Deficiency Indices of a Symmetric Ordinary Differential Operator with Infinitely Many Degeneration Points

Let $H$ be the minimal symmetric operator in $L^2(\mathbb{R})$ generated by the differential expression $(-1)^n(c(x)f^{(n)})^{(n)}$, $n\ge1$, with a real coefficient $c(x)$ that has countably many zeros without finite accumulation points and is infinitely smooth at all points $x\in\mathbb{R}$ with $c(x)\ne0$. We study the value $\operatorname{Def}H$ of the deficiency indices of $H$. It is shown that $\operatorname{Def}H=+\infty$ if infinitely many zeros of $c(x)$ have multiplicities $p$ satisfying the inequality $n-1/2<p<2n-1/2$. Our second result pertains to the case in which the set of zeros of $c(x)$ is bounded neither above nor below. Under this condition, $\operatorname{Def}H=0$ provided that the multiplicity of each zero is greater than or equal to $2n-1/2$. The multiplicities of zeros of $c(x)$ are understood in the paper in a broader sense than in the standard definition.