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 $$ y^{\prime\prime}+λp(x)y=0, \ \ 0\leq x \leq L<+\infty \eqno{(*)} $$ 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 $$ p(0)={1}{L^2μ_1}\mathop{\prod}\limits_{n=1}^{\infty}{λ_n^2}{μ_n μ_{n+1}} \eqno{(**)} $$ where $\{λ_k\}_{k=1}^{\infty}$ is the spectrum of the first boundary value problem and $\{μ_k\}_{k=1}^{\infty}$ 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 $$ p(0)=\lim\limits_{λ\to +\infty}({ϕ(L,-λ)}{λ^{{1}{2}}ψ(L,-λ)})^2 \eqno{(***}) $$ where $ϕ(x,λ)$ is the solution of (*) which satisfies the boundary conditions $ϕ(0)-1=ϕ^{\prime}(0)=0$ and $ψ(x,λ)$ is the solution of (*) which satisfies $ψ(0)=ψ^{\prime}(0)-1=0$. 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. $M^{\prime}(x)=p(x)\mathop{=}\limits^{a.e.}0$.

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