Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On essential self-adjointness of singular Sturm-Liouville operators   Considering singular Sturm--Liouville differential expressions of the type \[
\tau_{\alpha} = -(d/dx)x^{\alpha}(d/dx) + q(x), \quad x \in (0,b), \; \alpha
\in \mathbb{R}, \] 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, \[ \ln_1(x)
= |\ln(x)| = \ln(1/x), \quad \ln_{j+1}(x) = \ln(\ln_j(x)), \quad j \in
\mathbb{N}. \] 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 \[ - {\rm div} |x|^{\alpha} \nabla + q(|x|), \quad
\alpha \in \mathbb{R}, \; x \in B_n(0;R)\backslash\{0\}, \] 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 $R \in (0, \infty)$.

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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: cond-mat