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

Abstract

We consider essential self-adjointness on the space \(C_0^{\infty }((0,\infty ))\) of even-order, strongly singular, homogeneous differential operators associated with differential expressions of the type

$$\begin{aligned} \tau _{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}}, \quad x > 0, \; n \in {{\mathbb {N}}}, \; c \in {{\mathbb {R}}}, \end{aligned}$$

in \(L^2((0,\infty );dx)\). While the special case \(n=1\) is classical and it is well known that \(\tau _2(c)\big |_{C_0^{\infty }((0,\infty ))}\) is essentially self-adjoint if and only if \(c \ge 3/4\), 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

$$\begin{aligned} \begin{aligned}&\textit{there exists }c_n \in {{\mathbb {R}}}, n \in {{\mathbb {N}}}\textit{, such that} \\&\quad \tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))} \, \textit{ is essentially self-adjoint}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (*)\\ {}&\quad \textit{ if and only if } c \ge c_n. \end{aligned} \end{aligned}$$

As one of the principal results of this paper we indeed establish the existence of \(c_n\), satisfying \(c_n \ge (4n-1)!!\big /2^{2n}\), such that property (*) holds. In sharp contrast to the analogous lower semiboundedness question,

$$\begin{aligned} \textit{for which values of }c\textit{ is }\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}{} \textit{ bounded from below?}, \end{aligned}$$

which permits the sharp (and explicit) answer \(c \ge [(2n -1)!!]^{2}\big /2^{2n}\), \(n \in {{\mathbb {N}}}\), the answer for (*) is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly,

$$\begin{aligned} c_{1}&= 3/4, \quad c_{2 }= 45, \quad c_{3 } = 2240 \big (214+7 \sqrt{1009}\,\big )\big /27, \end{aligned}$$

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 \(n \ge 6\)).