Impenetrability condition for a~degenerate point of a~one-term symmetric differential operator of even order

Yu. B. Orochko

Sbornik. Mathematics, Tome 194 (2003) no. 5, pp. 745-774

Résumé

Let $a(x)\in C^\infty[0,h]$, $b(x)\in C^\infty[-h,0]$, $h>0$, be real functions not vanishing on their definition intervals. For fixed $p>0$ and $q>0$ one considers the differential expressions \begin{align*} s_p^+[f](x)=(-1)^n(x^pa(x)f^{(n)})^{(n)}(x), \\ s_q^-[f](x)=(-1)^n((-x)^qb(x)f^{(n)})^{(n)}(x) \end{align*} of arbitrary even order $2n$ degenerate at the point $x=0$. Let $H_p^+$ and $H_q^-$ be the minimal symmetric operators induced by $s_p^+[f](x)$ and $s_q^-[f](x)$ in the Hilbert spaces $L^2(0,h)$ and $L^2(-h,0)$, respectively. “Sewing together” the differential expressions $s_p^+[f](x)$ and $s_q^-[f](x)$ at $x=0$ one obtains a new differential expression $s_{pq}[f](x)$, $x\in[-h,h]$, which is degenerate at the same point, an interior point of $[-h,h]$. Under certain constraints on $p$ and $q$ the differential expression $s_{pq}[f](x)$ gives rise to a minimal symmetric operator $H_{pq}$ in $L^2(-h,h)$ which is a symmetric extension of the orthogonal sum $H_q^-\oplus H_p^+$. The point $x=0$ is called in this paper an interior barrier for $s_{pq}[f](x)$. Conditions ensuring the equality $H_{pq}=H_q\oplus H_p$ are found. It is natural to call an interior barrier an impenetrable interior interface if this equality holds and it is a penetrable interior interface if it fails. The main result of this paper is as follows: the point $x=0$ is an impenetrable interior interface if $p,q\geqslant 2n-\frac12$, and this condition is best possible in a certain sense.