The Classical Moment Problem and Generalized Indefinite Strings

Open Access


We show that the classical Hamburger moment problem can be included in the spectral theory of generalized indefinite strings. More precisely, we introduce the class of Krein–Langer strings and show that there is a bijective correspondence between moment sequences and this class of generalized indefinite strings. This result can be viewed as a complement to the classical results of Krein on the connection between the Stieltjes moment problem and Krein–Stieltjes strings and Kac on the connection between the Hamburger moment problem and \(2\times 2\) canonical systems with Hamburger Hamiltonians.


Hamburger moment problem Generalized indefinite strings 

Mathematics Subject Classification

Primary 44A60 34L05 Secondary 34B20 34B07 



Open access funding provided by Austrian Science Fund (FWF).


  1. 1.
    Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Oliver and Boyd Ltd, Edinburgh, London (1965)MATHGoogle Scholar
  2. 2.
    Beals, R., Sattinger, D.H., Szmigielski, J.: Multipeakons and the classical moment problem. Adv. Math. 154(2), 229–257 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, M., Liu, S., Zhang, Y.: A two-component generalization of the Camassa–Holm equation and its solutions. Lett. Math. Phys. 75(1), 1–15 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Constantin, A., Ivanov, R.I.: On an integrable two-component Camassa–Holm shallow water system. Phys. Lett. A 372(48), 7129–7132 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    de Branges, L.: Hilbert Spaces of Entire Functions. Prentice-Hall Inc, Englewood Cliffs, NJ (1968)MATHGoogle Scholar
  6. 6.
    Dym, H., McKean, H.P.: Gaussian Processes, Function Theory and The Inverse Spectral Problem, Probability and Mathematical Statistics 31. Academic Press, New York, London (1976)MATHGoogle Scholar
  7. 7.
    Eckhardt, J.: The inverse spectral transform for the conservative Camassa–Holm flow with decaying initial data. Arch. Ration. Mech. Anal. 224(1), 21–52 (2017)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Eckhardt, J., Gesztesy, F., Holden, H., Kostenko, A., Teschl, G.: Real-valued algebro-geometric solutions of the two-component Camassa–Holm hierarchy. Ann. Inst. Fourier (Grenoble) 67(3), 1185–1230 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Eckhardt, J., Kostenko, A.: An isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation. Commun. Math. Phys. 329(3), 893–918 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Eckhardt, J., Kostenko, A.: Quadratic operator pencils associated with the conservative Camassa–Holm flow. Bull. Soc. Math. France 145(1), 47–95 (2017)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Eckhardt, J., Kostenko, A.: The inverse spectral problem for indefinite strings. Invent. Math. 204(3), 939–977 (2016)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Eckhardt, J., Kostenko, A., Teschl, G.: The Camassa–Holm equation and the string density problem. Int. Math. Nachr. 233, 1–24 (2016)MATHGoogle Scholar
  13. 13.
    Fleige, A., Winkler, H.: An indefinite inverse spectral problem of Stieltjes type. Integr. Equ. Oper. Theory 87(4), 491–514 (2017)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gérard, P., Pushnitski, A.: An inverse problems for self-adjoint positive Hankel operators. Int. Math. Res. Notices 2015, 4505–4535 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hamburger, H.: Über eine Erweiterung des Stieltjesschen Momentenproblems. Math. Ann. 81(2–4), 235–319 (1920)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Holm, D.D., Ivanov, R.I.: Two-component CH system: inverse scattering, peakons and geometry. Inverse Probl. 27(4), 045013, 19 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kac, I.S.: Linear relations generated by canonical differential equations. Funct. Anal. Appl. 17(4), 315–317 (1983)MathSciNetGoogle Scholar
  18. 18.
    Kac, I.S.: The Hamburger power moment problem as part of spectral theory of canonical systems. Funct. Anal. Appl. 33(3), 228–230 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kac, I.S.: Inclusion Hamburger’s power moment problem in the spectral theory of canonical systems. J. Math. Sci. 110(5), 2991–3004 (1999)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kac, I.S., Krein, M.G.: On the spectral functions of the string. Am. Math. Soc. Transl. Ser. 2(103), 19–102 (1974)MATHGoogle Scholar
  21. 21.
    Kaltenbäck, M., Winkler, H., Woracek, H.: Strings, dual strings, and related canonical systems. Math. Nachr. 280(13–14), 1518–1536 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kotani, S., Watanabe, S.: Kreĭn’s spectral theory of strings and generalized diffusion processes. In: Functional Analysis in Markov Processes (Katata/Kyoto, 1981), Lecture Notes in Mathematics, vol. 923, pp. 235–259. Springer, Berlin (1982)Google Scholar
  23. 23.
    Kreĭn, M.G.: On a generalization of investigation of Stieltjes. Dokl. Akad. Nauk SSSR 87(6), 881–884 (1952). (in Russian) MathSciNetMATHGoogle Scholar
  24. 24.
    Kreĭn, M.G.: On a difficult problem of operator theory and its relation to classical analysis In: Gorbachuk, M.L., Gorbachuk, V.I. (eds) M.G. Krein’s Lectures on Entire Operators. Birhäuser, Basel-Boston-Berlin (1997)Google Scholar
  25. 25.
    Kreĭn, M.G., Langer, H.: On some extension problems which are closely connected with the theory of Hermitian operators in a space \(\Pi _\kappa \). III. Indefinite analogues of the Hamburger and Stieltjes moment problems. Part I. Beiträge Anal. 14, 25–40 (1979)MathSciNetGoogle Scholar
  26. 26.
    Kreĭn, M.G., Langer, H.: On some extension problems which are closely connected with the theory of Hermitian operators in a space \(\Pi _\kappa \) III Indefinite analogues of the Hamburger and Stieltjes moment problems Part II. Beiträge Anal. 15, 27–45 (1980)MathSciNetGoogle Scholar
  27. 27.
    Langer, H.: Spektralfunktionen einer Klasse von Differentialoperatoren zweiter Ordnung mit nichtlinearem Eigenwertparameter. Ann. Acad. Sci. Fenn. Ser. A I Math. 2, 269–301 (1976)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Langer, H., Winkler, H.: Direct and inverse spectral problems for generalized strings. Integr. Equ. Oper. Theory 30(4), 409–431 (1998)MathSciNetCrossRefMATHGoogle Scholar
  29. 28.
    Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137, 82–203 (1998)MathSciNetCrossRefMATHGoogle Scholar
  30. 29.
    Stieltjes, T.-J.: Recherches sur les Fractions Continues. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 8(4), 1–122 (1894)MathSciNetMATHGoogle Scholar
  31. 31.
    Winkler, H.: The inverse spectral problem for canonical systems. Integr. Equ. Oper. Theory 22(3), 360–374 (1995)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Winkler, H.: Two-dimensional Hamiltonian systems. In: Alpay, D. (ed.) Operator Theory, pp. 525–547. Springer, Basel (2015)CrossRefGoogle Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.RUDN UniversityMoscowRussia

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