Abstract
We study the inverse spectral problem for the Sturm–Liouville operator whose piecewise constant coefficient A(x) has discontinuity points x k , k=1,...,n, and jumps A k =A(x k +0)/A(x k -0). We show that if the discontinuity points x 1,...,x n are noncommensurable, i.e., none of their linear combinations with integer coefficients vanishes; then the spectral function of the operator determines all discontinuity points x k and jumps A k uniquely. We give an algorithm for finding x k and A k in finitely many steps.
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Shestakov, A.I. The Inverse Spectral Problem for the Sturm-Liouville Operators with Discontinuous Coefficients. Siberian Mathematical Journal 44, 891–907 (2003). https://doi.org/10.1023/A:1025905307566
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DOI: https://doi.org/10.1023/A:1025905307566