On essential self-adjointness of singular Sturm–Liouville operators

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