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Impenetrability condition for a degenerate point of a one-term symmetric differential operator of even order
Yu. B. Orochko Moscow State Institute of Electronics and Mathematics
Abstract:
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.
UDC:
517.98
MSC: Primary
47E05; Secondary
34L05 Received: 30.10.2002
DOI:
10.4213/sm737
Fulltext:
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