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

We extend the classical boundary values \begin{align*} & g(a) = - W(u_{a}(λ_0,.), g)(a) = \lim_{x \downarrow a} \frac{g(x)}{\hat u_{a}(λ_0,x)}, \\ &g^{[1]}(a) = (p g')(a) = W(\hat u_{a}(λ_0,.), g)(a) = \lim_{x \downarrow a} \frac{g(x) - g(a) \hat u_{a}(λ_0,x)}{u_{a}(λ_0,x)} \end{align*} for regular Sturm-Liouville operators associated with differential expressions of the type $τ= r(x)^{-1}[-(d/dx)p(x)(d/dx) + q(x)]$ for a.e. $x\in[a,b] \subset \mathbb{R}$, to the case where $τ$ is singular on $(a,b) \subseteq \mathbb{R}$ and the associated minimal operator $T_{min}$ is bounded from below. Here $u_a(λ_0, \cdot)$ and $\hat u_a(λ_0, \cdot)$ denote suitably normalized principal and nonprincipal solutions of $τu = λ_0 u$ for appropriate $λ_0 \in \mathbb{R}$, 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.

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