Geodesic
Deficiency Indices of a Symmetric Ordinary Differential Operator with Infinitely Many Degeneration Points
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 2, pp. 55-64.

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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 $n-1/2$. 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. 
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     author = {Yu. B. Orochko},
     title = {Deficiency {Indices} of a {Symmetric} {Ordinary} {Differential} {Operator} with {Infinitely} {Many} {Degeneration} {Points}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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     number = {2},
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     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_2_a4/}
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Yu. B. Orochko. Deficiency Indices of a Symmetric Ordinary Differential Operator with Infinitely Many Degeneration Points. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 2, pp. 55-64. http://geodesic.mathdoc.fr/item/FAA_2004_38_2_a4/

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