Essential self-adjointness of strongly singular homogeneous polyharmonic operators
Fritz Gesztesy, Markus Hunziker
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 $c=0$, $(-Δ)^2\big|_{C_0^{\infty}(\mathbb{R}^n \setminus \{0\})}$ 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 $\left((-Δ)^m + c|x|^{-2m}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \setminus \{0\})}$ 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 $c\in [0,β]\cup[γ,\infty)$, where $β\approx 1.0436\times 10^{10}$ and $γ\approx 1.8324\times 10^{10}$ are the two real roots of a certain quartic equation with integer coefficients.