On string density at the origin

In [V. Barcilon Explicit solution of the inverse problem for a vibrating string. J. Math. Anal. Appl. {\bf 93} (1983) 222-234] two boundary value problems were considered generated by the differential equation of a string $$y^{\prime\prime}+\lambda p(x)y=0, \ \ 0\leq x \leq L<+\infty \eqno{(*)}$$ with continuous real function $p(x)$ (density of the string) and the boundary conditions $y(0)=y(L)=0$ the first problem and $y^{\prime}(0)=y(L)=0$ the second one. In the above paper the following formula was stated $$p(0)={1}{L^2\mu_1}\mathop{\prod}\limits_{n=1}^{\infty}{\lambda_n^2}{\mu_n \mu_{n+1}} \eqno{(**)}$$ where $\{\lambda_k\}_{k=1}^{\infty}$ is the spectrum of the first boundary value problem and $\{\mu_k\}_{k=1}^{\infty}$ of the second one. Rigorous proof of (**) was given in [C.-L. Shen On the Barcilon formula for the string equation with a piecewise continuous density function. Inverse Problems {\bf 21}, (2005) 635--655] under more restrictive conditions of piecewise continuity of $p^{\prime}(x)$. In this paper (**) was deduced using $$p(0)=\lim\limits_{\lambda\to +\infty}({\phi(L,-\lambda)}{\lambda^{{1}{2}}\psi(L,-\lambda)})^2 \eqno{(***})$$ where $\phi(x,\lambda)$ is the solution of (*) which satisfies the boundary conditions $\phi(0)-1=\phi^{\prime}(0)=0$ and $\psi(x,\lambda)$ is the solution of (*) which satisfies $\psi(0)=\psi^{\prime}(0)-1=0$. In our paper we prove that (***) is true for the so-called M.G. Krein's string which may have any nondecreasing mass distribution function $M(x)$ with finite nonzero $M^{\prime}(0)$. Also we show that (**) is true for a wide class of strings including those for which $M(x)$ is a singular function, i.e. $M^{\prime}(x)=p(x)\mathop{=}\limits^{a.e.}0$.