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

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<L),\\ 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.