Skip to main content
SpringerLink
Log in

Dissipative and nonunitary solutions of operator commutation relations

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the (generalized) semi-Weyl commutation relations UgAU* g = g(A) on Dom(A), where A is a densely defined operator and G ∋ g ↦ Ug is a unitary representation of the subgroup G of the affine group G, the group of affine orientation-preserving transformations of the real axis. If A is a symmetric operator, then the group G induces an action/flow on the operator unit ball of contracting transformations from Ker(A* - iI) to Ker(A* + iI). We establish several fixed-point theorems for this flow. In the case of one-parameter continuous subgroups of linear transformations, self-adjoint (maximal dissipative) operators associated with the fixed points of the flow yield solutions of the (restricted) generalized Weyl commutation relations. We show that in the dissipative setting, the restricted Weyl relations admit a variety of representations that are not unitarily equivalent. For deficiency indices (1, 1), the basic results can be strengthened and set in a separate case.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Intersci. Monogr. Texts Phys. Astron., Vol. 26), Wiley, New York (1972).

  2. P. D. Lax and R. S. Phillips, Scattering Theory (Pure Appl. Math., Vol. 26), Acad. Press, Boston, Mass. (1989).

  3. G. Weyl, Z. Phys., 46, 1–46 (1928).

    Article  ADS  Google Scholar 

  4. J. von Neumann, Math. Ann., 104, 570–578 (1931).

    Article  MathSciNet  Google Scholar 

  5. M. Stone, Ann. Math., 33, 643–648 (1932).

    Article  Google Scholar 

  6. V. A. Zolotarev, Russian Acad. Sci. Sb. Math., 76, 99–122 (1993).

    MathSciNet  Google Scholar 

  7. V. K. Dubovoi, “The Weyl family of operator colligations and the corresponding open fields [in Russian],” in: Theory of Functions, Functional Analysis, and Their Applications, Vol. 14, Kharkov Univ. Press, Kharkov (1971), pp. 67–83.

    MathSciNet  Google Scholar 

  8. M. S. Livshits and A. A. Yantsevich, Theory of Operator Colligations in Hilbert Spaces [in Russian], Kharkov Univ. Press, Kharkov (1971).

    MATH  Google Scholar 

  9. A. P. Filimonov and È. R. Tsekanovskii, Funct. Anal. Appl., 21, 343–344 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  10. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics [in Russian], Vol. 3, Quantum Mechanics: Non-Relativistic Theory, Fizmatlit, Moscow (1989); English transl. prev. ed., Pergamon, London (1965).

  11. V. Efimov, Phys. Lett. B, 33, 563–564 (1970).

    Article  ADS  Google Scholar 

  12. S. P. Merkuriev and L. D. Faddeev, Quantum Scattering Theory for Several Particle Systems [in Russian], Nauka, Moscow (1985); English transl.: L. D. Faddeev and S. P. Merkuriev (Math. Phys. Appl. Math., Vol. 11), Kluwer, Dordrecht (1993).

  13. R. A. Minlos and L. D. Faddeev, Soviet Phys. Dokl., 6, 1072–1074 (1962).

    ADS  MathSciNet  Google Scholar 

  14. K. A. Makarov and V. V. Melezhik, Theor. Math. Phys., 107, 755–769 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Lang, SL2(ℝ) (Grad. Texts Math., Vol. 105), Springer, New York (1985).

  16. G. W. Mackey, Duke Math. J., 16, 313–326 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  17. M. S. Brodskii and M. S. Livšic, Doklady Akad. Nauk SSSR, n.s., 68, 213–216 (1949).

    MathSciNet  Google Scholar 

  18. P. E. T. Jorgensen, Bull. Amer. Math. Soc., 1, 266–269 (1979).

    Article  MathSciNet  Google Scholar 

  19. P. E. T. Jorgensen, Math. Z., 169, 41–62 (1979).

    Article  MathSciNet  Google Scholar 

  20. P. E. T. Jorgensen and P. S. Muhly, J. Analyse Math., 37, 46–99 (1980).

    Article  MathSciNet  Google Scholar 

  21. P. E. T. Jorgensen, J. Math. Anal. Appl., 73, 115–133 (1980).

    Article  MathSciNet  Google Scholar 

  22. P. E. T. Jorgensen, Amer. J. Math., 103, 273–287 (1981).

    Article  MathSciNet  Google Scholar 

  23. P. E. T. Jorgensen and R. T. Moore, Operator Commutation Relations (Math. Its Appl., Vol. 14), D. Reidel, Dordrecht (1984).

  24. K. Schmüdgen, J. Funct. Anal., 50, 8–49 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  25. K. Schmüdgen, Publ. Res. Inst. Math. Sci., 19, 601–671 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  26. S. G. Krein, Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1967); English transl., Amer. Math. Soc., Providence, R. I. (1971).

    Google Scholar 

  27. È. R. Tsekanovskii and Yu. L. Shmul’yan, Russian Math. Surv., 32, 73–131 (1977).

    Article  ADS  Google Scholar 

  28. M. S. Livšic, Amer. Math. Soc. Transl. (2), 13, 61–83 (1960).

    MathSciNet  Google Scholar 

  29. Yu. Arlinskii, S. Belyi, and E. Tsekanovskii, Conservative Realizations of Herglotz–Nevanlinna Functions (Oper. Theory Adv. Appl., Vol. 217), Birkhäuser, Basel (2011).

  30. K. A. Makarov and E. Tsekanovskii, Methods Func. Anal. Topology, 13, 181–186 (2007).

    MathSciNet  MATH  Google Scholar 

  31. M. G. Krein, Rec. Math. [Mat. Sbornik], n.s., 20(62), 431–495 (1947).

    MathSciNet  Google Scholar 

  32. T. Ando and K. Nishio, Tôhoku Math. J., 22, 65–75 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  33. L. Lusternik and V. Sobolev, Elements of Functional Analysis [in Russian], Nauka, Moscow (1965); English transl., Wiley, New York (1974).

    Google Scholar 

  34. J. Vick, Homology Theory: An Introduction to Algebraic Topology (Grad. Texts Math., Vol. 145), Springer, Berlin (1994).

  35. L. E. J. Brouwer, Mathematische Annalen, 71, 305–315 (1912).

    Article  MATH  Google Scholar 

  36. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space [in Russian], Nauka, Moscow (1966); English transl., Dover, New York (1993).

    Google Scholar 

  37. M. Reed and B. Simon, Methods of Modern Mathetical Physics, Vol. 2, Fourier Analysis, Self-Adjointness, Acad. Press, New York (1975).

  38. A. M. Perelomov and Y. B. Zel’dovich, QuantumMechanics: Selected Topics, World Scientific, Singapore (1998).

    Book  Google Scholar 

  39. A. M. Perelomov and V. S. Popov, Theor. Math. Phys., 4, 664–677 (1970).

    Article  Google Scholar 

  40. W. N. Everitt and H. Kalf, J. Comp. Appl. Math., 208, 3–19 (2007).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. F. Gesztesy and L. Pittner, Acta Phys. Austriaca, 51, 259–268 (1979).

    MathSciNet  Google Scholar 

  42. H. Kalf, J. London Math. Soc., 17, 511–521 (1978).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, Missouri, USA

    K. A. Makarov

  2. Department of Mathematics, Niagara University, Lewiston, New York, USA

    E. Tsekanovskii

Authors
  1. K. A. Makarov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Tsekanovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. A. Makarov.

Additional information

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 186, No. 1, pp. 51–75, January, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Makarov, K.A., Tsekanovskii, E. Dissipative and nonunitary solutions of operator commutation relations. Theor Math Phys 186, 41–60 (2016). https://doi.org/10.1134/S0040577916010049

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577916010049

Keywords

Search

Navigation