Essential self-adjointness of strongly singular homogeneous polyharmonic operators

Abstract

We consider essential self-adjointness of strongly singular, homogeneous, polyharmonic operators of the form \[ T_m = \left((-Δ)^m + c|x|^{-2m}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \setminus \{0\})}, \quad m,n\in\mathbb{N},\ n\ge 2,\ c\in\mathbb{R}, \] in $L^2({\mathbb{R}}^n;d^{n}x)$, with special emphasis on the biharmonic case $m=2$ and the case $m=3$. In the biharmonic case $m=2$ we prove the sharp result that $T_2$ is essentially self-adjoint if and only if \[ c \ge \begin{cases} 3(n+2)(6-n), & 2\le n\le 5,\\[4pt] -\dfrac{(n+4)n(n-4)(n-8)}{16}, & n\ge 6. \end{cases} \] In particular, in the special (nonsingular) case $c=0$, $(-\hat{I}”{)}^2{|}_{C_0^{\infty }({\mathbb{R}}^n\setminus \{0\})}$ is essentially self-adjoint in $L^2({\mathbb{R}}^n;d^{n}x)$ if and only if $n\ge 8$. Similarly, we derive the analogous sharp essential self-adjointness result for $T_3$ for all $n\ge 2$. Our methods extend to homogeneous polyharmonic differential operators, but certain nontrivial subtleties arise. In particular, the natural expectation that for each $m,n\in \mathbb{N}$, $n\ge 2$, there exists $c_{m,n}\in \mathbb{R}$ such that $\left((-\hat{I}”{)}^m+c|x{|}^{-2m}\right){|}_{C_0^{\infty }({\mathbb{R}}^n\setminus \{0\})}$ is essentially self-adjoint in $L^2({\mathbb{R}}^n;d^{n}x)$ if and only if $c\ge c_{m,n}$ is false. For example, for $n=20$ we prove that \[ \left((-Δ)^5 + c|x|^{-10}\right)\big|_{C_0^{\infty}(\mathbb{R}^{20} \setminus \{0\})} \] is essentially self-adjoint in $L^2({\mathbb{R}}^{20};d^{20}x)$ if and only if $c\in [0,{\hat{I}}^2]\cup [{\hat{I}}^3,\infty )$, where ${\hat{I}}^2\approx 1.0436\times 10^{10}$ and ${\hat{I}}^3\approx 1.8324\times 10^{10}$ are the two real roots of a certain quartic equation with integer coefficients.

31 pages, 4 figures