Brief communications

Power Asymptotics of Spectral Functions of Boundary Value Problems for Generalized Second-Order Differential Equations with Boundary Conditions at a Singular Endpoint

I. S. Kats

Odessa National Academy of Food Technology

Abstract: 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.

UDC: 517.91+517.43

Received: 13.07.2013

DOI: 10.4213/faa3179

 English version: Functional Analysis and Its Applications, 2015, 49:1, 67–71

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