On the Density of the Mass Distribution of a String at the Origin

Abstract

In V. Barcilon (J Math Anal Appl 93:222–234, 1983) two boundary value problems were considered generated by the differential equation of a string $$y^{\prime\prime} + \lambda p(x)y = 0, \,\, 0 \leq x \leq L < + \infty \quad \quad \quad (*)$$ with continuous real function p(x) (density of the string) and the boundary conditions y(0) = y(L) = 0 for the first problem and $${y^{\prime}(0) = y(L) = 0}$$ for the second one. In the above paper the following formula was stated $$p(0) = \frac{1}{L^2\mu_1} \mathop{\prod} \limits_{n=1}^{\infty}\frac{\lambda_n^2}{\mu_n \mu_{n+1}} \quad \quad \quad (**)$$ 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. A rigorous proof of (**) was given in C.-L. Shen (Inverse Probl 21:635–655, 2005) under the more restrictive conditions of piecewise continuity of $${p^{\prime}(x)}$$ . In this paper (**) was deduced using $$p(0)=\lim\limits_{\lambda\to +\infty} \left(\frac{\phi(L,-\lambda)} {\lambda^{\frac{1}{2}} \psi(L,-\lambda)} \right)^2 \quad \quad \quad \quad (\ast\ast\ast)$$ where $${\phi(x,\lambda)}$$ is the solution of (*) which satisfies the boundary conditions $${\phi(0) - 1 = \phi^{\prime}(0) = 0 \,\,{\rm and}\,\, \psi(x,\lambda)}$$ 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 strings 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. $${M^{\prime}(x) = p(x)\mathop{=} \limits^{a.e.}0}$$ .