Skip to main content
SpringerLink
Log in

Algebras with general commutation relations and their applications. I. Pseudodifferential equations with increasing coefficients

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

Operators of the regular representation for commutation relations of general form are constructed. On the basis of the technique developed, conditions for the existence of a mixed (with respect to smoothness and rate of growth at infinity) asymptotic expression for a solution of a pseudodifferential equation with increasing coefficients are obtained. The work contains a survey of the general theory of quasiinversion of functions of ordered operators.

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

Literature cited

  1. V. I. Arnold, Ergodic Problems of Classical Mechanics, W. A. Benjamin (1968).

  2. V. I. Arnold, “On the characteristic class contained in the quantization conditions,” Funkts. Anal. Prilozhen., No. 1, 1–14 (1967).

    Google Scholar 

  3. F. A. Berezin, The Method of Second Quantization [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  4. F. A. Berezin, “Wick and anti-Wick symbols of operators,” Mat. Sb.,86, No. 4, 578–610 (1971).

    Google Scholar 

  5. F. A. Berezin, “Quantization,” Izv. Akad. Nauk SSSR, Ser. Mat.,38, No. 5, 1116–1176 (1974).

    Google Scholar 

  6. F. A. Berezin and V. G. Kats, “Lie groups with commuting and anticommuting parameters,” Mat. Sb.,82, No. 3, 343–359 (1970).

    Google Scholar 

  7. V. S. Buslaev, “Quantization and the WKB method,” Tr. Mat. Inst. Akad. Nauk SSSR,110, 5–28 (1970).

    Google Scholar 

  8. M. I. Vishik and S. L. Sobolev, “A general formulation of some boundary value problems for elliptic partial differential equations,” Dokl. Akad. Nauk. SSSR,111, No. 3, 521–523 (1956).

    Google Scholar 

  9. K. Godiion, Differential Geometry and Analytic Mechanics [Russian translation], Mir, Moscow (1973).

    Google Scholar 

  10. V. V. Grushin, “On a class of hypoelliptic operators,” Mat. Sb.,83, No. 3, 456–473 (1970).

    Google Scholar 

  11. N. Dunford and J. T. Schwartz, Linear Operators. Spectral Theory, Self Adjoint Operators in Hilbert Space, Wiley (1961).

  12. M. V. Karasev andV. E. Nazaikinskii, “On the quantization of rapidly oscillating symbols,” Mat. Sb.,106, No. 2, 183–213 (1978).

    Google Scholar 

  13. M. V. Karasev, “Expansion of functions of noncommuting operators,” Dokl. Akad. Nauk SSSR,214, No. 6, 1254–1257 (1974).

    Google Scholar 

  14. M. V. Karasev, “Some formulas for functions of noncommuting operators,” Mat. Zametki,18, No. 2, 267–277 (1975).

    Google Scholar 

  15. A. A. Kirillov, Elements of the Theory of Representations [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  16. B. Kostant, “Quantization and unitary representations,” Usp. Mat. Nauk,28, No. 1, 163–225 (1973).

    Google Scholar 

  17. V. V. Kucherenko, “Asymptotic solutions of the Cauchy problem for equations with complex characteristics,” in: Sovrem. Probl. Mat., Vol. 8 (Itogi Nauki i Tekh. VINITI AN SSSR), Moscow (1977).

  18. V. V. Kucherenko, “Asymptotic solutions of equations with complex characteristics,” Mat. Sb.,95, No. 2, 163–213 (1974).

    Google Scholar 

  19. V. V. Kucherenko, “Asymptotic solutions of the system A(x-ih∂/∂x) as h→0 in the case of characteristics of variable multiplicity,” Izv. Akad. Nauk SSSR, Ser. Mat.,38, No. 3, 625–662 (1974).

    Google Scholar 

  20. B. M. Levitan, The Theory of Generalized Shift Operators [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  21. V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  22. V. P. Maslov, “Application of the method of ordered operators for obtaining exact solutions,” Teor. Mat. Fiz.,33, No. 2, 185–209 (1977).

    Google Scholar 

  23. V. P. Maslov, Perturbation Theory and Asymptotic Methods [in Russian], Moscow State Univ. (1965).

  24. V. P. Maslov and M. V. Karasev, J. Sov. Math.,15, No. 3 (1981).

  25. V. P. Maslov and M. V. Fedoryuk, The Quasiclassical Approximation for the Equations of Quantum Mechanics [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  26. A. S. Mishchenko, B. Yu. Sternin, and V. E. Shatalov, Lagrangian Manifolds and the Method of the Canonical Operator [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  27. A. S. Mishchenko, B. Yu. Sternin, and V. E. Shatalov, The Canonical Operator of Maslov. The Complex Theory [in Russian], Izd. MIÉM (1974).

  28. M. V. Mosolova, “A new formula for In eBeA in terms of commutators of the elements A and B,” Mat. Zametki,23, No. 6, 817–824 (1978).

    Google Scholar 

  29. V. E. Nazaikinskii, V. G. Oshmyan, B. Yu. Sternin, and V. E. Shatalov, “Fourier integral operators and the canonical operator,” Usp. Mat. Nauk (1979).

  30. M. A. Naimark, The Theory of Group Representations [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  31. W. Rudin, Functional Analysis, McGraw-Hill (1973).

  32. S. L. Sobolev, “A new method for solving the Cauchy problem for partial differential equations of normal hyperbolic type,” Mat. Sb.,1 (43), No. 1, 39 (1936).

    Google Scholar 

  33. B. Yu. Sternin and V. E. Shatalov, “On the propagation of electromagnetic waves in the short-wave range,” VINITI, No. 3333-78 DEP (1978).

  34. L. Hörmander, “On the existence and regularity of solutions of linear pseudodifferential equations,” Usp. Mat. Nauk,28, No. 6, 109–164 (1973).

    Google Scholar 

  35. M. A. Shubin, Pseudodifferential Operators and Spectral Theory [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  36. R. F. V. Anderson, “The Weyl functional calculus,” J. Funct. Anal.,4, No. 2, 240–267 (1969).

    Google Scholar 

  37. C. Chevalley, Fundamental Concepts of Algebra, New York (1956).

  38. C. H. Cook and H. R. Fisher, “Uniform convergence structures,” Math. Ann.,173, 290–306 (1967).

    Google Scholar 

  39. L. Corwin, Y. Neeman, and S. Sternberg, “Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry)” Rev. Mod. Phys.,47, No. 3, 573–603 (1975).

    Google Scholar 

  40. J. Duistermaat and L. Hormander, “Fourier integral operators. II,” Acta Math.,128, No.3–4, 183–269 (1972).

    Google Scholar 

  41. R. Feynman, “An operator calculus having applications in quantum electrodynamics,” Phys. Rev.,84, No. 2, 108–128 (1951).

    Google Scholar 

  42. H. R. Ficher, “Limesräume,” Math. Ann.,137, 269–303 (1959).

    Google Scholar 

  43. L. Hörmander, “Fourier integral operators, I,” Acta Math.,127, No. 1–2 (1971).

  44. B. Kostant, “Graded manifolds, graded Lie theory and prequantization,” Preprint, New York (1977).

  45. J. Leray, “Probleme de Cauchy. I–IV,” Bull. Soc. Math. France,85 (1957);86 (1958); 87 (1958);90, (1962).

  46. V. P. Maslov, Operational Methods [Russian translation], Mir, Moscow (1976).

    Google Scholar 

  47. V. P. Maslov, “The characteristics of pseudodifferential operators and difference schemes,” Actes Congres Int. Math. Nice, Vol. 2 (1970), pp. 755–769.

    Google Scholar 

  48. A. Melin and J. Sjöstrand, “Fourier integral operators with complex-valued phase functions,” Springer Lecture Notes,459, 120–233 (1975).

    Google Scholar 

  49. A. Melin and J. Sjostrand, “Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem,” Commun. Partial Diff. Eqs.1 (4), 313–400 (1976).

    Google Scholar 

  50. L. P. Rotschild and E. M. Stein, “Hypoelliptic differential operators and nilpotent groups,” Acta Math.,137, 3–4, 247–320 (1976).

    Google Scholar 

  51. A. Yoshikawa, “On Maslov's canonical operator,” Hokkaido Math. J.,4, No. 1, 8–38 (1975).

    Google Scholar 

Download references

Authors
  1. V. P. Maslov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. E. Nazaikinskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Itogi Nauk i Tekhniki, Sovremennye Problemy Matematiki, Vol. 13, pp. 5–144, 1979.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maslov, V.P., Nazaikinskii, V.E. Algebras with general commutation relations and their applications. I. Pseudodifferential equations with increasing coefficients. J Math Sci 15, 167–273 (1981). https://doi.org/10.1007/BF01083678

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01083678

Keywords

Search

Navigation