Abstract
In this paper, we consider the spectral problem for a Stieltjes string with both ends fixed and with one-dimensional damping at an interior point. We solve an inverse problem of recovering parameters of the Stieltjes string using partial information on the string and partial information on the spectrum. First we prove uniqueness and give an algorithm of recovering the unknown parameters of the string by virtue of a finite number of distinct not pure imaginary complex eigenvalues and a certain even number of real eigenvalues. For the case where only complex eigenvalues are used we solve the existence problem, i.e. propose conditions on a set of complex numbers necessary and sufficient for these numbers to be a subset of the spectrum of the problem.
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