## Survey article: Self-adjoint ordinary differential operators and their spectrum.(English)Zbl 1369.47057

Summary: We survey the theory of ordinary self-adjoint differential operators in Hilbert space and their spectrum. Such an operator is generated by a symmetric differential expression and a boundary condition. We discuss the very general modern theory of these symmetric expressions which enlarges the class of these expressions by many dimensions and eliminates the smoothness assumptions required in the classical case as given, e.g., in the celebrated books by E. A. Coddington and N. Levinson [Theory of ordinary differential equations. Toronto, London: McGill-Hill Book Company, Inc (1955; Zbl 0064.33002)] and N. Dunford and J. T. Schwartz [Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space. London: Interscience Publishers, a division of John Wiley and Sons 1963 (1963; Zbl 0128.34803)]. The boundary conditions are characterized in terms of square-integrable solutions for a real value of the spectral parameter, and this characterization is used to obtain information about the spectrum. Many of these characterizations are quite recent and widely scattered in the literature, some are new. A comprehensive review of the deficiency index (which determines the number of independent boundary conditions required in the singular case) is also given for an expression $$M$$ and for its powers. Using the modern theory mentioned above, these powers can be constructed without any smoothness conditions on the coefficients.

### MSC:

 47E05 General theory of ordinary differential operators 47-02 Research exposition (monographs, survey articles) pertaining to operator theory

### Keywords:

 [1] N.I. Akhiezer and I.M. Glazman, Theory of linear operators in Hilbert space , II, Ungar, New York, 1963. · Zbl 0098.30702 [2] J. Ao, J. Sun and A. Zettl, Matrix representations of fourth order boundary value problems with finite spectrum , Linear Alg. Appl. 436 (2012), 2359-2365. · Zbl 1244.34019 [3] —-, Equivalence of fourth order boundary value problems and matrix eigenvalue problems , Res. Math. 63 (2013), 581-595. · Zbl 1267.34036 [4] F.V. Atkinson, Discrete and continuous boundary value problems , Academic Press, New York, 1964. · Zbl 0117.05806 [5] P.B. Bailey, W.N. Everitt and A. Zettl, The SLEIGN$$2$$ Sturm-Liouville code , ACM TOMS, Trans. Math. Software 21 (2001), 143-192. · Zbl 1070.65576 [6] J.H. Barrett, Oscillation theory of ordinary linear differential equations , Adv. Math. (1969), 415-509. · Zbl 0213.10801 [7] H. Behncke, Spectral theory of higher order differential operators , Proc. Lond. Math. Soc. 3 , (2006), 139-160. · Zbl 1099.34073 [8] —-, Spectral analysis of fourth order operators II, Math. Nach. (2007). [9] H. Behncke and D.B. Hinton, Eigenfunctions, deficiency indices and spectra of odd order differential operators , Proc. Lond. Math. Soc. (2008). · Zbl 1158.34050 [10] —-, Spectral theory of higher order operators II, pre-print. [11] Z.J. Cao, On self-adjoint domains of second order differential operators in limit-circle case , Acta Math. Sinica 1 (1985), 175-180. [12] —-, On self-adjoint extensions in the limit-circle case of differential operators of order $$n$$ , Acta Math. Sinica 28 (1985), 205-217 (in Chinese). · Zbl 0633.34024 [13] —-, Ordinary differential operators , Shanghai Science Tech. Press, 1987 (in Chinese). [14] —-, Analytic descriptions of self-adjoint ordinary differential operators , J. Inner Mongolia Univ. 18 (1987), 393-401 (in Chinese). · Zbl 1332.47018 [15] Z.J. Cao and J.L. Liu, On the deficiency index theory for singular symmetric differential operators , Adv. Math. 12 (1983), 161-178 (in Chinese). [16] Z.J. Cao and J. Sun, Self-adjoint operators generated by symmetric quasi-differential expressions , Acta Sci. Natur., Univ. NeiMongol (1986), 7-15 (in Chinese). [17] —-, Collection of differential operators , Inner Mongolia Univ. Press, Hohhot, 1992. [18] —-, On completion and prolongation of classical Sturm-Liouville theory , Adv. Math. (China) 22 (1993), 97-117 (in Chinese). · Zbl 0776.34020 [19] Z.J. Cao, J. Sun and D.E. Edmunds, On self-adjointness of the product of two $$2$$-order differential operators , Acta Math. Sinica, English series, 15 (1999), 375-386. · Zbl 0934.34017 [20] J. Chaudhuri and W.N. Everitt, On the square of formally self-adjoint differential expressions , J. Lond. Math. Soc. 1 (1969), 661-673. · Zbl 0191.38402 [21] E.A. Coddington and N. Levinson, Theory of ordinary differential equations , McGraw-Hill, New York, 1955. · Zbl 0064.33002 [22] E.A. Coddington and A. Zettl, Hermitian and anti-hermitian properties of Green’s matrices , Pac. J. Math. (1966), 451-454. · Zbl 0152.08501 [23] R. Del Rio, On a problem of P. Hartman and A. Wintner , Aport. Matem. 16 (1995), 119-123. [24] N. Dunford and J.T. Schwartz, Linear operators , II, Wiley, New York, 1963. · Zbl 0084.10402 [25] M.S.P. Eastham, The limit-$$3$$ case of self-adjoint differential expressions of the fourth order with oscillating coefficients , J. Lond. Math. Soc. 8 (1974), 427-437. · Zbl 0288.34012 [26] M.S.P. Eastham and A. Zettl, Second order differential expressions whose squares are limit-$$3$$ , Proc. Roy. Soc. Edinb. 76 (1977), 233-238. · Zbl 0345.34010 [27] W.D. Evans, M.K. Kwong and A. Zettl, Lower bounds for the spectrum of ordinary differential operators , J. Diff. Equat. 48 (1983), 123-155. · Zbl 0573.34020 [28] W.D. Evans and E.I. Sobhy, Boundary conditions for general ordinary differential operators and their adjoints , Proc. Roy. Soc. Edinb. 114 (1990), 99-117. · Zbl 0704.34034 [29] W.D. Evans and A. Zettl, On the deficiency indices of powers of real 2nth order symmetric differential expressions , J. Lond. Math. Soc. 2 (1976), 543-556. · Zbl 0335.34007 [30] —-, Levinson’s limit-point criterion and powers , J. Funct.l Anal. Appl. 62 (1978), 629-639. · Zbl 0384.34017 [31] W.N. Everitt, A note on the self-adjoint domains of $$2$$nth-order differential equations , Quart. J. Math. 14 (1963), 41-45. · Zbl 0115.30003 [32] —-, Integrable-square solutions of ordinary differential equations (III), Quart. J. Math. 14 (1963), 170-180. · Zbl 0123.05001 [33] —-, Singular differential equations I: The even case , Math. Ann. 156 (1964), 9-24. · Zbl 0145.10902 [34] —-, Singular differential equations II: Some self-adjoint even case , Quart. J. Math. 18 (1967), 13-32. · Zbl 0153.40402 [35] W.N. Everitt and M. Giertz, On some properties of the powers of a formally self-adjoint differential expression , Proc. Lond. Math. Soc. 74 (1972), 149-170. · Zbl 0243.34046 [36] —-, On the deficiency indices of powers of formally symmetric differential expression , Lect. Notes Math. 448 , Springer Verlag, New York, 1975. · Zbl 0315.34010 [37] —-, On the deficiency indices of powers of formally symmetric differential expressions , Lect. Notes Math. 448 , Springer Verlag, New York, 1975. · Zbl 0315.34010 [38] W.N. Everitt and V.K. Kumar, On the Titchmarsh-Weyl theory of ordinary symmetric differential expressions I: The general theory , Nieuw Arch. Wisk. 34 (1976), 1-48. · Zbl 0339.34019 [39] —-, On the Titchmarsh-Weyl theory of ordinary symmetric differential expressions II: The odd order case , Nieuw Arch. Wisk. 34 (1976), 109-145. · Zbl 0338.34012 [40] W.N. Everitt and L. Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators , Math. Surv. Mono. 61 , American Mathematical Society, 1999. · Zbl 0909.34001 [41] —-, Complex symplectic geometry with applications to ordinary differential operators , Trans. Amer. Math. Soc. 351 (1999), 4905-4945. · Zbl 0936.34005 [42] W.N. Everitt and F. Neuman, A concept of adjointness and symmetry of differential expressions based on the generalized Lagrange identity and Green’s formula , Lect. Notes Math. 1032 , Springer-Verlag, Berlin, 1983. · Zbl 0531.34002 [43] W.N. Everitt and R. Race, On necessary and sufficient conditions for caratheodory type solutions of ordinary differential equations , Quaest. Math. (1978), 507-512. · Zbl 0392.34002 [44] W.N. Everitt and A. Zettl, The number of integrable-square solutions of products of differential expressions , Proc. Roy. Soc. Edinb. 76 (1977), 215-226. · Zbl 0347.34007 [45] —-, Generalized symmetric ordinary differential expressions I: The general theory , Nieuw Arch. Wisk. 27 (1979), 363-397. · Zbl 0451.34009 [46] W.N. Everitt and A. Zettl, Sturm-Liouville differential operators in direct sum spaces , Rocky Mountain J. Mathematics (1986), 497-516. · Zbl 0624.34020 [47] —-, Differential operators generated by a countable number of quasi-differential expressions on the line , Proc. Lond. Math. Soc. 64 (1992), 524-544. · Zbl 0723.34022 [48] H. Frentzen, Equivalence, adjoints and symmetry of quasi-differential expressions with matrix-valued coefficients and polynomials in them , Proc. Roy. Soc. Edinb. 92 (1982), 123-146. · Zbl 0506.34020 [49] K. Friedrichs, Spektraltheorie halbbeschränkter operatoren und anwendungen auf die spektralzerlegung von differentialoperatoren I, II, Math. Ann. 109 (1933), 467-487; corrections, ibid, 110 (1934), 777-779. · Zbl 0008.39203 [50] —-, Über die ausgezeichnete randbedingung in der spectraltheorie gewöhnlicher differential operatoren zweiter ordnung , Math. Ann. 112 (1935), 1-23. [51] S.Z. Fu, On the self-adjoint extensions of symmetric ordinary differential operators in direct sum spaces , J. Diff. Equat. 100 (1992), 269-291. · Zbl 0768.34052 [52] R.C. Gilbert, Asymptotic formulas for solutions of a singular ordinary differential equation , Proc. Roy. Soc. Edinb. 81 (1978), 57-70. · Zbl 0414.34060 [53] —-, A class of symmetric ordinary differential operators whose deficeinecy numbers differ by an integer , Proc. Roy. Soc. Edinb. 82 (1979), 117-134. · Zbl 0406.34033 [54] I.M. Glazman, On the theory of singular differential operators , Uspek. Math. Nauk. (1950), 102-135; Amer. Math. Soc. Trans. 4 (1962), 331-372. · Zbl 0052.34304 [55] —-, Direct methods of qualitative spectral analysis of singular differential operators , Israel Program for Scientific Translation, 1965. [56] A. Gorianov, V. Mikhailets and K. Pankrashkin, Formally self-adjoint quasi-differential operators and boundary value problems , arXiv.12051810v3. [57] I. Halperin, Closures and adjoints of linear differential operators , Annals Math. (1937), 880-919. · Zbl 0018.36603 [58] X. Hao, J. Sun, A. Wang and A. Zettl, Characterization of domains of self-adjoint ordinary differential operators II, Res. Math. 61 (2012), 255-281. · Zbl 1290.47046 [59] X. Hao, J. Sun and A. Zettl, Real-parameter square-integrable solutions and the spectrum of differential operators , J. Math. Anal. Appl. 376 (2011), 696-712. · Zbl 1210.47087 [60] —-, The spectrum of differential operators and square-integrable solutions , J. Funct. Anal. 262 (2012), 1630-1644. · Zbl 1271.34084 [61] —-, Canonical forms of self-adjoint boundary conditions for differential operators of order four , J. Math. Anal. Appl. 387 (2012), 1176-1187. · Zbl 1245.34018 [62] —-, Fourth order canonical forms of singular self-adjoint singular boundary conditions , Linear Alg. Appl. 437 (2012), 899-916. · Zbl 1254.34041 [63] P. Hartman and A. Winter, A separation theorem for continuous spectra , Amer. J. Math. 71 (1949), 650-662. · Zbl 0033.27301 [64] D.B. Hinton, Disconjugacy properties of a system of differential equations , J. Diff. Equat. (1966), 420-437. · Zbl 0161.27904 [65] —-, Limit-point criteria for differential equations , Canad. J. Math. 24 (1972), 293-305. · Zbl 0229.34024 [66] P.D. Hislop and I.M. Sigal, Introduction to spectral theory, with applications to Schrōdinger operators , Appl. Math. Sci. 113 , Springer-Verlag, New York, 1966. [67] R.M. Kauffman, Polynomials and the limit-point condition , Trans. Amer. Math. Soc. 201 , (1975), 347-366. · Zbl 0274.34019 [68] —-, A rule relating the deficiency index of $$L^{j}$$ to that of $$L^{k}$$ , Proc. Roy. Soc. Edinb. 74 (1976), 115-118. · Zbl 0333.34003 [69] R.M. Kauffman, T.T. Read and A. Zettl, The defiency index problem for powers of ordinary differential expressions , Lect. Notes Math. 621 , Springer Verlag, New York, 1977. · Zbl 0367.34014 [70] K. Kodaira, On ordinary differential equations of any even order and the corresponding eigenfunction expansions , Amer. J. Math. (1950), 502-544. · Zbl 0054.03903 [71] V.I. Kogan and F.S. Rofe-Beketov, On the question of deficiency indices of differential operators with complex coefficients , Proc. Roy. Soc. Edinb. 74 (1973), 281-298. · Zbl 0372.47025 [72] Q. Kong, H. Volkmer and A. Zettl, Matrix representations of Sturm-Liouville problems , Res. Math. 54 (2009), 103-116. · Zbl 1185.34032 [73] M.K. Kwong and A. Zettl, Discreteness conditions for the spectrum of ordinary differential operators , J. Diff. Equat. 40 (1981), 53-70. · Zbl 0486.34018 [74] W.M. Li, The high order differential operators in direct sum spaces , J. Diff. Equat. 84 (1990), 273-289. · Zbl 0726.47033 [75] —-, The domains of self-adjoint enxtensions of symmetric ordinary differential operators with two singular points , J. Inner Mongolia University (Natural Science) 20 (1989), in Chinese. [76] L.L. Littlejohn and A. Zettl, The Legendre equation and its self-adjoint operators , Electr. J. Differ. Equat. 2011 (69), 1-33. · Zbl 1417.34206 [77] N. Macrae and John von Neumann, American Mathematical Society, 1992. [78] M. Marletta and A. Zettl, The Friedrichs extension of singular differential operators , J. Differ. Equat. 160 (2000), 404-421. · Zbl 0954.47012 [79] J.B. McLeod, The number of integrable-square solutions of ordinary differential equations , Quart. J. Math (1966), 285-290. · Zbl 0148.06003 [80] M. Möller, On the unboundedness below of the Sturm-Liouville operator , Proc. Roy. Soc. Edinb. 129 (1999), 1011-1015. · Zbl 0947.34010 [81] M. Möller and A. Zettl, Weighted norm-inequalities for quasi-derivatives , Res. Math. 24 (1993), 153-160. · Zbl 0787.34016 [82] —-, Symmetric differential operators and their Friedrichs extension , J. Diff. Equat. 115 (1995), 50-69. · Zbl 0817.34048 [83] —-, Semi-boundedness of ordinary differential operators , J. Diff. Equat. 115 (1995), 24-49. · Zbl 0817.34047 [84] M.A. Naimark, Linear differential operators , Ungar, New York, 1968. · Zbl 0227.34020 [85] J.W. Neuberger, Concerning boundary value problems , Pacific J. Math. 10 (1960), 1385-1392. · Zbl 0097.07103 [86] H.-D. Niessen and A. Zettl, The Friedrichs extension of regular ordinary differential operators , Proc. Roy. Soc. Edinb. 114 (1990), 229-236. · Zbl 0712.34020 [87] —-, Singular Sturm-Liouville problems : The Friedrichs extension and comparison of eigenvalues , Proc. Lond. Math. Soc. 64 (1992), 545-578. · Zbl 0768.34015 [88] S.A. Orlov, On the deficiency indices of differential operators , Dokl. Akad. Nauk 92 (1953), 483-486. · Zbl 0161.27904 [89] J. Qi and S. Chen, On an open problem of J. Weidmann : Essential spectra and square-integrable solutions , Proc. Roy. Soc. Edinb. 141 (2011), 417-430. · Zbl 1237.47046 [90] T.T. Read, Sequences of deficiency indices , Proc. Roy. Soc. Edinb. (1976), 157-164. · Zbl 0334.34018 [91] W.T. Reid, Oscillation criteria for self-adjoint differential systems , Trans. Amer. Math. Soc. (1961), 91-106. · Zbl 0114.29202 [92] C. Remling, Essential spectrum and $$L_{2}$$-solutions of one dimensional Schrödinger operators , Proc. Amer. Math. Soc. (1996), 2097-2100. · Zbl 0858.34073 [93] F.S. Rofe-Beketov and A. Kholkin, Spectral analysis of differential operators, Interplay between spectral and oscillatory properties , World Sci. Mono. Math. 7 (2005). · Zbl 1090.47030 [94] Z.J. Shang, On $$J$$-selfadjoint extensions of $$J$$-symmetric ordinary differential operators , J. Diff. Equat. 73 (1988), 153-177. · Zbl 0664.34037 [95] Z.J. Shang and R.Y. Zhu, The domains of self-adjoint extensions of ordinary symmetric differential operators over $$(-\infty ,\infty )$$ , J. Inner Mongolia Univ. 17 (1986), 17-28. [96] D. Shin, On the solutions in $$L^{2}(0,\infty )$$ of the self-adjoint differential equation $$u^{(n)}=lu,I(l)=0$$ , Dokl. Akad. Nauk 18 (1938), 519-522. · Zbl 0019.21403 [97] —-, On quasi-differential operators in Hilbert space , Dokl. Akad. Nauk 18 (1938), 523-526. · Zbl 0019.21404 [98] —-, On the solutions of linear quasi-differential equations of the $$nth$$ order , Mat. Sbor. 7 (1940), 479-532. [99] —-, On quasi-differential operators in Hilbert space , Mat. Sbor. 13 (1943), 39-70. · Zbl 0061.26108 [100] M.H. Stone, Linear transformations in Hilbert space and their applications in analysis , Amer. Math. Soc. Colloq. 15 (1932). · Zbl 0005.40003 [101] J. Sun, On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices , Acta Math. Sinica 2 (1986), 152-167. · Zbl 0615.34015 [102] J. Sun, A. Wang and A. Zettl, Continuous spectrum and square-integrable solutions of differential operators with intermediate deficiency index , J. Funct. Anal. 255 (2008), 3229-3248. · Zbl 1165.34052 [103] J. Sun and Z. Wang, The spectrum analysis of linear oparator , Science Press, Beijing, 2005. [104] E.C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations $$[J]$$ , Part I, Second edition, Oxford, 1962. · Zbl 0099.05201 [105] P. Walker, A vector-matrix formulation for formally symmetric ordinary differential equations with applications to solutions of integrable-square , J. Lond. Math. Soc. (1974), 151-159. · Zbl 0308.34011 [106] A. Wang, J. Ridenhour and A. Zettl, Construction of regular and singular Green’s functions , Proc. Roy. Soc. Edinb. 142 (2012), 171-198. · Zbl 1251.34043 [107] A. Wang, J. Sun and A. Zettl, Characterization of domains of self-adjoint ordinary differential operators , J. Diff. Equat. 246 (2009), 1600-1622. · Zbl 1169.47033 [108] —-, The classification of self-adjoint boundary conditions : Separated, coupled, and mixed , J. Funct. Anal. (2008), 1554-1573. · Zbl 1170.34017 [109] —-, The classification of self-adjoint boundary conditions of differential operators with two singular endpoints , J. Math. Anal. Appl. 378 (2011), 493-506. · Zbl 1221.34048 [110] A. Wang and A. Zettl, The Green’s function of two-interval Sturm-Liouville problems , Electr. J. Diff. Equat. 76 (2013), 1-13. · Zbl 1290.34038 [111] —-, Self-adjoint Sturm-Liouville problems with discontinuous boundary conditions , Math. Meth. Appl. Anal. 22 (2015), 37-66. · Zbl 1331.34040 [112] W.Y. Wang and J. Sun, Complex $$J$$-symplectic geometry characterization for $$J$$-symmetric extensions of $$J$$-symmetric differential operators , Adv. Math. 32 (2003), 481-484. [113] —-, The self-adjoint extensions of singular differential operators with a real regularity point , J. Spectral Math. Appl. (2006). [114] —-, $$J$$-self-adjoint extensions of $$J$$-symmetric operators with interior singular points , J. Nanjing Univ. Sci. Tech. 31 (2007), 673-678. [115] J. Weidmann, Linear operators in Hilbert spaces , Grad. Texts Math., Springer-Verlag, Berlin, 1980. [116] —-, Spectral theory of ordinary differential operators , Lect. Notes Math. 1258 , Springer-Verlag, Berlin, 1987. [117] H. Weyl, Über Gewönliche Differentialgleichungen mit Singularitäten und die zugehörige Entwicklung willkürlicher Funktionen , Math. Ann. (1910), 220-269. · JFM 41.0343.01 [118] W. Windau, On linear differential equations of the fourth order with singularities, and the related representations of arbitrary functions , Math. Annal. 83 (1921), 256-279. [119] S. Yao, J. Sun and A. Zettl, Self-adjoint domains, symplectic geometry and limit-circle solutions , J. Math. Anal. Appl. 397 (2013), 644-657. · Zbl 1262.34030 [120] —-, Self-adjoint domains, symplectic geometry, and limit-circle solutions , J. Math. Anal. Appl. 386 (2013), 644-657. · Zbl 1262.34030 [121] —-, Symplectic geometry and dissipative differential operators , J. Math. Anal. Appl. 419 (2014), 218-230. · Zbl 1312.47047 [122] A. Zettl, The limit-point and limit-circle cases for polynomials in a differential expression , Proc. Roy. Soc. Edinb. 72 (1974), 219-224. · Zbl 0334.34025 [123] —-, Adjoint linear differential operators , Proc. Amer. Math. Soc. 16 (1965), 1239-1241. · Zbl 0139.03801 [124] A. Zettl, Formally self-adjoint quasi-differential operators , Rocky Mountain J. Mathematics 5 (1975), 453-474. · Zbl 0443.34019 [125] —-, Adjoint and self-adjoint problems with interface conditions , SIAM J. Appl. Math. 16 (1968), 851-859. · Zbl 0162.11201 [126] —-, Adjointness in non-adjoint boundary value problems , SIAM J. Appl. Math. 16 (1969), 1268-1279. · Zbl 0186.41401 [127] —-, The lack of self-adjointness in three point boundary value problems , Proc. Amer. Math. Soc. 17 (1966), 368-371. · Zbl 0144.10203 [128] —-, Formally self-adjoint quasi-differential operators , Rocky Mountain J. Math. 5 (1975), 453-474. · Zbl 0443.34019 [129] —-, Powers of symmetric differential expressions without smoothness assumptions , Quaest. Math. (1976), 83-94. · Zbl 0343.34010 [130] —-, On the Friedrichs extension of singular differential operators , Comm. Appl. Anal. 2 (1998), 31-36. · Zbl 0895.34018 [131] —-, Sturm-Liouville theory , AMS Surv. Mono. 121 , American Mathematical Society, Philadelphia, 2005. · Zbl 1103.34001