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Deficiency Indices of a Symmetric Ordinary Differential Operator with Infinitely Many Degeneration Points
Yu. B. Orochko Moscow State Institute of Electronics and Mathematics
Abstract:
Let
$H$ be the minimal symmetric operator in
$L^2(\mathbb{R})$ generated by the differential expression
$(-1)^n(c(x)f^{(n)})^{(n)}$,
$n\ge1$, with a real coefficient
$c(x)$ that has countably many zeros without finite accumulation points and is infinitely smooth at all points
$x\in\mathbb{R}$ with
$c(x)\ne0$. We study the value
$\operatorname{Def}H$ of the deficiency indices of
$H$. It is shown that
$\operatorname{Def}H=+\infty$ if infinitely many zeros of
$c(x)$ have multiplicities
$p$ satisfying the inequality
$n-1/2<p<2n-1/2$. Our second result pertains to the case in which the set of zeros of
$c(x)$ is bounded neither above nor below. Under this condition,
$\operatorname{Def}H=0$ provided that the multiplicity of each zero is greater than or equal to
$2n-1/2$. The multiplicities of zeros of
$c(x)$ are understood in the paper in a broader sense than in the standard definition.
Keywords:
symmetric operator, deficiency indices, degenerate ordinary differential operator.
UDC:
517.98 Received: 30.12.2002
DOI:
10.4213/faa107
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