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تسجيل مستخدم جديدOn essential self-adjointness of singular Sturm-Liouville operators
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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 $alphain (-infty,2)$ and there exist $Ninmathbb{N}$, 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} &quadquadquad +[(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 [ - Div |x|^{alpha} abla + q(|x|), quad alpha in mathbb{R}, ; x in B_n(0;R) backslash{0}, ] with $B_n(0;R)$ the open ball in $mathbb{R}^n$, $nin mathbb{N}$, $n geq 2$, centered at $x=0$ of radius $R in (0, infty)$.
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