Essential self-adjointness of strongly singular homogeneous polyharmonic operators

We consider essential self-adjointness of strongly singular, homogeneous, polyharmonic operators of the form $$T_m=((-\mathrm{\Delta }{)}^m+c\mathrm{\mid }x{\mathrm{\mid }}^{-2m}){\mid }_{C_0^{\mathrm{\infty }}({\mathbb{R}}^n\setminus \{0\})},m,n\in \mathbb{N},n\ge 2,c\in \mathbb{R},$$ 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 $$c\ge \{\begin{array}{ll}3(n+2)(6-n),& 2\le n\le 5,\\ -{\displaystyle \frac{(n+4)n(n-4)(n-8)}{16}},& n\ge 6.\end{array}$$ In particular, in the special (nonsingular) case $c=0$, $(-\mathrm{\Delta }{)}^2{\mid }_{C_0^{\mathrm{\infty }}({\mathbb{R}}^n\setminus \{0\})}$ 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 $((-\mathrm{\Delta }{)}^m+c\mathrm{\mid }x{\mathrm{\mid }}^{-2m}){\mid }_{C_0^{\mathrm{\infty }}({\mathbb{R}}^n\setminus \{0\})}$ 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 $$((-\mathrm{\Delta }{)}^5+c\mathrm{\mid }x{\mathrm{\mid }}^{-10}){\mid }_{C_0^{\mathrm{\infty }}({\mathbb{R}}^{20}\setminus \{0\})}$$ is essentially self-adjoint in $L^2({\mathbb{R}}^{20};d^{20}x)$ if and only if $c\in [0,\beta ]\cup [\gamma ,\mathrm{\infty })$, where $\beta \approx 1.0436\times {10}^{10}$ and $\gamma \approx 1.8324\times {10}^{10}$ are the two real roots of a certain quartic equation with integer coefficients.