On essential self-adjointness of singular Sturm-Liouville operators

Considering singular Sturm–Liouville differential expressions of the type $${\tau }_{\alpha }=-(d\mathrm{/}dx)x^{\alpha }(d\mathrm{/}dx)+q(x),x\in (0,b),\alpha \in \mathbb{R},$$ 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, $$q(x)\ge [(3\mathrm{/}4)-(\alpha \mathrm{/}2)]x^{\alpha -2},$$ or, if $\alpha \in (-\mathrm{\infty },2)$ and there exist $N\in \mathbb{N}$, and $\epsilon >0$ 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, $${\ln }_1(x)=\mathrm{\mid }\ln (x)\mathrm{\mid }=\ln (1\mathrm{/}x),{\ln }_{j+1}(x)=\ln ({\ln }_j(x)),j\in \mathbb{N}\mathrm{.}$$ 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 $$-\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }\mathrm{\nabla }+q(\mathrm{\mid }x\mathrm{\mid }),\alpha \in \mathbb{R},x\in B_n(0;R)\mathrm{\backslash }\{0\},$$ with $B_n(0;R)$ the open ball in ${\mathbb{R}}^n$, $n\in \mathbb{N}$, $n\ge 2$, centered at $x=0$ of radius $R\in (0,\mathrm{\infty })$.