Essential Self-Adjointness of Even-Order, Strongly Singular, Homogeneous Half-Line Differential Operators

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