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Characterization of domains of self-adjoint ordinary differential operators. (English) Zbl 1169.47033
The authors give a characterization of domains for selfadjoint ordinary differential operators. They give a representation in terms of certain solutions for real λ. This leads to a classification of solutions as limit-point or limit-circle in analogy with the Weyl classification in the second-order case.

47E05Ordinary differential operators
34B20Weyl theory and its generalizations
34B24Sturm-Liouville theory
47B25Symmetric and selfadjoint operators (unbounded)
[1]Cao, Z. J.: On self-adjoint domains of second order differential operators in limit-circle case, Acta math. Sinica (N.S.) 1, No. 3, 175-180 (1985) · Zbl 0609.34036 · doi:10.1007/BF02564818
[2]Cao, Z. J.: On self-adjoint extensions in the limit-circle case of differential operators of order n, Acta math. Sinica 28, No. 2, 205-217 (1985)
[3]Cao, Z. J.; Sun, J.: Self-adjoint operators generated by symmetric quasi-differential expressions, Acta sci. Natur. univ. Neimongol 17, No. 1, 7-15 (1986)
[4]Cao, Z. J.: Ordinary differential operators, (1987)
[5]Coddington, E. A.; Levinson, N.: Theory of ordinary differential equations, (1955) · Zbl 0064.33002
[6]Dunford, N.; Schwartz, J. T.: Linear operators, vol. II, (1963)
[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, Lecture notes in math. 1032, 161-169 (1983) · Zbl 0531.34002
[8]Evans, W. D.; Sobhy, E. I.: Boundary conditions for general ordinary differential operators and their adjoints, Proc. roy. Soc. Edinburgh sect. A 114, 99-117 (1990) · Zbl 0704.34034 · doi:10.1017/S030821050002429X
[9]Everitt, W. N.; Zettl, A.: Generalized symmetric ordinary differential expressions, I: The general theory, Nieuw arch. Wiskd. (3) 27, 363-397 (1979) · Zbl 0451.34009
[10]Everitt, W. N.: A note on the self-adjoint domains of 2nth-order differential equations, Q. J. Math. (Oxford) 14, No. 2, 41-45 (1963) · Zbl 0115.30003 · doi:10.1093/qmath/14.1.41
[11]Everitt, W. N.: Integrable-square solutions of ordinary differential equations (III), Q. J. Math. (Oxford) 14, No. 2, 170-180 (1963) · Zbl 0123.05001 · doi:10.1093/qmath/14.1.170
[12]Everitt, W. N.: Singular differential equations, I: The even case, Math. ann. 156, 9-24 (1964) · Zbl 0145.10902 · doi:10.1007/BF01359977
[13]Everitt, W. N.: Singular differential equations, II: Some self-adjoint even case, Q. J. Math. (Oxford) 18, No. 2, 13-32 (1967) · Zbl 0153.40402 · doi:10.1093/qmath/18.1.13
[14]Everitt, W. N.; Kumar, V. K.: On the titchmarsh – Weyl theory of ordinary symmetric differential expressions, I: The general theory, Nieuw arch. Wiskd. 34, No. 3, 1-48 (1976) · Zbl 0339.34019
[15]Everitt, W. N.; Kumar, V. K.: On the titchmarsh – Weyl theory of ordinary symmetric differential expressions, II: The odd order case, Nieuw arch. Wiskd. 34, No. 3, 109-145 (1976) · Zbl 0338.34012
[16]Everitt, W. N.; Markus, L.: Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, Math. surveys monogr. 61 (1999) · Zbl 0909.34001
[17]Everitt, W. N.; Markus, L.: Complex symplectic geometry with applications to ordinary differential operators, Trans. amer. Math. soc. 351, No. 12, 4905-4945 (1999) · Zbl 0936.34005 · doi:10.1090/S0002-9947-99-02418-6
[18]Frentzen, H.: Equivalence, adjoints and symmetry of quasi-differential expressions with matrix-valued coefficients and polynomials in them, Proc. roy. Soc. Edinburgh sect. A 92, 123-146 (1982) · Zbl 0506.34020 · doi:10.1017/S0308210500019995
[19]Fu, S. Z.: On the self-adjoint extensions of symmetric ordinary differential operators in direct sum spaces, J. differential equations 100, No. 2, 269-291 (1992) · Zbl 0768.34052 · doi:10.1016/0022-0396(92)90115-4
[20]Kauffman, R. M.; Read, T. T.; Zettl, A.: The deficiency index problem for powers of ordinary differential expressions, Lecture notes in math. 621 (1977) · Zbl 0367.34014
[21]Li, W. M.: The high order differential operators in direct sum spaces, J. differential equations 84, No. 2, 273-289 (1990) · Zbl 0726.47033 · doi:10.1016/0022-0396(90)90079-5
[22]Möller, M.: On the unboundedness below of the Sturm – Liouville operator, Proc. roy. Soc. Edinburgh sect. A 129, 1011-1015 (1999)
[23]Möller, M.; Zettl, A.: Weighted norm-inequalities for quasi-derivatives, Results math. 24, 153-160 (1993)
[24]Möller, M.; Zettl, A.: Symmetric differential operators and their Friedrichs extension, J. differential equations 115, 50-69 (1995) · Zbl 0817.34048 · doi:10.1006/jdeq.1995.1003
[25]Möller, M.; Zettl, A.: Semi-boundedness of ordinary differential operators, J. differential equations 115, 24-49 (1995) · Zbl 0817.34047 · doi:10.1006/jdeq.1995.1002
[26]Naimark, M. A.: Linear differential operators, (1968) · Zbl 0227.34020
[27]Niessen, H. -D.; Zettl, A.: The Friedrichs extension of regular ordinary differential operators, Proc. roy. Soc. Edinburgh sect. A 114, 229-236 (1990) · Zbl 0712.34020 · doi:10.1017/S0308210500024409
[28]Orlov, S. A.: On the deficiency indices of differential operators, Dokl. akad. Nauk SSSR 92, 483-486 (1953)
[29]Shang, Z. J.: On J-selfadjoint extensions of J-symmetric ordinary differential operators, J. differential equations 73, 153-177 (1988) · Zbl 0664.34037 · doi:10.1016/0022-0396(88)90123-4
[30]Shin, D.: On the solutions in L2(0,) of the self-adjoint differential equation u(n)=lu,I(l)=0, Dokl. akad. Nauk SSSR 18, 519-522 (1938)
[31]Shin, D.: On quasi-differential operators in Hilbert space, Dokl. akad. Nauk SSSR 18, 523-526 (1938) · Zbl 0019.21404
[32]Shin, D.: On the solutions of linear quasi-differential equations of the nth order, Mat. sb. 7, No. 49, 479-532 (1940)
[33]Shin, D.: On quasi-differential operators in Hilbert space, Mat. sb. 13, No. 55, 39-70 (1943) · Zbl 0061.26108
[34]Sun, J.: On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices, Acta math. Sinica (N.S.) 2, No. 2, 152-167 (1986) · Zbl 0615.34015 · doi:10.1007/BF02564877
[35]Titchmarsh, E. C.: Eigenfunction expansions associated with second-order differential equations, part I, (1962) · Zbl 0099.05201
[36]A.P. Wang, J. Sun, The self-adjoint extensions of singular differential operators with a real regularity point, preprint
[37]Wang, W. Y.; Sun, J.: Complex J-symplectic geometry characterization for J-symmetric extensions of J-symmetric differential operators, Adv. math. 32, No. 4, 481-484 (2003)
[38]Weidmann, J.: Linear operators in Hilbert spaces, Grad texts in math. (1980)
[39]Weidmann, J.: Spectral theory of ordinary differential operators, Lecture notes in math. 1258 (1987) · Zbl 0647.47052
[40]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)
[41]Zettl, A.: Adjoint linear differential operators, Proc. amer. Math. soc. 16, 1239-1241 (1965) · Zbl 0139.03801 · doi:10.2307/2035906
[42]Zettl, A.: Formally self-adjoint quasi-differential operators, Rocky mountain J. Math. 5, 453-474 (1975) · Zbl 0443.34019 · doi:10.1216/RMJ-1975-5-3-453
[43]Zettl, A.: Sturm – Liouville theory, Math. surveys monogr. 121 (2005)