Generalization of an asymptotic formula of V.\,A.~Marchenko for spectral functions of a~second-order boundary value problem

I. S. Kats

Izvestiya. Mathematics , Tome 7 (1973) no. 2, pp. 424-438

Résumé

It is established that the spectral functions $\tau(\lambda)$ of the second-order boundary value problem \begin{gather*} -\frac d{dM(x)}\biggl[y^-(x)-\int_{-0}^{x-0}y(s)\,dQ(s)\biggr]-\lambda y(x)=0\qquad(0\le x),\\ y^-(0)=m,\qquad y(0)=n, \end{gather*} possess power asymptotics $\tau(\lambda)\sim C\lambda^\nu$ as $\lambda\uparrow+\infty$, when the function $M(x)$ possesses power asymptotics as $x\downarrow0$. A partial converse of this fact is also obtained.