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

Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 1, pp. 82-87

Résumé

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.

Détail

Keywords: string, boundary value problem, singular endpoint, spectral function.