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

Abstract

We extend the classical boundary values \begin{align*} & g(a) = - W(u_{a}(\lambda_0,.), g)(a) = \lim_{x \downarrow a} \frac{g(x)}{\hat u_{a}(\lambda_0,x)}, \\ &g^{[1]}(a) = (p g')(a) = W(\hat u_{a}(\lambda_0,.), g)(a) = \lim_{x \downarrow a} \frac{g(x) - g(a) \hat u_{a}(\lambda_0,x)}{u_{a}(\lambda_0,x)} \end{align*} 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. $x\in[a,b] \subset \mathbb{R}$, to the case where $\tau$ is singular on $(a,b) \subseteq \mathbb{R}$ and the associated minimal operator $T_{min}$ is bounded from below. Here $u_a(\lambda_0, \cdot)$ and $\hat u_a(\lambda_0, \cdot)$ denote suitably normalized principal and nonprincipal solutions of $\tau u = \lambda_0 u$ for appropriate $\lambda_0 \in \mathbb{R}$, respectively. 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.