Essential Self-Adjointness of Strongly Singular Homogeneous Polyharmonic Operators

Abstract We consider essential self-adjointness of strongly singular, homogeneous, polyharmonic operators of the form $$ T_m = \left( (-\Delta )^m +c|x|^{-2\,m}\right) \big |_{C_0^{\infty }({\mathbb {R}}^n \backslash \{0\})}, \quad m \in {\mathbb {N}}, \; n \in {\mathbb {N}}, n\ge 2, \; c \in {\mathbb {R}}, $$ T m = ( - Δ ) m + c | x | - 2 m | C 0 ∞ ( R n \ { 0 } ) , m ∈ N , n ∈ N , n ≥ 2 , c ∈ R , in $$L^2({\mathbb {R}}^n; d^n x)$$ L 2 ( R n ; d n x ) , with special emphasis on the biharmonic case $$m=2$$ m = 2 and the case $$m=3$$ m = 3 . For instance, in the biharmonic case $$m=2$$ m = 2 we prove the sharp result $$\begin{aligned}&T_2 \text { is essentially self-adjoint if and only if} \\&\quad c\ge {\left\{ \begin{array}{ll} 3(n+2)(6-n) & \text { for } 2\le n\le 5; \\ {\displaystyle -\frac{(n+4)n(n-4)(n-8)}{16}}& \text {for } n\ge 6. \end{array}\right. } \end{aligned}$$ T 2 is essentially self-adjoint if and only if c ≥ 3 ( n