Journal d’Analyse Mathématique

, Volume 37, Issue 1, pp 46–99 | Cite as

Self adjoint extensions satisfying the Weyl operator commutation relations

  • Palle T. JØrgensen
  • Paul S. Muhly
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. I. Akhiezer and I. M. Glazman,Theory of Linear Operators in Hilbert Space, Vol. II, Frederick Ungar, New York, 1969.Google Scholar
  2. 2.
    Wm. B. Arveson,On groups of automorphisms of operator algebras, J. Functional Analysis15 (1974), 217–243.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    R. Beals,Topics in Operator Theory, The University of Chicago Press, Chicago, 1971.MATHGoogle Scholar
  4. 4.
    P. Cartier,Quantum mechanical commutation relations and theta functions, Proc. Symp. Pure Math. No. 9, Amer. Math. Soc., 1966, pp. 361–383.Google Scholar
  5. 5.
    J. Dixmier,Sur la relation i(PQ-QP = I, Compositio Math.13 (1958), 263–270.MATHMathSciNetGoogle Scholar
  6. 6.
    J. Dixmier,Les Algèbres d’Opérateurs dans l’Espace Hilbertien, 2i⨼e éd., Gauthier-Villars, Paris, 1969.MATHGoogle Scholar
  7. 7.
    R. G. Douglas,Structure theory for operators I, J. Reine Angew. Math.232 (1968), 180–193.MATHMathSciNetGoogle Scholar
  8. 8.
    N. Dunford and J. T. Schwartz,Linear Operators, Vol. II, Interscience Publ. (J. Wiley and Sons), New York, 1963.MATHGoogle Scholar
  9. 9.
    C. Foias,On the Lax-Phillips nonconservative scattering theory, J. Functional Analysis19 (1975), 273–301.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    C. Foias and L. Geher,über die Weylsche Vertauschungs relationen, Acta Sci. Math. (Szeged)24 (1963), 97–102.MATHMathSciNetGoogle Scholar
  11. 11.
    B. Fuglede,On the relation PQ-OP = -iI, Math. Scand.20 (1967), 79–88.MATHMathSciNetGoogle Scholar
  12. 12.
    L. Gårding,Note on continuous representations of Lie groups, Proc. Nat. Acad. Sci. U.S.A.33 (1947), 331–332.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    H. Helson,Lectures on Invariant Subspaces, Academic Press, New York, 1964.MATHGoogle Scholar
  14. 14.
    J. W. Helton,Unitary operators on a space with an indefinite inner product, J. Functional Analysis6 (1970), 412–440.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    E. Hille and R. S. Phillips,Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ.31, Providence, 1957.Google Scholar
  16. 16.
    P. T. JØrgensen,Perturbations of homogeneous Lebesgue spectrum, preprint, Stanford University. 1978.Google Scholar
  17. 17.
    P. T. JØrgensen and R. T. Moore,Commutation relations for operators, semigroups and resolvents in mathematical physics and group representations, preprint, 650 pp., submitted.Google Scholar
  18. 18.
    T. Kato,On the commutation relation AB-BA = c, Arch. Rational Mech. Anal.10 (1962), 273–275.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    T. Kato,Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.MATHGoogle Scholar
  20. 20.
    P. D. Lax and R. S. Phillips,Scattering Theory, Academic Press, New York, 1967.MATHGoogle Scholar
  21. 21.
    P. D. Lax and R. S. Phillips,Scattering theory for dissipative hyperbolic systems, J. Functional Analysis14 (1973), 172–235.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    G. W. Mackey,The theory of unitary group representations, The University of Chicago Press, Chicago, 1976.MATHGoogle Scholar
  23. 23.
    M. A. Naimark,On commuting unitary operators in spaces with indefinite metric, Acta Sci. Math. (Szeged)24 (1963), 177–189.MATHMathSciNetGoogle Scholar
  24. 24.
    E. Nelson,Analytic vectors, Ann. Math.70 (1959), 572–615.CrossRefGoogle Scholar
  25. 25.
    J. von Neumann,Mathematische Begründung der Quantenmechanik, Nachr. Gesell. Wiss. Göttingen. Math-Phys. KI. (1927), 1–57.Google Scholar
  26. 26.
    J. von Neumann,Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann.102 (1929-30), 49–131.CrossRefGoogle Scholar
  27. 27.
    J. von Neumann,Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann.104 (1931), 570–578.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    B. Oersted,Induced representations and a new proof of the imprimitivity theorem, J. Functional Analysis31 (1979), 355–359.CrossRefGoogle Scholar
  29. 29.
    R. S. Phillips,On a theorem due to Sz.-Nagy, Pacific J. Math.9 (1959), 169–173.MATHMathSciNetGoogle Scholar
  30. 30.
    R. S. Phillips,The extension of dual subspaces invariant under an algebra, Proc. Internat. Symp. on Linear Spaces, Israel, 1960, pp. 366–398.Google Scholar
  31. 31.
    R. S. Phillips,On dissipative operators, InLecture Series in Differential Equations (A. K. Aziz, ed.), Van Nostrand Math. Studies No. 19, New York, 1969.Google Scholar
  32. 32.
    N S. Poulsen,On the canonical commutation relations, Math. Scand.32 (1973), 112–122.MATHMathSciNetGoogle Scholar
  33. 33.
    C. R. Putnam,Commutation Properties of Hilbert Space Operator, Ergebnisse Math. Grenzgebiete, N. F. 36, Springer, Berlin, 1967.Google Scholar
  34. 34.
    F. Rellich,Der Eindeutigkeitssatz für die Lösungen der quantenmechanischen Vertauschungsrelationen, Nachr. Akad. Wiss. Göttingen, Math.-Phys. K1. 11A (1946), 107–115.MathSciNetGoogle Scholar
  35. 35.
    W. Rudin,Functional Analysis, McGraw Hill, New York, 1973.MATHGoogle Scholar
  36. 36.
    D. Sarason,On spectral sets having connected complement, Acta Sci. Math. (Szeged)26 (1965), 289–300.MATHMathSciNetGoogle Scholar
  37. 37.
    I. E. Segal,Foundations of the theory of dynamical systems of infinitely many degrees of freedom, I. Mat. Fys. Medd. Danske Vid. Selsk.31, No. 12 (1959), 39 pp.Google Scholar
  38. 38.
    Ja. G. Sinai,Dynamical systems with countable Lebesgue spectrum I, Izv. Akad. Nauk SSSR25 (1961), 899–924.MathSciNetGoogle Scholar
  39. 39.
    I. M. Singer,Lie Algebras of Unbounded Operatöts, thesis, The University of Chicago, 1950 (unpublished).Google Scholar
  40. 40.
    S. P. Slinker,On commuting self-adjoint extensions of unbounded operators, to appear in Indiana Univ. Math. J.Google Scholar
  41. 41.
    M. H. Stone,Linear transformations in Hilbert space and their applications to analysis, Amer. Math. Soc. Colloq. Publ., Vol. 25, Providence, R.I., 1932.Google Scholar
  42. 42.
    B. Sz.- Nagy,Extensions of Linear Transformations in Hilbert Space which Extend Beyond this Space, F. Ungar, New York, 1960.Google Scholar
  43. 43.
    B. Sz.- Nagy and C. Foias,Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing Co., Amsterdam, 1970.Google Scholar
  44. 44.
    H. Weyl,The Theory of Groups and Quantum Mechanics, Dover Publ., New York, 1950.Google Scholar

Copyright information

© Hebrew University of Jerusalem 1980

Authors and Affiliations

  • Palle T. JØrgensen
    • 1
    • 2
    • 3
  • Paul S. Muhly
    • 1
    • 2
    • 3
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Mathematics InstituteAarhus UniversityAarhusDenmark
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations