Essential self-adjointness of (² +c|x|^-4) |₂䃐^ (Rⁿ \{0\) }

Let n, n 2. We prove that the strongly singular differential operator \ (² +c|x|^-4) |₂䃐^ (Rⁿ \{0\) }, c R, \ is essentially self-adjoint in L² (Rⁿ; dⁿ x) if and only if cases3 (n+2) (6-n) \\[5pt -n (n+4) (n-4) (n-8) {16}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, n 2, there exist c₌, ₍ such that (ᵐ+c|x|^-2m) |₂䃐^ (Rⁿ \{0\) } is essentially self-adjoint in L² (Rⁿ; dⁿ x) if and only if c c₌, ₍, turns out to be false. Indeed, for n=20, we prove that the differential operator \ ( (-) ⁵+c|x|^-10) |₂䃐^ (R^{20 \0\) }, c R, \ is essentially self-adjoint in L² (R^20; d^20 x) if and only if c 0, [, ), where 1. 0436 10^10, and 1. 8324 10^10 are the two real roots of the quartic equation align*&3125z⁴-83914629120000z³+429438995162964368031744 z²\\&+1045471534388841527438982355353600z\\& +629847004905001626921946285352115240960000=0. align*