New York: John Wiley 1969

    MATH  Google Scholar 

  82. Weidmann, J.: Oszillationsmethoden für Systeme gewöhnlicher Differentialgleichungen. Math. Z. 119, 349–373 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  83. Wienholtz, E.: Halbbeschränkte partielle Differentialoperatoren zweiter Ordnung vom elliptischen Typus. Math. Ann. 135, 50–80 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  84. Wienholtz, E.: Bemerkungen über elliptische Differentialoperatoren. Arch. Math. 10, 126–133 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  85. Wightman, A.S.: The problem of existence of solutions in quantum field theory. Proc. 5th Annual Eastern Theor. Phys. Conference 1966. New York: Benjamin 1967

    Google Scholar 

  86. Witte, J.: Über die Regularität der Spektralschar eines singulären elliptischen Differentialoperators. Math. Z. 107, 116–126 (1968)

    Article  MATH  Google Scholar 

  87. Witte, J.: Über das Verhalten der Spektralschar eines elliptischen Differentialoperators in der Umgebung der Singularität des Potentials q(x) = |x|−α. Math. Z. 115, 140–152 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  88. Witte, J.: Über Regularitätseigenschaften der Potenzräume eines Schrödinger-Operators mit singulärem Potential. Math. Z. 128, 199–205 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  89. Wüst, R.: Generalizations of Rellich’s theorem on perturbation of (essentially) self-adjoint operators. Math. Z. 119, 276–280 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  90. Wüst, R.: A convergence theorem for selfadjoint operators applicable to Dirac operators with cutoff potentials. Math. Z. 131, 339–349 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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Authors
  1. H. Kalf
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  2. U.-W. Schmincke
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  3. J. Walter
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  4. R. Wüst
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William N. Everitt

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Kalf, H., Schmincke, UW., Walter, J., Wüst, R. (1975). On the spectral theory of schrödinger and dirac operators with strongly singular potentials. In: Everitt, W.N. (eds) Spectral Theory and Differential Equations. Lecture Notes in Mathematics, vol 448. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067087

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