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This article is cited in 2 scientific papers (total in 2 papers)
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.
Received: 30.10.2002
Citation:
Yu. B. Orochko, “Impenetrability condition for a degenerate point of a one-term symmetric differential operator of even order”, Sb. Math., 194:5 (2003), 745–774
Linking options:
https://www.mathnet.ru/eng/sm737https://doi.org/10.1070/SM2003v194n05ABEH000737 https://www.mathnet.ru/eng/sm/v194/i5/p109
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Abstract page: | 501 | Russian version PDF: | 196 | English version PDF: | 18 | References: | 67 | First page: | 1 |
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