Sbornik. Mathematics, Tome 196 (2005) no. 5, pp. 673-702
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Yu. B. Orochko. Deficiency indices of a one-term symmetric differential operator of even order degenerate in the interior of an interval. Sbornik. Mathematics, Tome 196 (2005) no. 5, pp. 673-702. http://geodesic.mathdoc.fr/item/SM_2005_196_5_a2/
@article{SM_2005_196_5_a2,
author = {Yu. B. Orochko},
title = {Deficiency indices of a~one-term symmetric differential operator of even order degenerate in the interior of an interval},
journal = {Sbornik. Mathematics},
pages = {673--702},
year = {2005},
volume = {196},
number = {5},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2005_196_5_a2/}
}
TY - JOUR
AU - Yu. B. Orochko
TI - Deficiency indices of a one-term symmetric differential operator of even order degenerate in the interior of an interval
JO - Sbornik. Mathematics
PY - 2005
SP - 673
EP - 702
VL - 196
IS - 5
UR - http://geodesic.mathdoc.fr/item/SM_2005_196_5_a2/
LA - en
ID - SM_2005_196_5_a2
ER -
%0 Journal Article
%A Yu. B. Orochko
%T Deficiency indices of a one-term symmetric differential operator of even order degenerate in the interior of an interval
%J Sbornik. Mathematics
%D 2005
%P 673-702
%V 196
%N 5
%U http://geodesic.mathdoc.fr/item/SM_2005_196_5_a2/
%G en
%F SM_2005_196_5_a2
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.
