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Inverse Stieltjes-like Functions and Inverse Problems for Systems with Schrödinger Operator

  • Sergey V. Belyi
  • Eduard R. Tsekanovskii
Chapter
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Part of the Operator Theory: Advances and Applications book series (OT, volume 197)

Abstract

A class of scalar inverse Stieltjes-like functions is realized as linear-fractional transformations of transfer functions of conservative systems based on a Schrödinger operator T h in L 2au][a,+∞) with a non-selfadjoint boundary condition. In particular it is shown that any inverse Stieltjes function of this class can be realized in the unique way so that the main operator \( \mathbb{A} \) possesses a special semi-boundedness property. We derive formulas that restore the system uniquely and allow to find the exact value of a non-real boundary parameter h of the operator T h as well as a real parameter μ that appears in the construction of the elements of the realizing system. An elaborate investigation of these formulas shows the dynamics of the restored parameters h and μ in terms of the changing free term α from the integral representation of the realizable function.

Mathematics Subject Classification (2000)

Primary 47A10 47B44 Secondary 46E20 46F05 

Keywords

Operator colligation conservative system transfer (characteristic) function 

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References

  1. [1]
    N.I. Akhiezer and I.M. Glazman. Theory of linear operators. Pitman Advanced Publishing Program, 1981.Google Scholar
  2. [2]
    D. Alpay, I. Gohberg, M.A. Kaashoek, A.L. Sakhnovich, “Direct and inverse scattering problem for canonical systems with a strictly pseudoexponential potential”, Math. Nachr. 215 (2000), 5–31.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    D. Alpay and E.R. Tsekanovskiî, “Interpolation theory in sectorial Stieltjes classes and explicit system solutions”, Lin. Alg. Appl., 314 (2000), 91–136.zbMATHCrossRefGoogle Scholar
  4. [4]
    Yu. M. Arlinskiî. On regular (*)-extensions and characteristic matrix-valued functions of ordinary differential operators. Boundary value problems for differential operators, Kiev, 3–13, 1980.Google Scholar
  5. [5]
    Yu. Arlinskiî and E. Tsekanovskiî. Regular (*)-extension of unbounded operators, characteristic operator-functions and realization problems of transfer functions of linear systems. Preprint, VINITI, Dep.-2867, 72p., 1979.Google Scholar
  6. [6]
    Yu. M. Arlinskiî and E.R. Tsekanovskiî, “Linear systems with Schrödinger operators and their transfer functions”, Oper. Theory Adv. Appl., 149, 2004, 47–77.Google Scholar
  7. [7]
    D. Arov, H. Dym, “Strongly regular J-inner matrix-valued functions and inverse problems for canonical systems”, Oper. Theory Adv. Appl., 160, Birkhäuser, Basel, (2005), 101–160.CrossRefGoogle Scholar
  8. [8]
    D. Arov, H. Dym, “Direct and inverse problems for differential systems connected with Dirac systems and related factorization problems”, Indiana Univ. Math. J. 54 (2005), no. 6, 1769–1815.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J.A. Ball and O.J. Staffans, “Conservative state-space realizations of dissipative system behaviors”, Integr. Equ. Oper. Theory, 54 (2006), no. 2, 151–213.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    H. Bart, I. Gohberg, and M.A. Kaashoek, Minimal Factorizations of Matrix and Operator Functions, Operator Theory: Advances and Applications, Vol. 1, Birkhäuser, Basel, 1979.Google Scholar
  11. [11]
    S.V. Belyi and E.R. Tsekanovskiî, “Realization theorems for operator-valued R-functions”, Oper. Theory Adv. Appl., 98 (1997), 55–91.Google Scholar
  12. [12]
    S.V. Belyi and E.R. Tsekanovskiî, “On classes of realizable operator-valued R-functions”, Oper. Theory Adv. Appl., 115 (2000), 85–112.Google Scholar
  13. [13]
    S.V. Belyi, S. Hassi, H.S.V. de Snoo, and E.R. Tsekanovskiî, “On the realization of inverse of Stieltjes functions”, Proceedings of MTNS-2002, University of Notre Dame, CD-ROM, 11p., 2002.Google Scholar
  14. [14]
    S.V. Belyi and E.R. Tsekanovskiî, “Stieltjes-like functions and inverse problems for systems with Schrödinger operator”, Operators and Matrices., vol. 2, no. 2, (2008), pp. 265–296.zbMATHMathSciNetGoogle Scholar
  15. [15]
    Brodskiî, M.S., Triangular and Jordan Representations of Linear Operators, Amer. Math. Soc., Providence, RI, 1971.Google Scholar
  16. [16]
    M.S. Brodskiî, M.S. Livšsic. Spectral analysis of non-selfadjoint operators and intermediate systems, Uspekhi Matem. Nauk, XIII, no. 1 (79)), au][1958], 3–84.Google Scholar
  17. [17]
    V.A. Derkach, M.M. Malamud, and E.R. Tsekanovskiî, Sectorial extensions of a positive operator, and the characteristic function, Sov. Math. Dokl. 37, 106–110 (1988).zbMATHGoogle Scholar
  18. [18]
    F. Gesztesy and E.R. Tsekanovskiî, “On matrix-valued Herglotz functions”, Math. Nachr., 218 (2000), 61–138.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    F. Gesztesy, N.J. Kalton, K.A. Makarov, E. Tsekanovskiî, “Some Applications of Operator-Valued Herglotz Functions”, Operator Theory: Advances and Applications, 123, Birkhäuser, Basel, (2001), 271–321.Google Scholar
  20. [20]
    S. Khrushchev, “Spectral Singularities of dissipative Schrödinger operator with rapidly decreasing potential”, Indiana Univ. Math. J., 33 no. 4, (1984), 613–638.CrossRefMathSciNetGoogle Scholar
  21. [21]
    I.S. Kac and M.G. Krein, R-functions — analytic functions mapping the upper half-plane into itself, Amer. Math. Soc. Transl. (2) 103, 1–18 (1974).Google Scholar
  22. [22]
    Kato T.: Perturbation Theory for Linear Operators, Springer-Verlag, 1966Google Scholar
  23. [23]
    B.M. Levitan, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987.zbMATHGoogle Scholar
  24. [24]
    M.S. Livšic, Operators, oscillations, waves, Moscow, Nauka, 1966 (Russian).Google Scholar
  25. [25]
    M.A. Naimark, Linear Differential Operators II, F. Ungar Publ., New York, 1968.zbMATHGoogle Scholar
  26. [26]
    O.J. Staffans, “Passive and conservative continuous time impedance and scattering systems, Part I: Well-posed systems”, Math. Control Signals Systems, 15, (2002), 291–315.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    O.J. Staffans, Well-posed linear systems, Cambridge University Press, Cambridge, 2005.zbMATHCrossRefGoogle Scholar
  28. [28]
    E.R. Tsekanovskiî, “Accretive extensions and problems on Stieltjes operator-valued functions realizations”, Oper. Theory Adv. Appl., 59 (1992), 328–347.Google Scholar
  29. [29]
    E.R. Tsekanovskiî. “Characteristic function and sectorial boundary value problems”, Investigation on geometry and math. analysis, Novosibirsk, 7, (1987), 180–194.Google Scholar
  30. [30]
    E.R. Tsekanovskiî and Yu.L. Shmul’yan, “The theory of bi-extensions of operators on rigged Hilbert spaces. Unbounded operator colligations and characteristic functions”, Russ. Math. Surv., 32 (1977), 73–131.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Sergey V. Belyi
    • 1
  • Eduard R. Tsekanovskii
    • 2
  1. 1.Department of MathematicsTroy State UniversityTroyUSA
  2. 2.Department of MathematicsNiagara UniversityNYUSA

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