Spectral Analysis of Singular Matrix-Valued Sturm–Liouville Operators

  • Bilender P. AllahverdievEmail author


In the Hilbert space \(L_{W}^{2}([a,b);E)\) (\(-\infty<a<b\le +\infty ,\) \(\dim E=N<+\infty ,\) \(W>0\)) a space of boundary values of the symmetric singular matrix-valued Sturm–Liouville operator with maximal deficiency indices (2N, 2N) (in limit-circle case at singular end point b) is constructed. With the help of the boundary conditions at a and b, all maximal dissipative, maximal accumulative and self-adjoint extensions of such a symmetric operator are established. In particular, the maximal dissipative operators with separated boundary conditions, called ‘dissipative at a’ and ‘self-adjoint at b’ are investigated. A self-adjoint dilation of the dissipative operator is constructed and then its incoming and outgoing spectral representations are determined. This representation allows us to determine the scattering matrix of the dilation with the help of the Weyl matrix-valued function of a self-adjoint matrix-valued Sturm–Liouville operator. Further a functional model of the dissipative operator is determined and its characteristic function in terms of the scattering matrix of the dilation (or of the Weyl function) is established. Finally, a theorem on completeness of the system of root vectors of the dissipative operator is proved.


Singular matrix-valued Sturm–Liouville equation symmetric operator space of boundary values self-adjoint and nonself-adjoint extensions self-adjoint dilation scattering matrix functional model characteristic function completeness of the system of root vectors 

Mathematics Subject Classification

Primary 34B40 34B24 34L10 47A20 Secondary 47A40 47A45 47E05 47B25 47B44 



  1. 1.
    Allahverdiev, B.P.: Dissipative Schrödinger operators with matrix potentials. Potential Anal. 20(4), 303–315 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Allahverdiev, B.P.: Nonselfadjoint Sturm–Liouville operators in limit-circle case. Taiwan. J. Math. 16(6), 2035–2052 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Allahverdiev, B.P.: Extensions, dilations and spectral analysis of singular Sturm–Liouville operators. Math. Rep. 19(2), 225–243 (2017)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Allakhverdiev, B.P.: On the theory of nonselfadjoint operators of Schrödinger type with a matrix potential. Russ. Acad. Sci. Izv. Math. 41(2), 193–205 (1993). (transl. from Izv. Ross. Akad. Nauk, Ser. Mat. 56(5), 920-933 (1992))Google Scholar
  5. 5.
    Anderson, R.L.: Limit point and limit circle criteria for a class of singular symmetric differential operators. Can. J. Math. 28, 905–914 (1976)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Atkinson, F.V.: Discrete and Continuous Boundary Problems. Academic Press, New York (1964)zbMATHGoogle Scholar
  7. 7.
    Baro, M., Neidhardt, H.: Dissipative Schrödinger-type operators as a model for generation and recombination. J. Math. Phys. 44, 2373–2401 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bhagat, B.: On the \(L^{2}\) classification of a second-order matrix differential equation. Indian J. Pure Appl. Math. 10, 804–809 (1979)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bhagat, B., Swesi, G.: Deficiency indices of second-order matrix differential operators. Indian J. Pure Appl. Math. 13, 433–439 (1982)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bruk, V.M.: On a class of boundary-value problems with a spectral parameter in the boundary conditions. Mat. Sb. 100, 210–216 (1976). (English. transl. Mat USSR Sb. 28, 186-192 (1976))MathSciNetGoogle Scholar
  11. 11.
    Clark, S., Gesztesy, F.: Weyl–Titchmarsh \(M\)-function asymptotics for matrix-valued Schrödinger operators. Proc. Lond. Math. Soc. 82(3), 701–724 (2001)CrossRefGoogle Scholar
  12. 12.
    Dunford, N., Schwartz, J.T.: Linear Operators. Part II. Interscience, New York (1964)Google Scholar
  13. 13.
    Eastham, M.S.P.: The deficiency index of a second-order differential system. J. Lond. Math. Soc. II. Ser. 23, 311–320 (1981)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Eastham, M.S.P., Gould, K.J.: Square-integrable solutions of a matrix differential expression. J. Math. Anal. Appl. 91, 424–433 (1983)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ginzburg, YuP, Talyush, N.A.: Exceptional sets of analytical matrix-functions, contracting and dissipative operators. Izv. Vyssh. Uchebn. Zaved. Mat. 267, 9–14 (1984). (English transl. Soviet Math. (Izv. VUZ) 28, 10-16 (1984))zbMATHGoogle Scholar
  16. 16.
    Gorbachuk, M.L., Gorbachuk, V.I., Kochubei, A.N.: The theory of extensions of symmetric operators and boundary-value problems for differential equations. Ukrain. Mat. Zh. 41, 1299–1312 (1989). (English transl. Ukrainian Mat. J. 41,1117-1129 (1989))MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gorbachuk, V.I., Gorbachuk, M.L.: Boundary Value Problems for Operator Differential Equations. Naukova Dumka, Kiev (1984). (English transl. Kluwer Acad. Publ., Dordrecht (1991))zbMATHGoogle Scholar
  18. 18.
    Kaiser, H-Ch., Neidhardt, H., Rehberg, J.: On 1-dimensional dissipative Schrödinger-type operators their dilations and eigenfunction expansions. Math. Nachr. 252, 51–69 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kochubei, A.N.: Extensions of symmetric operators and symmetric binary relations. Mat. Zametki 17, 41–48 (1975). (English transl. Math. Notes 17, 25-28 (1975))MathSciNetGoogle Scholar
  20. 20.
    Kogan, V.I., Rofe-Beketov, F.S.: On square-integrable solutions of symmetric systems of differential equations or arbitrary order. Proc. R. Soc. Edinb. Sect. A 74, 5–40 (1974)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lax, P.D., Phillips, R.S.: Scattering Theory. Academic Press, New York (1967)zbMATHGoogle Scholar
  22. 22.
    Lesch, M., Malamud, M.: On the deficiency indices and self-adjointness of symmetric Hamiltonian systems. J. Differ. Equ. 189, 556–615 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lidskiĭ, V.B.: On the number of square-integrable solutions of the system of differential equations \(-y^{\prime \prime }+p(t)y=\lambda y\) (Russian). Dokl. Akad. Nauk SSSR 95, 217–220 (1954)Google Scholar
  24. 24.
    Sz.-Nagy, B., Foiaş, C.: Analyse Harmonique des Opérateurs de L’espace de Hilbert. Masson, Paris, and Akad. Kiadó, Budapest (1967). (English transl. North-Holland, Amsterdam, and Akad. Kiado, Budapest (1970))Google Scholar
  25. 25.
    Naimark, M.A.: Linear Differential Operators, 2nd edn. Nauka, Moscow (1969). (English transl. of 1st ed. Vols. 1, 2, Ungar, New York (1967, 1968))Google Scholar
  26. 26.
    Nikol’skii, N.K.: Treatise on the Shift Operator. Nauka, Moscow (1980). (English transl. Springer-Verlag, Berlin (1986))Google Scholar
  27. 27.
    von Neumann, J.: Allgemeine Eigenwerttheorie Hermitischer funktionaloperatoren. Math. Ann. 102, 49–131 (1929)CrossRefGoogle Scholar
  28. 28.
    Pavlov, B.S.: On conditions for separation of the spectral components of a dissipative operator. Izv. Akad. Nauk SSSR, Ser. Matem. 39, 123–148 (1975). (English transl.: Math USSR Izvestiya 9, 113–137 (1976))MathSciNetGoogle Scholar
  29. 29.
    Pavlov, B.S.: Dilation theory and spectral analysis of nonselfadjoint differential operators. In : Proceedings of 7th Winter School, Drobobych 1974, 3–69 (1976). (English transl: Transl. II. Ser., Am. Math. Soc. 115, 103–142 (1981))Google Scholar
  30. 30.
    Pavlov, B.S.: Spectral analysis of a dissipative singular Schrödinger operator in terms of a functional model. Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fundam. Napravleniya 65, 95–163 (1991). (English transl. Partial Differential Equations VIII. Encyc. Math. Sci. 65, 87-163 (1996))zbMATHGoogle Scholar
  31. 31.
    Pavlov, B.S.: Irreversibility, Lax–Phillips approach to resonance scattering and spectral analysis of non-self-adjoint operators in Hilbert space. Int. J. Theor. Phys. 38, 21–45 (1999)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Phillips, R.S.: Dissipative operators and hyperbolic systems of partial differential equations. Trans. Am. Math. Soc. 90, 193–254 (1959)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Rofe-Beketov, F.S.: Self-adjoint extensions of differential operators in a space of vector-valued functions. Dokl. Akad. Nauk SSSR 184, 1034–1037 (1969). (English transl. Soviet Math. Dokl. 10, 188-192(1969))MathSciNetGoogle Scholar
  34. 34.
    Rofe-Beketov, F.S., Khol’kin, A.M.: Spectral Analysis of Differential Operators (Russian). Mariupol (2002). (English transl. Spectral Analysis of Differential Operators: Interplay Between Spectral and Oscillatory Properties. World Scientific Monograph Series in Mathematics v.7 (2005))Google Scholar
  35. 35.
    Ronkin, L.I.: Introduction to the Theory of Entire Functions of Several Variables. Nauka, Moscow (1971). (English transl. Amer. Math. Soc., Providence, RI (1974))zbMATHGoogle Scholar
  36. 36.
    Stone, M.H.: Linear Transformation in Hilbert Space and Their Applications to Analysis, vol. 15. American Mathematical Society Colloquium Publications, Providence (1932)zbMATHGoogle Scholar
  37. 37.
    Valeev, N.F.: On the density of discrete spectra Sturm–Liouville singular operators. Mat. Zametki 71, 307–3011 (2002). (English transl. Math. Notes 71, 276-280 (2002))CrossRefGoogle Scholar
  38. 38.
    Weidmann, J.: Spectral theory of ordinary differential operators. Lecture Notes in Mathematics, vol. 1258. Springer, Berlin (1987)Google Scholar
  39. 39.
    Weyl, H.: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörligen Entwicklungen willkürlicher Funktionen. Math. Ann. 68, 222–269 (1910)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsSuleyman Demirel UniversityIspartaTurkey

Personalised recommendations