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Integral Equations and Operator Theory

, Volume 77, Issue 3, pp 303–354 | Cite as

On Titchmarsh–Weyl Functions and Eigenfunction Expansions of First-Order Symmetric Systems

  • Sergio Albeverio
  • Mark Malamud
  • Vadim MogilevskiiEmail author
Article

Abstract

We study general (not necessarily Hamiltonian) first-order symmetric system J y′(t)−B(t)y(t) = Δ(t) f(t) on an interval \({\mathcal{I}=[a,b) }\) with the regular endpoint a. It is assumed that the deficiency indices n ±(T min) of the minimal relation T min associated with this system in \({L^2_\Delta(\mathcal{I})}\) satisfy \({n_-(T_{\rm min})\leq n_+(T_{\rm min})}\). We are interested in boundary conditions playing the role similar to that of separated self-adjoint boundary conditions for Hamiltonian systems. Instead we define λ-depending boundary conditions with Nevanlinna type spectral parameter τ = τ(λ) at the singular endpoint b. With this boundary value problem we associate the matrix m-function m(·) of the size \({N_\Sigma = {\rm dim} {\rm ker} (iJ+I)}\). Its role is similar to that of the Titchmarsh–Weyl coefficient for the Hamiltonian system. In turn, it allows one to define the Fourier transform \({V: L^2_\Delta(\mathcal{I}) \to L^2(\Sigma)}\) where Σ (·) is a spectral matrix function of m(·). If V is an isometry, then the (exit space) self-adjoint extension \({\tilde{T}}\) of T min induced by the boundary problem is unitarily equivalent to the multiplication operator in L 2(Σ). Hence the multiplicity of spectrum of \({\tilde{T}}\) does not exceed N Σ. We also parameterize a set of spectral functions Σ(·) by means of the set of boundary parameters τ. Similar parameterizations for various classes of boundary value problems have earlier been obtained by Kac and Krein, Fulton, Hinton and Shaw, and others.

Mathematics Subject Classification (2010)

34B08 34B20 34B40 34L10 47A06 47B25 

Keywords

First-order symmetric system nevanlinna boundaryconditions m-function spectral function of a boundary problem fourier transform 

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References

  1. 1.
    Akhiezer N.I., Glazman I.M.: Theory of Linear Operators in Hilbert Space. Dover Publications, New York (1993)zbMATHGoogle Scholar
  2. 2.
    Atkinson F.V.: Discrete and Continuous Boundary Problems. Academic Press, New York (1963)Google Scholar
  3. 3.
    Behrndt J., Hassi S., de Snoo H., Wiestma R.: Square-integrable solutions and Weyl functions for singular canonical systems. Math. Nachr. 284(11–12), 1334–1383 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berezanskii, Yu.M.: Expansions in eigenfunctions of selfadjoint operators. American Mathematical Society, Providence (1968) (Russian edition: Naukova Dumka, Kiev, 1965)Google Scholar
  5. 5.
    Bruk V.M.: On a class of boundary value problems with spectral parameter in the boundary condition. Math. USSR Sbornik. 29(2), 186–192 (1976)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Derkach V.A., Hassi S., Malamud M.M., de Snoo H.S.V.: Generalized resolvents of symmetric operators and admissibility. Methods Funct. Anal. Topol. 6(3), 24–55 (2000)zbMATHGoogle Scholar
  7. 7.
    Derkach V.A., Hassi S., Malamud M.M., de Snoo H.S.V.: Boundary relations and generalized resolvents of symmetric operators. Russ. J. Math. Phys. 16,–11760 (2009)Google Scholar
  8. 8.
    Derkach V.A., Malamud M.M.: Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal. 95, 1–95 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Derkach V.A., Malamud M.M.: The extension theory of Hermitian operators and the moment problem. J. Math. Sci. 73(2), 141–242 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dijksma A., Langer H., de Snoo H.S.V.: Hamiltonian systems with eigenvalue depending boundary conditions. Oper. Theory Adv. Appl. 35, 37–83 (1988)Google Scholar
  11. 11.
    Dijksma A., Langer H., de Snoo H.S.V.: Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary conditions. Math. Nachr. 161, 107–153 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dunford N., Schwartz J.T.: Linear Operators. Part 2. Spectral Theory. Interscience Publishers, New York (1963)Google Scholar
  13. 13.
    Fulton Ch.T.: Parametrizations of Titchmarsh’s m(ł)-functions in the limit circle case. Trans. Am. Math. Soc. 229, 51–63 (1977)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gohberg, I., Krein, M.G.: Theory and applications of Volterra operators in Hilbert space, Transl. Math. Monographs, vol. 24. American Mathematical Society, Providence (1970)Google Scholar
  15. 15.
    Gorbachuk M.L.: On spectral functios of a differential equation of the second order with operator-valued coefficients. Ukrain. Mat. Zh. 18(2), 3–21 (1966)CrossRefzbMATHGoogle Scholar
  16. 16.
    Gorbachuk, V.I., Gorbachuk, M.L.: Boundary problems for differential-operator equations. Kluver Academic Publishers, Dordrecht (1991) (Russian edition: Naukova Dumka, Kiev, 1984)Google Scholar
  17. 17.
    Hassi S., de Snoo H.S.V., Winkler H.: Boundary-value problems for two-dimensional canonical systems. Integr. Equ. Oper. Theory 36, 445–479 (2000)CrossRefzbMATHGoogle Scholar
  18. 18.
    Hinton, D.B., Shaw, J.K.: Parameterization of the M(ł) function for a Hamiltonian system of limit circle type. Proc. R Soc. Edinb. Sect. A 93(3–4), 349–360 (1982/1983)Google Scholar
  19. 19.
    Hinton D.B., Schneider A.: On the Titchmarsh–Weyl coefficients for singular S-Hermitian systems I. Math. Nachr. 163, 323–342 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hinton D.B., Schneider A.: Titchmarsh–Weyl coefficients for odd order linear Hamiltonian systems. J. Spectr. Math. 1, 1–36 (2006)Google Scholar
  21. 21.
    Hinton D.B., Schneider A.: On the Spectral Representation for Singular Selfadjoint Boundary Eigenfunction Problems. Oper. Theory: Advances and Applications, vol. 106. Birkhauser, Basel (1998)Google Scholar
  22. 22.
    Khol’kin A.M.: Description of selfadjoint extensions of differential operators of an arbitrary order on the infinite interval in the absolutely indefinite case. Teor. Funkcii Funkcional. Anal. Prilozhen. 44, 112–122 (1985)zbMATHGoogle Scholar
  23. 23.
    Kats, I.S.: On Hilbert spaces generated by monotone Hermitian matrix-functions. Khar’kov. Gos. Univ. Uchen. Zap. 34, 95–113 (1950) [Zap. Mat. Otdel. Fiz. -Mat. Fak. i Khar’kov. Mat. Obshch. 22(4), 95–113 (1950)]Google Scholar
  24. 24.
    Kats I.S.: Linear relations generated by the canonical differential equation of phase dimension 2, and eigenfunction expansion. St. Petersburg Math. J. 14, 429–452 (2003)MathSciNetGoogle Scholar
  25. 25.
    Kac, I.S., Krein, M.G.: On Spectral Functions of a String. Supplement to the Russian edition of F.V. Atkinson. Discrete and continuous boundary problems, Mir, Moscow (1968)Google Scholar
  26. 26.
    Khrabustovsky V.I.: On the characteristic operators and projections and on the solutions of Weyl type of dissipative and accumulative operator systems. 3. Separated boundary conditions. J. Math. Phys. Anal. Geom. 2(4), 449–473 (2006)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Kogan, V.I., Rofe-Beketov, F.S.: On square-integrable solutions of symmetric systems of differential equations of arbitrary order. Proc. R Soc. Edinb. Sect. A 74, 5–40 (1974/1975)Google Scholar
  28. 28.
    Kovalishina I.V.: Analytic theory of a class of interpolation problems. Izv. Akad. Nauk SSSR Ser. Mat. 47(3), 455–497 (1983)MathSciNetGoogle Scholar
  29. 29.
    Krall A.M.: M(ł)-theory for singular Hamiltonian systems with one singular endpoint. SIAM J. Math. Anal. 20, 664–700 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Langer H., Textorious B.: On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space. Pac. J. Math. 72(1), 135–165 (1977)CrossRefzbMATHGoogle Scholar
  31. 31.
    Langer H., Textorius B.: L-resolvent matrices of symmetric linear relations with equal defect numbers; appliccations to canonical differential relations. Integr. Equ. Oper. Theory 5, 208–243 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lesch M., Malamud M.M.: On the deficiency indices and self-adjointness of symmetric Hamiltonian systems. J. Differ. Equ. 189, 556–615 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Malamud M.M.: On the formula of generalized resolvents of a nondensely defined Hermitian operator. Ukr. Math. Zh. 44(12), 1658–1688 (1992)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Malamud M.M., Malamud S.M.: Spectral theory of operator measures in Hilbert space. St. Petersburg Math. J. 15(3), 323–373 (2003)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Malamud M.M., Mogilevskii V.I.: Krein type formula for canonical resolvents of dual pairs of linear relations. Methods Funct. Anal. Topol. 8(4), 72–100 (2002)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Malamud M., Neidhardt H.: Sturm–Liouville boundary value problems with operator potentials and unitary equivalence. J. Differ. Equ. 252, 5875–5922 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Mogilevskii V.I.: Nevanlinna type families of linear relations and the dilation theorem. Methods Funct. Anal. Topol. 12(1), 38–56 (2006)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Mogilevskii V.I.: Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers. Methods Funct. Anal. Topol. 12(3), 258–280 (2006)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Mogilevskii V.I.: Description of spectral functions of differential operators with arbitrary deficiency indices. Math. Notes 81(4), 553–559 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Mogilevskii V.I.: Boundary triplets and Titchmarsh–Weyl functions of differential operators with arbitrary deficiency indices. Methods Funct. Anal. Topol. 15(3), 280–300 (2009)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Mogilevskii V.I.: Boundary pairs and boundary conditions for general (not necessarily definite) first-order symmetric systems with arbitrary deficiency indices. Math. Nachr. 285(14–15), 1895–1931 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Naimark M.A.: Linear Differential Operators, vols. 1, 2. Harrap, London (1967)Google Scholar
  43. 43.
    Orcutt, B.C.: Canonical differential equations. Dissertation, University of Virginia (1969)Google Scholar
  44. 44.
    Rofe-Beketov F.S.: Self-adjoint extensions of differential operators in the space of vector-valued functions. Teor. Funkcii Funkcional. Anal. Prilozhen. 8, 3–23 (1969)MathSciNetzbMATHGoogle Scholar
  45. 45.
    S̆traus A.V.: On generalized resolvents and spectral functions of differential operators of an even order. Izv. Akad. Nauk. SSSR Ser. Mat. 21, 785–808 (1957)MathSciNetGoogle Scholar
  46. 46.
    Qi J.: Non-limit-circle criteria for singular Hamiltonian differential systems. Math. Anal. Appl. 305, 599–616 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Qi J.: Limit-point criteria for semi-degenerate singular Hamiltonian differential systems with perturbation terms. Math. Anal. Appl. 334, 983–997 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Weidmann J.: Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics, vol. 1258. Springer, Berlin (1987)Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Sergio Albeverio
    • 1
    • 2
  • Mark Malamud
    • 3
  • Vadim Mogilevskii
    • 4
    Email author
  1. 1.Institut für Angevandte Mathematic HCM, IZKS, SFB611Universität BonnBonnGermany
  2. 2.CERFIMLocarnoSwitzerland
  3. 3.Institute of Applied Mathematics and MechanicsNAS of UkraineDonetskUkraine
  4. 4.Department of Mathematical AnalysisLugans’k National UniversityLugans’kUkraine

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