Essential self-adjointness of $\left(\Delta^2 +c|x|^{-4}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \backslash \{0\})}

• Main Authors: Gesztesy, Fritz, Hunziker, Markus
• Format: Journal Article
• Language: English
• Published: 11.03.2024
• Subjects: Mathematics - Classical Analysis and ODEs; Mathematics - Spectral Theory
• DOI: 10.48550/arxiv.2403.07160

Let $n\in\mathbb{N}, n\geq 2$. We prove that the strongly singular differential operator \[\left(\Delta^2 +c|x|^{-4}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \backslash \{0\})}, \quad c \in \mathbb{R}, \] is essentially self-adjoint in $L^2(\mathbb{R}^n; d^n x)$ if and only if \[c\geq \begin{cases}3(n+2)(6-n)&\mbox{for $2\leq n\leq 5$};\\[5pt] {\displaystyle -\frac{n(n+4)(n-4)(n-8)}{16}}&\mbox{for $n\geq 6$}.\end{cases}\] Via separation of variables, our proof reduces to studying the essential self-adjointness on the space $C_0^{\infty}((0,\infty))$ of fourth-order Euler-type differential operators of the form \[ \frac{d^4}{dr^4}+c_1\left(\frac{1}{r^2}\frac{d^2}{dr^2}+\frac{d^2}{dr^2}\frac{1}{r^2}\right)+\frac{c_2}{r^4},\quad r\in(0,\infty),\quad(c_1,c_2)\in \mathbb{R}^2,\] in $L^2((0,\infty);dr)$. Our methods generalize to differential operators related to higher-order powers of the Laplacian, however, there are some nontrivial subtleties that arise. For example, the natural expectation that for $m,n\in\mathbb{N}$, $n \geq 2$, there exist $c_{m,n}\in\mathbb{R}$ such that $\left(\Delta^m+c|x|^{-2m}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \backslash \{0\})}$ is essentially self-adjoint in $L^2(\mathbb{R}^n; d^n x)$ if and only if $c \geq c_{m,n}$, turns out to be false. Indeed, for $n=20$, we prove that the differential operator \[ \left((-\Delta)^5+c|x|^{-10}\right)\big|_{C_0^{\infty}(\mathbb{R}^{20} \backslash \{0\})}, \quad c \in \mathbb{R},\] is essentially self-adjoint in $L^2\big( \mathbb{R}^{20}; d^{20} x\big)$ if and only if $c\in [0,\beta]\cup [\gamma,\infty)$, where $\beta\approx 1.0436\times 10^{10}$, and $\gamma\approx 1.8324\times 10^{10}$ are the two real roots of the quartic equation \begin{align*}&3125z^4-83914629120000z^3+429438995162964368031744 z^2\\&\quad+1045471534388841527438982355353600z\\&\quad +629847004905001626921946285352115240960000=0.\end{align*}