Skip to main content
SpringerLink
Log in

Spectral theory of a string

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

In this survey, we present the principal results of Krein's spectral theory of a string and describe its development by other authors.

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.

Similar content being viewed by others

References

  1. M. S. Birman and V. V. Borzov, “On the asymptotics of discrete spectra of some singular differential operators,”Probl. Mat. Fiz., Issue 5, 24–38 (1971).

    Google Scholar 

  2. V. V. Borzov, “On the quantitative characteristics of singular measures,”Probl. Mat. Fiz., Issue 4, 42–47 (1970).

    Google Scholar 

  3. L. De Branges,Hilbert Spaces of Entire Functions, Prentice-Hall, London (1968).

    Google Scholar 

  4. S. Watanabe, “On time inversion of one-dimensional diffusion processes,”Z. Wahrscheinlichkeitstheor. verw. Geb.,31, 115–124 (1975).

    Google Scholar 

  5. F. R. Gantmakher and M. G. Krein,Oscillation Matrices and Kernels. Small Oscillations of Mechanical Systems [in Russian], Gostekhteorizdat, Moscow-Leningrad (1950).

    Google Scholar 

  6. I. Ts. Gokhberg and M. G. Krein,Theory of Volterra Operators in Hilbert Spaces and Its Applications [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  7. H. Dym and H. P. McKean,Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Academic Press, New York (1976).

    Google Scholar 

  8. E. B. Dynkin,Markov Processes [in Russian], Fizmatgiz, Moscow (1963).

    Google Scholar 

  9. K. Itô,Probability Processes [Russian translation], Issue 2, Inostr. Lit., Moscow (1963).

    Google Scholar 

  10. K. Itô and H. P. McKean,Diffusion Processes and Their Sample Paths, Springer-Verlag, Berlin-Heidelberg-New York (1965).

    Google Scholar 

  11. Y. Kasahara, “Spectral theory of generalized second order differential operators and its applications to Markov processes,”Jpn. J. Math.,1, No. 1, 67–84 (1975).

    Google Scholar 

  12. Y. Kasahara, “Limit theorems of occupation times for Markov processes,”Publ. RIMS, Kyoto Univ.,12, 801–818 (1977).

    Google Scholar 

  13. Y. Kasahara, S. Kotani, and H. Watanabe, “On the Green functions of 1-dimensional diffusion processes,”Publ. RIMS, Kyoto Univ.,16, 175–188 (1980).

    Google Scholar 

  14. I. S. Kats, “On the existence of spectral functions of singular second-order differential systems,”Dokl. Akad. Nauk SSSR,106, No. 1, 15–18 (1956).

    Google Scholar 

  15. I. S. Kats, “On the behavior of spectral functions of second-order differential systems,”Dokl. Akad. Nauk SSSR,106, No. 2, 183–186 (1956).

    Google Scholar 

  16. I. S. Kats, “General theorems on the behavior of spectral functions of second-order differential systems,”Dokl. Akad. Nauk SSSR,122, No. 6, 974–977 (1958).

    Google Scholar 

  17. I.S. Kats, “On the denseness of the spectrum of a string,”Dokl. Akad. Nauk SSSR,126, No. 6, 1180–1182 (1959).

    Google Scholar 

  18. I. S. Kats, “On the type of the spectrum of a singular string,”Izv. Vyssh. Uchebn. Zaved., Mat., No. 1 (26), 57–64 (1962).

    Google Scholar 

  19. I. S. Kats, “Two general theorems on the asymptotic behavior of spectral functions of second-order differential systems,”Izv. Akad. Nauk SSSR, Ser. Mat.,26, No. 1, 53–78 (1962).

    Google Scholar 

  20. I. S. Kats, “On the multiplicity of the spectrum of a second-order differential operator,”Dokl. Akad. Nauk SSSR,145, No. 3, 510–513 (1962).

    Google Scholar 

  21. I. S. Kats, “Multiplicity of the spectrum of a-second-order differential operator and expansion in eigenfunctions,”Izv. Akad. Nauk SSSR, Ser. Mat.,27, No. 5, 1081–1112 (1963).

    Google Scholar 

  22. I. S. Kats, “On the behavior of spectral functions of second-order differential systems with a boundary condition at a singular end,”Dokl. Akad. Nauk SSSR,157, No. 1, 34–37 (1964).

    Google Scholar 

  23. I. S. Kats, “Existence of spectral functions of generalized second-order differential systems with boundary conditions at a singular end,”Mat. Sb.,68 (110), No. 2, 174–227 (1965).

    Google Scholar 

  24. I. S. Kats, “Some cases of the uniqueness of a solution of the inverse problem for strings with a boundary condition at a singular end,”Dokl. Akad. Nauk SSSR,164, No. 5, 975–978 (1965).

    Google Scholar 

  25. I. S. Kats, “Integral characteristics of the growth of spectral functions of the generalized second-order boundary-value problems with boundary conditions at a regular end,”Izv. Akad. Nauk. SSSR, Ser. Mat.,35, No. 1, 175–205 (1971).

    Google Scholar 

  26. I. S. Kats, “Power asymptotics of the spectral functions of generalized second-order boundary-value problems,”Dokl. Akad. Nauk SSSR,203, No. 4, 752–755 (1972).

    Google Scholar 

  27. I. S. Kats, “Generalization of the Marchenko asymptotic formula for spectral functions of a second-order boundary-value problem,”Izv. Akad. Nauk SSSR, Ser. Mat.,37, No. 2, 422–436 (1973).

    Google Scholar 

  28. I. S. Kats, “Denseness of the spectrum of a string,”Dokl. Akad. Nauk SSSR,221, No. 3, 16–19 (1973).

    Google Scholar 

  29. I. S. Kats, “Description of the set of spectral functions of a regular string bearing a concentrated mass at its end which is free of boundary conditions,”Izv. Vyssh. Uchebn. Zaved., Mat., No. 7 (146), 27–33 (1974).

    Google Scholar 

  30. I. S. Kats, “Some general theorems on the denseness of the spectrum of a string,”Dokl. Akad. Nauk SSSR, 238, No. 4, 785–788 (1978).

    Google Scholar 

  31. I. S. Kats, “Integral estimates for the growth of spectral functions of a string,”Ukr. Mat. Th.,34, No. 3, 296–302 (1982).

    Google Scholar 

  32. I. S. Kats, “Integral estimates for the distribution of the spectrum of a string,”Sib. Mat. Zh.,27, No. 2, 62–74 (1986).

    Google Scholar 

  33. I. S. Kats, “Denseness of the spectrum of a singular string,”Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 23–30 (1990).

    Google Scholar 

  34. I. S. Kats and M. G. Krein, “A criterion of the discreteness of the spectrum of a singular string,”Izv. Vyssh. Uchebn. Zaved., Mat., No. 2 (3), 136–153 (1958).

    Google Scholar 

  35. I. S. Kats and M. G. Krein, “R-functions are analytic functions that map the upper half plane onto itself,” Appendix 1 in: F. Atkinson,Discrete and Continuous Boundary-Value Problems [Russian translation], Mir, Moscow (1968).

    Google Scholar 

  36. I. S. Kats and M. G. Krein, “On spectral functions of a string,” Appendix 2 in: F. Atkinson,Discrete and Continuous Boundary-Value Problems [Russian translation], Mir, Moscow (1968).

    Google Scholar 

  37. L. P. Klotz and H. Langer, “Generalized resolvents and spectral functions of a matrix generalization of the Krein — Feller second order derivative,”Math. Nachr.,100, 163–186 (1981).

    Google Scholar 

  38. S. Kotani, “On a generalized Sturm — Liouville operator with a singular boundary,”J. Math. Kyoto Univ.,15, No. 2, 423–454 (1975).

    Google Scholar 

  39. S. Kotani and S. Watanabe, “Krein's spectral theory of strings and generalized diffusion processes,” Springer Lecture Notes Math.,923, 235–259 (1982).

    Google Scholar 

  40. M. G. Krein, “On the representations of functions by Fourier — Stieltjes integrals,”Uchen. Zapiski Kuibyshev. Ped. Inst., Issue 7, 123–148 (1943).

  41. M. G. Krein, “On the problem of extension of spiral arcs in Hilbert spaces,”Dokl. Akad. Nauk SSSR,45, No. 4, 147–150 (1944).

    Google Scholar 

  42. M. G. Krein, “General method for the decomposition of positive definite kernels into elementary products,”Dokl. Akad. Nauk SSSR,53, No. 1, 3–6 (1946).

    Google Scholar 

  43. M. G. Krein, “To the theory of entire functions of exponential type,”Izv. Akad. Nauk SSSR, Ser. Mat.,11, 309–326 (1947).

    Google Scholar 

  44. M. G. Krein, “On Hermitian operators with guiding functionals,”Sb. Tr. Inst. Mat. Akad. Nauk Ukr. SSR,10, 83–106 (1948).

    Google Scholar 

  45. M. G. Krein, “Solution of the inverse Sturm — Liouville problem,”Dokl. Akad. Nauk SSSR,76, No. 1, 21–24 (1951).

    Google Scholar 

  46. M. G. Krein, “Determination of the density of an inhomogeneous symmetric string from the spectrum of its frequencies,”Dokl. Akad. Nauk SSSR,76, No. 3, 345–348 (1951).

    Google Scholar 

  47. M. G. Krein, “On certain problems of maximum and minimum for characteristic numbers and on the Lyapunov regions of stability,”Prikl. Mat. Mekh.,15, Issue 3, 323–348 (1951).

    Google Scholar 

  48. M. G. Krein, “On the indefinite case of the Sturm — Liouville boundary-value problem on the interval (0, •),”Izv. Akad. Nauk. SSSR, Ser. Mat.,16, No. 2, 293–324 (1952).

    Google Scholar 

  49. M. G. Krein, “On inverse problems for inhomogeneous strings,”Dokl. Akad. Nauk SSSR,82, No. 5, 669–672 (1952).

    Google Scholar 

  50. M. G. Krein, “On a generalization of Stieltjes results,”Dokl. Akad. Nauk SSSR,87, No. 6, 881–884 (1952).

    Google Scholar 

  51. M. G. Krein, “New problems in the theory of oscillations of Sturm systems,”Prikl. Mat. Mekh.,16, Issue 5, 555–568 (1952).

    Google Scholar 

  52. M. G. Krein, “On the transition function of a second-order one-dimensional boundary-value problem,”Dokl. Akad. Nauk SSSR,88, No. 3, 405–408 (1953).

    Google Scholar 

  53. M. G. Krein, “An analog of the Chebyshev — Markov inequality in a one-dimensional boundary-value problem,”Dokl. Akad. Nauk SSSR,89, No. 1, 5–8 (1953).

    Google Scholar 

  54. M. G. Krein, “Some cases of effective determination of the density of an inhomogeneous string from its spectral function,”Dokl. Akad. Nauk SSSR,93, No. 4, 617–620 (1953).

    Google Scholar 

  55. M. G. Krein, “On the inverse problems in filter theory and λ-domains of stability,”Dokl. Akad. Nauk SSSR,93, No. 5, 767–770 (1953).

    Google Scholar 

  56. M. G. Krein, “Principal approximation problem in the theory of extrapolation and filtration of stationary random processes,”Dokl. Akad. Nauk SSSR,94, No. 1, 13–16 (1954).

    Google Scholar 

  57. M. G. Krein, “An effective method for the solution of inverse boundary-value problems,”Dokl. Akad. Nauk SSSR,97, No. 6, 987–990 (1954).

    Google Scholar 

  58. M. G. Krein, “Integral equations that generate second-order differential equations,”Dokl. Akad. Nauk SSSR,97, No. 4, 21–24 (1954).

    Google Scholar 

  59. M. G. Krein, “Determination of the potential of a particle according to itsS-function,”Dokl. Akad. Nauk SSSR,105, No. 3, 433–436 (1955).

    Google Scholar 

  60. M. G. Krein, “Continual analogs of the assertions on polynomials orthogonal on the unit circle,”Dokl. Akad. Nauk SSSR,105, No. 4, 637–640 (1955).

    Google Scholar 

  61. M. G. Krein, “On the theory of accelerants andS-matrices of canonical differential systems,”Dokl. Akad. Nauk SSSR,111, No. 6, 1167–1170 (1956).

    Google Scholar 

  62. M. G. Krein, “Chebyshev-Markov inequalities in the theory of spectral functions of a string,”Mat. Issled.,5, No. 1, 77–101 (1970).

    Google Scholar 

  63. U. Küchler, “Some asymptotic properties of the transition densities of one-dimensional quasidiffusions,”Publ. RIMS, Kyoto Univ.,16, No. 1, 245–268 (1980).

    Google Scholar 

  64. H. Langer,Zur Spectraltheorie Verallgemeinerter Cewönnlicher Differenzialoperatoren Zweiter Ordung mit Einer Nichtmonotonen Gewichtfunktion, University of Jyväskylä, Department of Mathematics, Report 14 (1972).

  65. H. Langer, L. Partzsch, and D. Schütze, “Über verallgemeinerte gewönliche Differentialoperatoren mit nichtlokalen Randbedingungen und die von ihnen erzeugten Markov-Processe,”Publ. RIMS, Kyoto Univ.,7, No. 3, 660–702 (1972).

    Google Scholar 

  66. B. M. Levitan, “On the asymptotic behavior of the spectral function of a self-adjoint second-order differential equation,”Izv. Akad. Nauk SSSR, Ser. Mat.,16, 325–352 (1952).

    Google Scholar 

  67. V. A. Marchenko, “Some problems in the theory of one-dimensional second-order linear differential operators. I,”Tr. Mosk. Mat. Obshch.,1, 381–422 (1952).

    Google Scholar 

  68. H. P. McKean and D. B. Ray, “Spectral distribution of a differential operator,”Duke Math. J.,29, 281–292 (1962).

    Google Scholar 

  69. M. A. Naimark,Linear Differential Operators [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  70. M. Tomisaki, “On the asymptotic behavior of transition probability densities of one-dimensional diffusion processes,”Publ. RIMS Kyoto Univ.,12, 819–834 (1977).

    Google Scholar 

  71. T. Uno and I. Hong, “Some consideration of eigenvalues for the equationd 2 u/dx 2+ λρ(x)u=0”Jpn. J. Math.,29, 152–164 (1959).

    Google Scholar 

  72. W. Feller, “On second order differential operators,”Ann. Math.,61, 90–105 (1955).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Odessa Technological Institute of Food Industry, Odessa

    I. S. Kats

Authors
  1. I. S. Kats
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 155–176, March, 1994.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kats, I.S. Spectral theory of a string. Ukr Math J 46, 159–182 (1994). https://doi.org/10.1007/BF01062233

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01062233

Keywords

Search

Navigation