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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.

MSC:

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.)

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