Collectanea Mathematica

, Volume 68, Issue 2, pp 251–263 | Cite as

De Branges functions of Schroedinger equations

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Abstract

We characterize the Hermite–Biehler (de Branges) functions E which correspond to Schroedinger operators with \(L^2\) potential on the finite interval. From this characterization one can easily deduce a recent theorem by Horvath. We also obtain a result about location of resonances.

References

  1. 1.
    Abakumov, E., Baranov, A., Belov, Y.: Localization of zeros for Cauchy transforms. Int. Math. Res. Not. 2015, 6699–6733 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chelkak, D.: An application of the fixed point theorem to the inverse Sturm–Liouville problem. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 370 (2009). [English transl. in J. Math. Sci. 166(1), 118–126 (2010)]Google Scholar
  3. 3.
    de Branges, L.: Hilbert Spaces of Entire Functions. Prentice-Hall, Englewood Cliffs (1968)MATHGoogle Scholar
  4. 4.
    Dym, H., McKean, H.P.: Gaussian Processes, Function Theory and the Inverse Spectral Problem. Academic Press, New York (1976)MATHGoogle Scholar
  5. 5.
    Hitrik, M.: Bounds on scattering poles in one dimension. Commun. Math. Phys. 208(2), 381–411 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Horváth, M.: Inverse spectral problems and closed exponential systems. Ann. Math. 162(2), 885–918 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Iantchenko, A., Korotyaev, E.: Resonances for Dirac operators on the half-line. J. Math. Anal. Appl. 420(1), 279–313 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Korotyaev, E.: Inverse resonance scattering on the half line. Asymptot. Anal. 37(3–4), 215–226 (2004)MathSciNetMATHGoogle Scholar
  9. 9.
    Lagarias, J.: Zero spacing distributions for differenced L-functions. Acta Arith. 120(2), 159–184 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lagarias, J.: The Schrödinger operator with Morse potential on the right half line. Commun. Number Theory Phys. 3(2), 323–361 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Levin, B.Ya.: Distribution, of Zeros of Entire Functions. Am. Math. Soc., Providence (1964). [Revised edition: Am. Math. Soc., Providence (1980)]Google Scholar
  12. 12.
    Marchenko, V.A.: Certain problems in the theory of second-order differential operators. Dokl. Akad. Nauk SSSR 72, 457–460 (1950). (Russian)Google Scholar
  13. 13.
    Marchenko, V.A.: Some questions in the theory of one-dimensional linear differential operators of the second order. I, Trudy Moskov. Mat. Ob. 1, 327–420 (1952). [English transl. in Am. Math. Soc. Transl. (2) 101, 1–104 (1973)]Google Scholar
  14. 14.
    Makarov, N., Poltoratski, A.: Meromorphic inner functions, Toeplitz kernels, and the uncertainty principle. In: Perspectives in Analysis, Carleson’s Special Volume. Springer, Berlin (2005)Google Scholar
  15. 15.
    Poshel, J., Trubowitz, E.: Inverse Spectral Theory. Academic Press, New York (1987)Google Scholar
  16. 16.
    Remling, C.: Schrödinger operators and de Branges spaces. J. Funct. Anal. 196, 323–394 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Romanov, R.: Canonical systems and de Branges spaces. arXiv:1408.6022
  18. 18.
    Romanov, R.: Order problem for canonical systems and a conjecture of Valent. Trans. Am. Math. Soc. arXiv:1502.04402
  19. 19.
    Simon, B.: Resonances in one dimension and Fredholm determinants. J. Funct. Anal. 178(2), 396–420 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2016

Authors and Affiliations

  1. 1.Department of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Chebyshev LaboratorySt. Petersburg State UniversitySt. PetersburgRussia
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA

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