Essential self-adjointness of $(Δ^2 +c|x|⁻⁴)|_{C_0∞(R^n \{0\})}$

Let 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 L²(Rⁿ; dⁿ x) if and only if \[c≥ {cases}3(n+2)(6-n)&for 2≤ n≤ 5;\\[5pt] { -n(n+4)(n-4)(n-8)/16}&for n≥ 6.{cases}\] Via separation of variables, our proof reduces to studying the essential self-adjointness on the space C₀∞((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 L²((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 ≥ 2, there exist cm,n such that (Δᵐ+c|x|⁻²ᵐ)|_C₀∞(Rⁿ \0\) is essentially self-adjoint in L²(Rⁿ; dⁿ x) if and only if c ≥ cm,n, turns out to be false. Indeed, for $n=20$, we prove that the differential operator \[ ((-Δ)^5+c|x|⁻¹⁰)|_{C_0∞(R²⁰ \{0\})}, c ∈ R,\] is essentially self-adjoint in L²( R²⁰; d²⁰ x) if and only if c∈ [0,β]∪ [γ,∞), where β≈ 1.0436× 10¹⁰, and γ≈ 1.8324× 10¹⁰ are the two real roots of the quartic equation {align*}&3125z^4-83914629120000z^3+429438995162964368031744 z^2\&+1045471534388841527438982355353600z\& +629847004905001626921946285352115240960000=0.{align*}