Geodesic
Power Asymptotics of Spectral Functions of Boundary Value Problems for Generalized Second-Order Differential Equations with Boundary Conditions at a Singular Endpoint
Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 1, pp. 82-87 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $I=(-\infty,b)$, where $b\le +\infty$, and let $M(x)$, $x\in I$, be a nondecreasing function on $I$ such that $M(x)>0$ for $x\in I$. In the middle of the past century, it was proved that, in the case where $M(x)$ is Lebesgue integrable on the interval $(-\infty, c)$, $c\in I$, the boundary value problem $-\frac{d}{dM(x)} y^+ (x)=\lambda y(x)$, $x\in I$, $\lim_{x\to -\infty}y(x)=1$ is uniquely solvable for any complex $\lambda$ and has at least one spectral function $\tau (\lambda)$ ("${}^+$" denotes right derivative). A result relating the asymptotic behavior of $M(x)$ as $x \to -\infty$ to that of $\tau(\lambda)$ as $\lambda \to +\infty$ is announced. Similar results are also announced for two other boundary value problems with boundary conditions at a singular endpoint.
Keywords: string, boundary value problem, singular endpoint, spectral function. 
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     author = {I. S. Kats},
     title = {Power {Asymptotics} of {Spectral} {Functions} of {Boundary} {Value} {Problems} for {Generalized} {Second-Order} {Differential} {Equations} with {Boundary} {Conditions} at a {Singular} {Endpoint}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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I. S. Kats. Power Asymptotics of Spectral Functions of Boundary Value Problems for Generalized Second-Order Differential Equations with Boundary Conditions at a Singular Endpoint. Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 1, pp. 82-87. http://geodesic.mathdoc.fr/item/FAA_2015_49_1_a8/

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