An inverse eigenvalue problem for a vibrating string with two Dirichlet spectra (Q2841014)

Let \(\lambda_0<\lambda_1<\dots\) denote the eigenvalues of \(y''+\lambda\rho y=0\) on \((0,1)\) with \(y(0)=0=y(1)\), where \(\rho\) is a positive absolutely continuous function on \([0,1]\), and \(\mu_0<\mu_1<\dots\) the eigenvalues when \(\rho\) is replaced by \(\rho +\delta\) where \(\delta_0\) is absolutely continuous and \(\rho+\delta\) is positive on \([0,1]\). The question posed by the authors is whether knowing the sequences \((\lambda_j)\) and \((\mu_j)\) for some (known) non-identically zero function \(\delta\) is sufficient to determine \(\rho\) uniquely. Without additional assumptions, it is shown that the answer is negative. Examples of additional assumptions on \(\delta\) which ensure a positive answer are given as well as some numerical experiments.