On self-adjoint boundary conditions for singular Sturm-Liouville operators bounded from below

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Gesztesy, Fritz¹  Littlejohn, Lance L.¹  Nichols, Roger² 

[1] Baylor Univ, Dept Math, One Bear Pl 97328, Waco, TX 76798 USA

[2] Univ Tennessee, Dept Math, 415 EMCS Bldg,Dept 6956,615 McCallie Ave, Chattanooga, TN 37403 USA

关键词: Singular Sturm-Liouville operators;    Boundary values;    Boundary conditions;    Weyl m-functions;   

DOI  :  10.1016/j.jde.2020.05.005

来源: Elsevier

【 摘 要 】

We extend the classical boundary values g(a) =- W (u(a) (lambda(0), . ), g)(a) = lim(x down arrow a) g(x)/(u) over cap (a) (lambda(0) , x), g([1]) (a) = (pg')(a) = W((u) over cap (a) (lambda(0),.),g)(a) = lim(x down arrow a) g(x)- g(a)(u) over cap (a) (lambda(0),x)/u(a)(lambda(0),x) (0.1) for regular Sturm-Liouville operators associated with differential expressions of the type tau = r(x)(-1) [- (d/dx)p(x)(d/dx) + q (x) ] for a.e. x is an element of (a, b) subset of R, to the case where tau is singular on (a, b) subset of R and the associated minimal operator T-min is bounded from below. Here u(a) (lambda(0) ,.) and (u) over cap (a) (lambda(0) , .) denote suitably normalized principal and nonprincipal solutions of tau(u) = lambda(0)u for appropriate lambda(0) is an element of R, respectively. Our approach to deriving the analog of (0.1) in the singular context employing principal and nonprincipal solutions of tau(u) = lambda(0)u is closely related to a seminal 1992 paper by Niessen and Zettl [58]. We also recall the well-known fact that the analog of the boundary values in (0.1) characterizes all self-adjoint extensions of T-min in the singular case in a manner familiar from the regular case. We briefly discuss the singular Weyl-Titchmarsh-Kodaira m-function and finally illustrate the theory in some detail with the examples of the Bessel, Legendre, and Kummer (resp., Laguerre) operators. (c) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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