Deficiency indices of a~one-term symmetric differential operator of even order degenerate in the interior of an interval

Yu. B. Orochko

Sbornik. Mathematics, Tome 196 (2005) no. 5, pp. 673-702

Résumé

Let $a(x)\in C^\infty[-h,h]$, $h>0$, be a real function such that $a(x)\ne 0$ for $x\in[-h,h]$. Consider the differential expression $s_p[f]=(-1)^n(x^pa(x)f^{(n)})^{(n)}$ of arbitrary order $2n\geqslant 2$, which depends on the positive integer $p$ and is degenerate for $x=0$. Let $H_p$ be the real symmetric operator in $L^2(-h,h)$ corresponding to $s_p[f]$ and let $\operatorname{Def}H_p$ be its deficiency index in the upper (or lower) half-plane. The proof of the formula $\operatorname{Def}H_p=2n+p$, $1\leqslant p\leqslant n$, is presented. It complements the formulae $\operatorname{Def}H_p=2n$ for $p\geqslant 2n$ and $\operatorname{Def}H_p=4n-p$ for $p=2n-2,2n-1$ obtained by the same author before.