On essential self-adjointness of singular Sturm-Liouville operators

Abstract

Considering singular Sturm--Liouville differential expressions of the type \[ τ_α = -(d/dx)x^α(d/dx) + q(x), \quad x \in (0,b), \; α\in \mathbb{R}, \] 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, \[ q(x) \geq [(3/4)-(α/2)]x^{α-2}, \] or, if $\alpha \in (-\infty ,2)$ and there exist $N\in \mathbb{N}$, and $\epsilon >0$ such that for $0<x$ sufficiently small, \begin{align*} &q(x)\geq[(3/4)-(α/2)]x^{α-2} - (1/2) (2 - α) x^{α-2} \sum_{j=1}^{N}\prod_{\ell=1}^{j}[\ln_{\ell}(x)]^{-1} \\ &\quad\quad\quad +[(3/4)+\varepsilon] x^{α-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, \[ \ln_1(x) = |\ln(x)| = \ln(1/x), \quad \ln_{j+1}(x) = \ln(\ln_j(x)), \quad j \in \mathbb{N}. \] 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 \[ - {\rm div} |x|^α \nabla + q(|x|), \quad α\in \mathbb{R}, \; x \in B_n(0;R)\backslash\{0\}, \] with $B_n(0;R)$ the open ball in ${\mathbb{R}}^n$, $n\in \mathbb{N}$, $n\ge 2$, centered at $x=0$ of radius $R\in (0,\infty )$.

21 pages, small corrections made and a reference added