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Journal of Mathematical Sciences

, Volume 243, Issue 5, pp 783–807 | Cite as

The Wave Model of the Sturm–Liouville Operator on an Interval

  • S. A. SimonovEmail author
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In the paper the wave functional model of a symmetric restriction of the regular Sturm-Liouville operator on an interval is constructed. The model is based upon the notion of the wave spectrum and is constructed according to an abstract scheme, which was proposed earlier. The result of the construction is a differential operator of the second order on an interval, which differs from the original operator only by a simple transformation.

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References

  1. 1.
    M. I. Belishev, “A unitary invariant of a semi-bounded operator in reconstruction of manifolds,” J. Operator Theory, 69, No. 2, 299–326 (2013).MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. I. Belishev, “Boundary control in reconstruction of manifolds and metrics (the BC method),” Inverse Problems, 13, No. 5, 1–45 (1997).MathSciNetCrossRefGoogle Scholar
  3. 3.
    M. I. Belishev, “On the Kac problem of the domain shape reconstruction via the Dirichlet problem spectrum,” J. Soviet Math., 55, No. 3, 1663–1672 (1991).MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. I. Belishev, “Recent progress in the boundary control method,” Inverse Problems, 23, No. 5, 1–67 (2007).MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. I. Belishev and M. N. Demchenko, “Dynamical system with boundary control associated with a symmetric semibounded operator,” J. Math. Sci., 194, No. 1, 8–20 (2013).MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. I. Belishev and M. N. Demchenko, “Elements of noncommutative geometry in inverse problems on manifolds,” J. Geom. Phys., 78, 29–47 (2014).MathSciNetCrossRefGoogle Scholar
  7. 7.
    M. I. Belishev and S. A. Simonov, “Wave model of the Sturm-Liouville operator on the half-line,” St. Petersburg Math. J., 29, No. 2, 227–248 (2018).MathSciNetCrossRefGoogle Scholar
  8. 8.
    G. Birkhoff, Lattice Theory, Providence, Rhode Island (1967).zbMATHGoogle Scholar
  9. 9.
    M. S. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D.Reidel Publishing Comp. (1987).Google Scholar
  10. 10.
    V. A. Derkach and M. M. Malamud, “The extension theory of Hermitian operators and the moment problem,” J. Math. Sci., 73, No. 2, 141–242 (1995).MathSciNetCrossRefGoogle Scholar
  11. 11.
    J. L. Kelley, General Topology, D. Van Nostrand Company, Inc Princeton, New Jersey, Toronto, London, New York (1957).zbMATHGoogle Scholar
  12. 12.
    J. M. Kim, “Compactness in \( \mathcal{B} \)(X),” J. Math. Anal. Appl., 320, 619–631 (2006).MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. N. Kochubei, “Extensions of symmetric operators and symmetric binary relations,” Math. Notes, 17, No. 1, 25–28 (1975).MathSciNetCrossRefGoogle Scholar
  14. 14.
    A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis. Vol. 1, Metric and Normed Spaces, Graylock Press (1957).Google Scholar
  15. 15.
    M. A. Naimark, Linear Differential Operators, WN Publishing, Gronnongen, The Netherlands (1970).Google Scholar
  16. 16.
    V. Ryzhov, “A general boundary value problem and its Weyl function,” Opuscula Math., 27, No. 2, 305–331 (2007).MathSciNetzbMATHGoogle Scholar
  17. 17.
    S. A. Simonov, “Wave model of the regular Sturm–Liouville operator,” in: Proceedings of 2017 Days on Diffraction (2017), pp. 300–303.Google Scholar
  18. 18.
    A. V. Strauss, “Functional models and generalized spectral functions of symmetric operators,” St. Petersbg. Math. J., 10, No. 5, 733–784 (1999).MathSciNetGoogle Scholar
  19. 19.
    M. I. Vishik, “On general boundary problems for elliptic differential equations,” Transl., Ser. 2, Amer. Math. Soc., 24, 107–172 (1963).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of SciencesSt. PetersburgRussia
  2. 2.St. Petersburg State UniversitySt. PetersburgRussia

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