On self-adjoint boundary conditions for singular Sturm-Liouville operators bounded from below. (English) Zbl 1458.34143

In his famous paper on second-order semibounded ordinary differential operators \(L\) [Math. Ann. 122, 343–368 (1951; Zbl 0044.31201)], F. Rellich combined the observations that non-oscillatory equations have a distinguished fundamental system of solutions and that semibounded operators have a distinguished self-adjoint realization. Loosely speaking, he showed that the domain of definition of the Friedrichs extension of \(L\) consists of those functions in the domain of definition of the maximal operator that behave like the principle solution at the singular endpoints. (See eq. (4.17) of the authors’ Theorem 4.5 for the precise result.)
In the course of an approximation argument Rellich had to use stronger regularity conditions on the coefficients of \(L\) than the (now) usual minimal ones, and this was remedied much later in a dissertation of R. Rosenberger [J. Lond. Math. Soc., II. Ser. 31, 501–510 (1985; Zbl 0615.34019)]. A different proof was given by H. D. Niessen and A. Zettl [Proc. Lond. Math. Soc. (3) 64, No. 3, 545–578 (1992; Zbl 0768.34015)], showing that a singular non-oscillatory limit-circle endpoint can be transformed into a regular one. This motivated the authors of the present paper to give a proof solely by working with the principal and non-principle solutions and the variation-of-constants formula. As mentioned before, (4.17) is Rellich’s result; the correlated relation (4.18), which was not previously brought to the fore in this generality, is, as the authors put it, “the new wrinkle in this context”. The reviewer also wishes to direct the reader’s attention to the long final section where (4.17–18) are applied to the equations of Bessel, Legendre and Laguerre in the course of which errors in some standard handbooks on special functions are pointed out.


34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B09 Boundary eigenvalue problems for ordinary differential equations
34B24 Sturm-Liouville theory
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34B20 Weyl theory and its generalizations for ordinary differential equations
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)


Full Text: DOI arXiv


[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1972), Dover: Dover New York · Zbl 0543.33001
[2] Akhiezer, N. I.; Glazman, I. M., Theory of Linear Operators in Hilbert Space, vol. II (1981), Pitman: Pitman Boston · Zbl 0467.47001
[3] Anan’eva, A. Yu.; Budyka, V. S., On the spectral theory of the Bessel operator on a finite interval and the half-line, Differ. Equ., 52, 1517-1522 (2016) · Zbl 06695433
[4] Ananieva, A. Yu.; Budyika, V. S., To the spectral theory of the Bessel operator on finite interval and half-line, J. Math. Sci., 211, 624-645 (2015) · Zbl 1354.47031
[5] Atkinson, F. V.; Fulton, C. T., Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity. I, Proc. R. Soc. Edinb., 99A, 51-70 (1994) · Zbl 0584.34015
[6] Balinsky, A. A.; Evans, W. D.; Lewis, R. T., The Analysis and Geometry of Hardy’s Inequality, Universitext (2015), Springer · Zbl 1332.26005
[7] Behrndt, J.; Hassi, S.; De Snoo, H., Boundary Value Problems, Weyl Functions, and Differential Operators, Monographs in Math., Vol. 108 (2020), Birkhäuser: Birkhäuser Springer · Zbl 1457.47001
[8] Berg, C., Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity, (Mateu, J.; Porcu, E., Positive Definite Functions. From Schoenberg to Space-Time Challenges (2008), University Jaume I: University Jaume I Castellon, Spain), 15-45
[9] C. Berg, private communication, October 22, 2019.
[10] Berg, C.; Koumandos, S.; Pedersen, H. L., Nielsen’s beta function and some infinitely divisible distributions, Math. Nachr. (2020), in press
[11] Brüning, J., Heat equation asymptotics for singular Sturm-Liouville operators, Math. Ann., 268, 173-196 (1984) · Zbl 0528.34020
[12] Bruneau, L.; Dereziński, J.; Georgescu, V., Homogeneous Schrödinger operators on half-line, Ann. Henri Poincaré, 12, 547-590 (2011) · Zbl 1226.47049
[13] Bulla, W.; Gesztesy, F., Deficiency indices and singular boundary conditions in quantum mechanics, J. Math. Phys., 26, 2520-2528 (1985) · Zbl 0583.35029
[14] Clark, S.; Gesztesy, F.; Nichols, R., Principal solutions revisited, (Bernido, C. C.; Carpio-Bernido, M. V.; Grothaus, M.; Kuna, T.; Oliveira, M. J.; da Silva, J. L., Stochastic and Infinite Dimensional Analysis. Stochastic and Infinite Dimensional Analysis, Trends in Mathematics (2016), Birkhäuser, Springer), 85-117
[15] Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1985), Krieger Publ.: Krieger Publ. Malabar, FL
[16] Dereziński, J.; Richard, S., On radial Schrödinger operators with a Coulomb potential, Ann. Henri Poincaré, 19, 2869-2917 (2018) · Zbl 1428.81077
[17] Derkach, V., Extensions of Laguerre operators in indefinite inner product spaces, Math. Notes, 63, 449-459 (1998) · Zbl 0923.47018
[18] Donoghue, W. F., Monotone Matrix Functions and Analytic Continuation (1974), Springer: Springer Berlin · Zbl 0278.30004
[19] Dunford, N.; Schwartz, J. T., Linear Operators. Part II: Spectral Theory (1988), Wiley, Interscience: Wiley, Interscience New York
[20] Eckhardt, J.; Gesztesy, F.; Nichols, R.; Teschl, G., Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials, Opusc. Math., 33, 467-563 (2013) · Zbl 1283.34022
[21] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F., Higher Transcendental Functions, vol. I (1953), McGraw-Hill: McGraw-Hill New York · Zbl 0051.30303
[22] Evans, W. D.; Lewis, R.t., On the Rellich inequality with magnetic potentials, Math. Z., 251, 267-284 (2005) · Zbl 1090.35153
[23] Everitt, W. N., A catalogue of Sturm-Liouville differential equations, (Amrein, W. O.; Hinz, A. M.; Pearson, D. B., Sturm-Liouville Theory: Past and Present (2005), Birkhäuser: Birkhäuser Basel), 271-331 · Zbl 1088.34017
[24] Everitt, W. N.; Kalf, H., The Bessel differential equation and the Hankel transform, J. Comput. Appl. Math., 208, 3-19 (2007) · Zbl 1144.34013
[25] Everitt, W. N.; Littlejohn, L. L.; Wellman, R., The Sobolev orthogonality and spectral analysis of the Laguerre polynomials \(\{ L_n^{- k} \}\) for positive integers k, J. Comput. Appl. Math., 171, 199-234 (2004) · Zbl 1063.33009
[26] Fulton, C. T., Parametrizations of Titchmarsh’s ‘\(m(\lambda)\)’-Functions in the Limit Circle Case (1973), Technical University of Aachen: Technical University of Aachen Germany, PhD thesis
[27] Fulton, C. T., Parametrizations of Titchmarsh’s \(m(\lambda)\)-functions in the limit circle case, Transl. Am. Math. Soc., 229, 51-63 (1977) · Zbl 0358.34021
[28] Fulton, C. T., Expansions in Legendre functions, Q. J. Math. Oxford (2), 33, 215-222 (1982) · Zbl 0513.34025
[29] Fulton, C. T., Titchmarsh-Weyl m-functions for second-order Sturm-Liouville problems with two singular endpoints, Math. Nachr., 281, 1418-1475 (2008) · Zbl 1165.34011
[30] C.T. Fulton, private communication, January 22, 2020.
[31] Fulton, C. T.; Langer, H., Sturm-Liouville operators with singularities and generalized Nevanlinna functions, Complex Anal. Oper. Theory, 4, 179-243 (2010) · Zbl 1214.34022
[32] Gal, C. G., Sturm-Liouville operator with general boundary conditions, Electron. J. Differ. Equ., 2005, 120, 1-17 (2005) · Zbl 1085.34022
[33] Gesztesy, F.; Pittner, L., On the Friedrichs extension of ordinary differential operators with strongly singular potentials, Acta Phys. Austriaca, 51, 259-268 (1979)
[34] Gesztesy, F.; Pittner, L., Two-body scattering for Schrödinger operators involving zero-range interactions, Rep. Math. Phys., 19, 143-154 (1984) · Zbl 0557.47006
[35] Gesztesy, F.; Zinchenko, M., On spectral theory for Schrödinger operators with strongly singular potentials, Math. Nachr., 279, 1041-1082 (2006) · Zbl 1108.34063
[36] Gitman, D. M.; Tyutin, I. V.; Voronov, B. L., Self-Adjoint Extensions in Quantum Mechanics. General Theory and Applications to Schrödinger and Dirac Equations with Singular Potentials, Progress in Math. Phys., vol. 62 (2012), Birkhäuser, Springer: Birkhäuser, Springer New York · Zbl 1263.81002
[37] Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products (1980), Academic Press · Zbl 0521.33001
[38] Hartman, P., Ordinary Differential Equations (2002), SIAM: SIAM Philadelphia · Zbl 0125.32102
[39] Hartman, P.; Wintner, A., On the assignment of asymptotic values for the solutions of linear differential equations of second order, Am. J. Math., 77, 475-483 (1955) · Zbl 0064.33205
[40] Jörgens, K.; Rellich, F., Eigenwerttheorie Gewöhnlicher Differentialgleichungen (1976), Springer-Verlag: Springer-Verlag Berlin · Zbl 0347.34013
[41] Kalf, H., On the characterization of the Friedrichs extension of ordinary or elliptic differential operators with a strongly singular potential, J. Funct. Anal., 10, 230-250 (1972) · Zbl 0237.35026
[42] Kalf, H., A characterization of the Friedrichs extension of Sturm-Liouville operators, J. London Math. Soc. (2), 17, 511-521 (1978) · Zbl 0406.34029
[43] Kalf, H., Ernst Mohrs Version der Weylschen Theorie der Sturm-Liouville-Operatoren, (Sitzber. Berliner Math. Ges., Jahrgänge 1988-1922 (1992)), 221-234 · Zbl 0849.46001
[44] Kaper, H. G.; Kwong, M. K.; Zettl, A., Characterizations of the Friedrichs extensions of singular Sturm-Liouville expressions, SIAM J. Math. Anal., 17, 772-777 (1986) · Zbl 0608.34022
[45] Kirsten, K.; Loya, P.; Park, J., The very unusual properties of the resolvent, heat kernel, and zeta function for the operator \(- d^2 / d r^2 - 1 /(4 r^2)\), J. Math. Phys., 47, Article 043506 pp. (2006) · Zbl 1111.58025
[46] Kodaira, K., The eigenvalue problem for ordinary differential equations of the second order and Heisenberg’s theory of S-matrices, Am. J. Math., 71, 921-945 (1949) · Zbl 0035.27101
[47] Kostenko, A.; Sakhnovich, A.; Teschl, G., Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials, Int. Math. Res. Not., 8, 1699-1747 (2012) · Zbl 1248.34027
[48] Kostenko, A.; Teschl, G., On the singular Weyl-Titchmarsh function of perturbed spherical Schrödinger operators, J. Differ. Equ., 250, 3701-3739 (2011) · Zbl 1219.34037
[49] Krall, A. M., Laguerre polynomial expansions in indefinite inner product spaces, J. Comput. Appl. Math., 70, 267-279 (1979) · Zbl 0416.46012
[50] Leighton, W.; Morse, M., Singular quadratic functionals, Transl. Am. Math. Soc., 40, 252-286 (1936) · JFM 62.0577.02
[51] Levin, B. Ja., Distribution of Zeros of Entire Functions, Transl. of Math. Monographs, vol. 5 (1980), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
[52] Littlejohn, L. L.; Zettl, A., The Legendre equation and its self-adjoint operators, Electron. J. Differ. Equ., 2011, 69, 1-33 (2011) · Zbl 1417.34206
[53] Magnus, W.; Oberhettinger, F.; Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics, Grundlehren, vol. 52 (1966), Springer: Springer Berlin · Zbl 0143.08502
[54] Marletta, M.; Zettl, A., The Friedrichs extension of singular differential operators, J. Differ. Equ., 160, 404-421 (2000) · Zbl 0954.47012
[55] Mohr, E., Eine Bemerkung zur Weylschen Theorie vom Grenzkreis- und Grenzpunktfall, Ann. Mat. Pura Appl., 129, 161-199 (1981) · Zbl 0501.34021
[56] Naimark, M. A., Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space (1968), Ungar Publishing: Ungar Publishing New York, Transl. by E.R. Dawson, Engl. translation edited by W.N. Everitt · Zbl 0227.34020
[57] Narnhofer, H., Quantum theory for \(1 / r^2\)-potentials, Acta Phys. Austriaca, 40, 306-322 (1974)
[58] Niessen, H.-D.; Zettl, A., Singular Sturm-Liouville problems: the Friedrichs extension and comparison of eigenvalues, Proc. London Math. Soc. (3), 64, 545-578 (1992) · Zbl 0768.34015
[59] (Olver, F. W.J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions (2010), National Institute of Standards and Technology (NIST), U.S. Dept. of Commerce, and Cambridge Univ. Press) · Zbl 1198.00002
[60] Pearson, D. B., Quantum Scattering and Spectral Theory (1988), Academic Press: Academic Press London · Zbl 0673.47011
[61] Pick, S., Hamiltonians with \(x^{- 2}\)-like singularity, J. Math. Phys., 18, 118-119 (1977)
[62] Pick, S., Singular potentials and perturbation theory, Acta Phys. Slovaca, 29, 25-30 (1979)
[63] Rellich, F., Die zulässigen Randbedingungen bei den singulären Eigenwertproblemen der mathematischen Physik. (Gewöhnliche Differentialgleichungen zweiter Ordnung), Math. Z., 49, 702-723 (1943/1944) · Zbl 0028.40803
[64] Rellich, F., Halbbeschränkte gewöhnliche Differentialoperatoren zweiter Ordnung, Math. Ann., 122, 343-368 (1951), (in German) · Zbl 0044.31201
[65] Rosenberger, R., A new characterization of the Friedrichs extension of semibounded Sturm-Liouville operators, J. London Math. Soc. (2), 31, 501-510 (1985) · Zbl 0615.34019
[66] Szmytkowski, R., Erratum to “Formulas and Theorems for the Special Functions of Mathematical Physics” by W. Magnus, F. Oberhettinger, R.P. Soni, Math. Comput., 82, 1709-1710 (2013), by · Zbl 1273.33002
[67] Temme, N. M., Special Functions. An Introduction to the Classical Functions of Mathematical Physics (1996), Wiley: Wiley New York · Zbl 0856.33001
[68] Teschl, G., Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators, Graduate Studies in Math., vol. 157 (2014), Amer. Math. Soc.: Amer. Math. Soc. RI · Zbl 1342.81003
[69] Titchmarsh, E. C., Eigenfunction Expansions Associated with Second-Order Differential Equations, Part I (1962), Oxford University Press: Oxford University Press Oxford · Zbl 0099.05201
[70] van Haeringen, H.; Kok, L. P., Math. Comput., 41, 775-780 (1983)
[71] Weidmann, J., Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, vol. 68 (1980), Springer: Springer New York
[72] Weidmann, J., Lineare Operatoren in Hilberträumen. Teil II: Anwendungen (2003), Teubner: Teubner Stuttgart · Zbl 1025.47001
[73] Yao, S.; Sun, J.; Zettl, A., The Sturm-Liouville Friedrichs extension, Appl. Math., 60, 299-320 (2015) · Zbl 1363.47039
[74] Zettl, A., Sturm-Liouville Theory, Mathematical Surveys and Monographs, vol. 121 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1074.34030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.