On string density at the origin

TL;DR

This paper extends the validity of a formula relating string density at the origin to spectral data, proving it for a broader class of strings including those with singular mass distributions.

Contribution

It proves the Barcilon formula for a wider class of strings, including those with singular mass distributions, using Krein's string theory.

Findings

The formula holds for M.G. Krein's strings with any nondecreasing mass distribution.

The formula is valid for strings with singular mass functions where the derivative is zero almost everywhere.

The proof generalizes previous results limited to piecewise continuous densities.

Abstract

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. $M^{\prime}(x)=p(x)\mathop{=}\limits^{a.e.}0$.