On the reconstruction of a string from spectral data

We consider the problem to reconstruct the mass distribution of a string wh ere the mass is concentrated in a finite number of points, or, equivalently , the problem to reconstruct a simply connected mass-spring system with unk nown masses and stiffness parameters if the following data are given: Problem 1: The spectra of the string and of a modification of the string, o r Problem 2: The spectra of two different modifications of the string. Here a modification of the string is a string which appears if we link the unknown string with another string of known mass distribution. The paper contains a necessary condition for the existence of a solution of Problem 1, and explicit formulas and an algorithm for the solutions of the Problems 1 and 2 under the condition that there exists a solution. For the case that the mass distribution of the unknown string is not discre te we consider the problem to find discrete approximations of this distribu tion from the respective spectral data. The methods are based on the spectral theory of generalized second order di fferential operators as developed by M. G. KREIN.