Essential self-adjointness of even-order, strongly singular, homogeneous half-line differential operators

We consider essential self-adjointness on the space $C_0^{\mathrm{\infty }}((0,\mathrm{\infty }))$ of even order, strongly singular, homogeneous differential operators associated with differential expressions of the type $${\tau }_{2n}(c)=(-1{)}^n\frac{d^{2n}}{d{x}^{2n}}+\frac{c}{x^{2n}},x>0,n\in \mathbb{N},c\in \mathbb{R},$$ in $L^2((0,\mathrm{\infty });dx)$. While the special case $n=1$ is classical and it is well-known that ${\tau }_2(c){\mid }_{C_0^{\mathrm{\infty }}((0,\mathrm{\infty }))}$ is essentially self-adjoint if and only if $c\ge 3\mathrm{/}4$, the case $n\in \mathbb{N}$, $n\ge 2$, is far from obvious. In particular, it is not at all clear from the outset that $$\begin{array}{cccc}& \text{ there exists }{c}_n\in \mathbb{R},n\in \mathbb{N},\text{ such that }{\tau }_{2n}(c){\mid }_{C_0^{\mathrm{\infty }}((0,\mathrm{\infty }))}\text{ is essentially self-adjoint if and only if }c\ge c_n\mathrm{.}\text{label}0.1& & \text{(*)}\end{array}$$ As one of the principal results of this paper we indeed establish the existence of $c_n$, satisfying $c_n\ge (4n-1)!!/2^{2n}$, such that property \eqref{0.1} holds. In sharp contrast to the analogous lower semiboundedness question, $$\text{ for which values of }c\mathit{\text{is}}{\tau }_{2n}(c){\mid }_{C_0^{\mathrm{\infty }}((0,\mathrm{\infty }))}\text{ bounded from below?},$$ which permits the sharp (and explicit) answer $c\ge [(2n-1)!!{]}^2/2^{2n}$, $n\in \mathbb{N}$, the answer for \eqref{0.1} is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly, $$c_1=3\mathrm{/}4,c_2=45,c_3=2240(214+7\sqrt{1009})/27,$$ and remark that $c_n$ is the root of a polynomial of degree $n-1$. We demonstrate that for $n=6,7$, $c_n$ are algebraic numbers not expressible as radicals over $\mathbb{Q}$ (and conjecture this is in fact true for general $n\ge 6$).