Advertisement

Results in Mathematics

, Volume 61, Issue 3–4, pp 255–281 | Cite as

Characterization of Domains of Self-Adjoint Ordinary Differential Operators II

  • Xiaoling Hao
  • Jiong Sun
  • Aiping Wang
  • Anton Zettl
Article

Abstract

We characterize the self-adjoint domains of general even order linear ordinary differential operators in terms of real-parameter solutions of the differential equation. This for endpoints which are regular or singular and for arbitrary deficiency index. This characterization is obtained from a new decomposition of the maximal domain in terms of limit-circle solutions. These are the solutions which contribute to the self-adjoint domains in analogy with the celebrated Weyl limit-circle solutions in the second order Sturm–Liouville case. As a special case we obtain the previously known characterizations when one or both endpoints are regular.

Mathematics Subject Classification (2010)

Primary 34B20 34B24 Secondary 47B25 

Keywords

Differential operators deficiency index self-adjoint domains real-parameter solutions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cao Z.J.: On self-adjoint domains of second order differential operators in limit-circle case. Acta Math. Sin. 1(3), 175–180 (1985)Google Scholar
  2. 2.
    Cao Z.J.: On self-adjoint extensions in the limit-circle case of differential operators of order n. Acta Math. Sin. 28(2), 205–217 (1985) (Chinese)zbMATHGoogle Scholar
  3. 3.
    Cao Z.J., Sun J.: Self-adjoint operators generated by symmetric quasi-differential expressions. Acta Sci. Natur. Univ. NeiMongol 17(1), 7–15 (1986)MathSciNetGoogle Scholar
  4. 4.
    Cao, Z.J.: Ordinary Differential Operators. Shanghai Science and Technology Press (1987, in Chinese)Google Scholar
  5. 5.
    Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)zbMATHGoogle Scholar
  6. 6.
    Dunford N., Schwartz J.T.: Linear Operators, vol. II. Wiley, New York (1963)Google Scholar
  7. 7.
    Everitt, W.N., Neuman, F.: A concept of adjointness and symmetry of differential expressions based on the generalized Lagrange identity and Green’s formula. In: Lecture Notes in Mathematics, vol. 1032, pp. 161–169. Springer-Verlag, Berlin (1983)Google Scholar
  8. 8.
    Evans W.D., Sobhy E.I.: Boundary conditions for general ordinary differential operators and their adjoints. Proc. R. Soc. Edinburgh 114A, 99–117 (1990)CrossRefGoogle Scholar
  9. 9.
    Everitt W.N., Zettl A.: Generalized symmetric ordinary differential expressions I: The general theory. Nieuw Archief voor Wiskunde 27(3), 363–397 (1979)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Everitt W.N.: A note on the self-adjoint domains of 2nth-order differential equations. Quart. J. Math. (Oxford) 14(2), 41–45 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Everitt W.N.: Integrable-square solutions of ordinary differential equations (III). Quart. J. Math. (Oxford) 14(2), 170–180 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Everitt W.N.: Singular differential equations I: the even case. Math. Ann. 156, 9–24 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Everitt W.N.: Singular differential equations II: some self-adjoint even case. Quart. J. Math. (Oxford) 18(2), 13–32 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Everitt W.N., Kumar V.K.: On the Titchmarsh-Weyl theory of ordinary symmetric differential expressions I: the general theory. Nieuw Archief voor Wiskunde 34(3), 1–48 (1976)Google Scholar
  15. 15.
    Everitt W.N., Kumar V.K.: On the Titchmarsh-Weyl theory of ordinary symmetric differential expressions II: the odd order case. Nieuw Archief voor Wiskunde 34(3), 109–145 (1976)Google Scholar
  16. 16.
    Everitt, W.N., Markus, L.: Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators. In: Mathematical Surveys and Monographs, vol. 61. American Mathematics Society (1999)Google Scholar
  17. 17.
    Everitt W.N., Markus L.: Complex symplectic geometry with applications to ordinary differential operators. Trans. Am. Math. Soc. 351(12), 4905–4945 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Frentzen H.: Equivalence, adjoints and symmetry of quasi-differential expressions with matrix-valued coefficients and polynomials in them. Proc. R. Soc. Edinburgh (A) 92, 123–146 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Fu S.Z.: On the self-adjoint extensions of symmetric ordinary differential operators in direct sum spaces. J. Differ. Equ. 100(2), 269–291 (1992)zbMATHCrossRefGoogle Scholar
  20. 20.
    Hao X., Sun J., Zettl A.: Real-parameter square-integrable solutions and the spectrum of differential operators. J. Math. Anal. Appl. 376, 696–712 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kauffman, R.M., Read, T.T., Zettl, A.: The deficiency index problem for powers of ordinary differential expressions. In: Lecture Notes in Mathematics, vol. 621. Springer-Verlag, New York (1977)Google Scholar
  22. 22.
    Li W.M.: The high order differential operators in direct sum spaces. J. Differ. Equ. 84(2), 273–289 (1990)zbMATHCrossRefGoogle Scholar
  23. 23.
    Li W.M.: The domains of self-adjoint extensions of symmetric ordinary differential operators with two singular points. J. Inner Mongolia University (Natural Science) 20(3), 291–298 (1989)Google Scholar
  24. 24.
    Möller M.: On the unboundedness below of the Sturm–Liouville operator. Proc. R. Soc. Edinburgh 129A, 1011–1015 (1999)CrossRefGoogle Scholar
  25. 25.
    Möller M., Zettl A.: Weighted norm-inequalities for quasi-derivatives. Results Math. 24, 153–160 (1993)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Möller M., Zettl A.: Symmetric differential operators and their Friedrichs extension. J. Differ. Equ. 115, 50–69 (1995)zbMATHCrossRefGoogle Scholar
  27. 27.
    Möller M., Zettl A.: Semi-boundedness of ordinary differential operators. J. Differ. Equ. 115, 24–49 (1995)zbMATHCrossRefGoogle Scholar
  28. 28.
    Macrae, N.: Jon von Neumann. American Mathematical Society (1992)Google Scholar
  29. 29.
    Naimark M.A.: Linear differential operators. English Transl, Ungar (1968)zbMATHGoogle Scholar
  30. 30.
    Niessen H.-D., Zettl A.: The Friedrichs extension of regular ordinary differential operators. Proc. R. Soc. Edinburgh 114A, 229–236 (1990)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Orlov S.A.: On the deficiency indices of differential operators. Doklady. Akad. Nauk SSSR 92, 483–486 (1953) (Russian)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Qi, J., Chen, S.: On an open problem of J. Weidmann: essential spectra and square-integrable solutions. Pre-printGoogle Scholar
  33. 33.
    Shang Z.J.: On J-selfadjoint extensions of J-symmetric ordinary differential operators. J. Differ. Equ. 73, 153–177 (1988)zbMATHCrossRefGoogle Scholar
  34. 34.
    Shin D.: On the solutions in L 2(0,∞) of the self-adjoint differential equation u (n) = lu,I(l) = 0. Doklady. Akad. Nauk SSSR 18, 519–522 (1938) (Russian)Google Scholar
  35. 35.
    Shin D.: On quasi-differential operators in Hilbert space. Doklady. Akad. Nauk SSSR 18, 523–526 (1938) (Russian)Google Scholar
  36. 36.
    Shin D.: On the solutions of linear quasi-differential equations of the nth order. Mat. Sbor. 7(49), 479–532 (1940) (Russian)Google Scholar
  37. 37.
    Shin D.: On quasi-differential operators in Hilbert space. Mat. Sbor. 13(55), 39–70 (1943) (Russian)Google Scholar
  38. 38.
    Sun J.: On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices. Acta Math. Sin. 2(2), 152–167 (1986)zbMATHCrossRefGoogle Scholar
  39. 39.
    Shang Z.J., Zhu R.Y.: The domains of self-adjoint extensions of ordinary symmetric differential operators over (−∞ ,∞). J. Inner Mongolia Univ. (Natural Science) 17(1), 17–28 (1986) (in Chinese)MathSciNetGoogle Scholar
  40. 40.
    Sun J., Wang A., Zettl A.: Continuous spectrum and square-integrable solutions of differential operators with intermediate deficiency index. J. Funct. Anal. 255, 3229–3248 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Wang A., Sun J.: J-Self-adjoint extensions of J-symmetric operators with interior singular points. J. Nanjing Univ. Sci. Technol. (Natural Science) 31(6), 673–678 (2007) (in Chinese)Google Scholar
  42. 42.
    Wang A., Sun J., Zettl A.: Characterization of domains of self-adjoint ordinary differential operators. J. Differ. Equ. 246, 1600–1622 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Weidmann, J.: Spectral theory of ordinary differential operators. In: Lecture Notes in Mathematics, vol. 1258. Springer-Verlag, Berlin (1987)Google Scholar
  44. 44.
    Windau W.: On linear differential equations of the fourth order with singularities, and the related representations of arbitrary functions. Math. Ann. 83, 256–279 (1921) (German)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Zettl A.: Formally self-adjoint quasi-differential operators. Rocky Mt. J. Math. 5, 453–474 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Zettl, A.: Sturm–Liouville theory. In: Mathematical Surveys and Monographs, vol. 121. American Mathematical Society (2005)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Xiaoling Hao
    • 1
  • Jiong Sun
    • 1
  • Aiping Wang
    • 2
  • Anton Zettl
    • 3
  1. 1.Mathematics DepartmentInner Mongolia UniversityHohhotChina
  2. 2.Mathematics DepartmentTianjin University of Science and TechnologyTianjinChina
  3. 3.Mathematics DepartmentNorthern Illinois UniversityDeKalbUSA

Personalised recommendations