Skip to main content
SpringerLink
Log in

Representations of groups in solution spaces of invariant differential equations

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

Abstract

We present different versions of the definition of invariant differential equations and introduce the notion of a spectrum for these equations. We give examples of an explicit description of the spectrum and introduce sufficient conditions for discreteness of the spectrum. As an example, we consider the representations of the Heisenberg group in solution spaces of invariant differential equations.

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. D. P. Zhelobenko, Compact Lie Groups and Their Representations [in Russian], Nauka, Moscow (1973); English transl., Amer. Math. Soc., Providence, R. I. (1973).

    Google Scholar 

  2. B. Kostant, “Verma modules and the existence of quasi-invariant differential operators,” in: Non-Commutative Harmonic Analysis (Lect. Notes Math., Vol. 466, J. Carmona, J. Dixmier, and M. Vergne, eds.), Springer, Berlin (1975), pp. 101–128.

    Google Scholar 

  3. V. I. Fushchich and A. G. Nikitin, Symmetries of Equations of Quantum Mechanics [in Russian], Nauka, Moscow (1990); English transl., Allerton, New York (1994).

    Google Scholar 

  4. B. Speh, J. Funct. Anal., 33, 95–118 (1979).

    Google Scholar 

  5. H. P. Yakobsen, Publ. RIMS Kyoto Univ., 22, 345–364 (1986).

    Google Scholar 

  6. V. F. Molchanov, Math. USSR-Sb., 10, 333–347 (1970).

    Google Scholar 

  7. V. F. Molchanov, “Maximal degenerate series representations of the universal covering of the group SU(n, n),” in: Lie Groups and Lie Algebras: Their Representations, Generalizations, and Applications (Math. Appl. Dordr., Vol. 433, B. P. Komrakov, J. S. Krasil’shchik, G. L. Litvinov, and A. B. Sossinsky, eds.), Kluwer, Dordrecht (1998), pp. 313–336.

    Google Scholar 

  8. H. P. Yakobsen and M. Vergne, J. Funct. Anal., 24, 52–106 (1977).

    Google Scholar 

  9. I. M. Gelfand, M. I. Graev, and N. Y. Vilenkin, Integral Geometry and Related Questions in Representation Theory [in Russian] (Vol. 5 of Generalized Functions and Operations on Them by I. M. Gelfand and G. E. Shilov), Fizmatgiz, Moscow (1959); English transl.: Integral Geometry and Representation Theory (Vol. 5 of Generalized Functions by I. M. Gelfand and G. E. Shilov), Acad. Press, New York (1966).

    Google Scholar 

  10. Yu. V. Egorov, “Microlocal analysis,” in: Differential Equations with partial derivatives-4 [in Russian] (Sovrem. Probl. Mat. Fund. Naprav., Vol. 33, Yu. V. Egorov and M. A. Shubin, eds., R. V. Gamkrelidze, series ed.), VINITI, Moscow (1988), pp. 5–156; English transl. in: Partial Differential Equations IV: Microlocal Analysis and Hyperbolic Equations (Encycl. Math. Sci., Vol. 33, Yu. V. Egorov and M. A. Shubin, eds., R. V. Gamkrelidze, series ed.), Springer, Berlin (1993), pp. 1–147.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bauman Moscow State Technical University, Moscow, Russia

    R. S. Ismagilov

Authors
  1. R. S. Ismagilov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 3, pp. 489–499, March, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ismagilov, R.S. Representations of groups in solution spaces of invariant differential equations. Theor Math Phys 142, 412–421 (2005). https://doi.org/10.1007/s11232-005-0032-y

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-005-0032-y

Keywords

Search

Navigation