A formula for the mass density of a vibrating string in terms of the trace (Q1262991)

The boundary value problem for u(x) satisfying the equation of the vibrating string \(u''+\lambda \rho u=0\) on \(0\leq x\leq 1\) with \(u(0)=u(1)=0\), has a sequence of eigenvalues \(0<\lambda_ 1<\lambda_ 2<... \). If the interval is shortened from [0,1] to [0,x], then the eigenvalues \(\lambda_ n\) are functions of x. In this paper it is shown that the mass density \(\rho\) (x) in the differential equation is related to the trace, \(tr(x)=\sum (1/\lambda_ n(x)),\) by the formula \(\rho (x)=tr''+(2/x)tr'.\) More precisely, this formula for \(\rho\) is proved to hold if \(\rho\) is positive, has a continuous first derivative, and has a second derivative in \(L^ 2\). This, in theory, solves the inverse problem to determine the mass density given the eigenvalues.