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Characterization of Domains of Self-Adjoint Ordinary Differential Operators II

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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.

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References

  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. 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)

    MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  4. Cao, Z.J.: Ordinary Differential Operators. Shanghai Science and Technology Press (1987, in Chinese)

  5. Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)

    MATH  Google Scholar 

  6. Dunford N., Schwartz J.T.: Linear Operators, vol. II. Wiley, New York (1963)

    Google Scholar 

  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)

  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)

    Article  Google Scholar 

  9. Everitt W.N., Zettl A.: Generalized symmetric ordinary differential expressions I: The general theory. Nieuw Archief voor Wiskunde 27(3), 363–397 (1979)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Everitt W.N.: Integrable-square solutions of ordinary differential equations (III). Quart. J. Math. (Oxford) 14(2), 170–180 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  12. Everitt W.N.: Singular differential equations I: the even case. Math. Ann. 156, 9–24 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  13. Everitt W.N.: Singular differential equations II: some self-adjoint even case. Quart. J. Math. (Oxford) 18(2), 13–32 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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)

  17. Everitt W.N., Markus L.: Complex symplectic geometry with applications to ordinary differential operators. Trans. Am. Math. Soc. 351(12), 4905–4945 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  22. Li W.M.: The high order differential operators in direct sum spaces. J. Differ. Equ. 84(2), 273–289 (1990)

    Article  MATH  Google Scholar 

  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. Möller M.: On the unboundedness below of the Sturm–Liouville operator. Proc. R. Soc. Edinburgh 129A, 1011–1015 (1999)

    Article  Google Scholar 

  25. Möller M., Zettl A.: Weighted norm-inequalities for quasi-derivatives. Results Math. 24, 153–160 (1993)

    MathSciNet  MATH  Google Scholar 

  26. Möller M., Zettl A.: Symmetric differential operators and their Friedrichs extension. J. Differ. Equ. 115, 50–69 (1995)

    Article  MATH  Google Scholar 

  27. Möller M., Zettl A.: Semi-boundedness of ordinary differential operators. J. Differ. Equ. 115, 24–49 (1995)

    Article  MATH  Google Scholar 

  28. Macrae, N.: Jon von Neumann. American Mathematical Society (1992)

  29. Naimark M.A.: Linear differential operators. English Transl, Ungar (1968)

    MATH  Google Scholar 

  30. Niessen H.-D., Zettl A.: The Friedrichs extension of regular ordinary differential operators. Proc. R. Soc. Edinburgh 114A, 229–236 (1990)

    Article  MathSciNet  Google Scholar 

  31. Orlov S.A.: On the deficiency indices of differential operators. Doklady. Akad. Nauk SSSR 92, 483–486 (1953) (Russian)

    MathSciNet  MATH  Google Scholar 

  32. Qi, J., Chen, S.: On an open problem of J. Weidmann: essential spectra and square-integrable solutions. Pre-print

  33. Shang Z.J.: On J-selfadjoint extensions of J-symmetric ordinary differential operators. J. Differ. Equ. 73, 153–177 (1988)

    Article  MATH  Google Scholar 

  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. Shin D.: On quasi-differential operators in Hilbert space. Doklady. Akad. Nauk SSSR 18, 523–526 (1938) (Russian)

    Google Scholar 

  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. Shin D.: On quasi-differential operators in Hilbert space. Mat. Sbor. 13(55), 39–70 (1943) (Russian)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Wang A., Sun J., Zettl A.: Characterization of domains of self-adjoint ordinary differential operators. J. Differ. Equ. 246, 1600–1622 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Weidmann, J.: Spectral theory of ordinary differential operators. In: Lecture Notes in Mathematics, vol. 1258. Springer-Verlag, Berlin (1987)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zettl A.: Formally self-adjoint quasi-differential operators. Rocky Mt. J. Math. 5, 453–474 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zettl, A.: Sturm–Liouville theory. In: Mathematical Surveys and Monographs, vol. 121. American Mathematical Society (2005)

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Authors and Affiliations

  1. Mathematics Department, Inner Mongolia University, Hohhot, 010021, China

    Xiaoling Hao & Jiong Sun

  2. Mathematics Department, Tianjin University of Science and Technology, Tianjin, 300457, China

    Aiping Wang

  3. Mathematics Department, Northern Illinois University, DeKalb, IL, 60115, USA

    Anton Zettl

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  1. Xiaoling Hao
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  2. Jiong Sun
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  3. Aiping Wang
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  4. Anton Zettl
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Correspondence to Anton Zettl.

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Hao, X., Sun, J., Wang, A. et al. Characterization of Domains of Self-Adjoint Ordinary Differential Operators II. Results. Math. 61, 255–281 (2012). https://doi.org/10.1007/s00025-011-0096-y

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  • DOI: https://doi.org/10.1007/s00025-011-0096-y

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