On essential self-adjointness of singular Sturm–Liouville operators

Abstract

Considering singular Sturm-Liouville differential expressions of the type
\[
\tau_{\alpha} = -(d/dx)x^{\alpha}(d/dx) + q(x),
\quad x \in (0,b), \, \alpha \in \mathbb{R},
\]
we employ some Sturm comparison-type results in the spirit of Kurss to derive criteria for $\tau_{\alpha}$ to be in the limit-point and limit-circle case at $x=0$. More precisely, if $\alpha \in \mathbb{R}$ and, for $0 < x$ sufficiently small,
\[
q(x) \geq [(3/4)-(\alpha/2)]x^{\alpha-2},
\]
or, if $\alpha\in (-\infty,2)$ and there exist $N\in\mathbb{N}$ and $\varepsilon>0$ such that, for $0 < x$ sufficiently small,
\begin{align*}
& q(x)\geq[(3/4)-(\alpha/2)]x^{\alpha-2} - (1/2) (2 - \alpha) x^{\alpha-2}
\sum_{j=1}^{N}\prod_{\ell=1}^{j}[\ln_{\ell}(x)]^{-1}
\\
& \quad\quad\quad +[(3/4)+\varepsilon] x^{\alpha-2}[\ln_{1}(x)]^{-2},
\end{align*}
then $\tau_{\alpha}$ is nonoscillatory and in the limit-point case at $x=0$. Here iterated logarithms for $0 < x$ sufficiently small are of the form
\[
\ln_1(x) = |\ln(x)| = \ln(1/x),
\qquad
\ln_{j+1}(x) = \ln(\ln_j(x)), \quad j \in \mathbb{N}.
\]
Analogous results are derived for $\tau_{\alpha}$ to be in the limit-circle case at $x=0$. We also discuss a multi-dimensional application to partial differential expressions of the type
\[
- \operatorname{div} |x|^{\alpha} \nabla + q(|x|),
\quad \alpha \in \mathbb{R}, \, x \in B_n(0;R) \backslash \{0\},
\]
with $B_n(0;R)$ the open ball in $\mathbb{R}^n$, $n\in \mathbb{N}$, $n \geq 2$, centered at $x=0$ of radius $R \in (0, \infty)$.