Essential self-adjointness of (Δ^2 +c|x|^-4)|_C_0^∞(ℝ^n \{0})

Let n∈ℕ, n≥ 2. We prove that the strongly singular differential operator (Δ^2 +c|x|^-4)|_C_0^∞(ℝ^n \{0}), c ∈ℝ, is essentially self-adjoint in L^2(ℝ^n; d^n x) if and only if c≥3(n+2)(6-n) ; -n(n+4)(n-4)(n-8)/16 . Via separation of variables, our proof reduces to studying the essential self-adjointness on the space C_0^∞((0,∞)) of fourth-order Euler-type differential operators of the form d^4/dr^4+c_1(1/r^2d^2/dr^2+d^2/dr^21/r^2)+c_2/r^4, r∈(0,∞), (c_1,c_2)∈ℝ^2, in L^2((0,∞);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, n ≥ 2, there exist c_m,n∈ℝ such that (Δ^m+c|x|^-2m)|_C_0^∞(ℝ^n \{0}) is essentially self-adjoint in L^2(ℝ^n; d^n x) if and only if c ≥ c_m,n, turns out to be false. Indeed, for n=20, we prove that the differential operator ((-Δ)^5+c|x|^-10)|_C_0^∞(ℝ^20\{0}), c ∈ℝ, is essentially self-adjoint in L^2( ℝ^20; d^20 x) if and only if c∈ [0,β]∪ [γ,∞), where β≈ 1.0436× 10^10, and γ≈ 1.8324× 10^10 are the two real roots of the quartic equation 3125z^4-83914629120000z^3+429438995162964368031744 z^2 +1045471534388841527438982355353600z +629847004905001626921946285352115240960000=0.