The Inverse Spectral Problem for the Sturm-Liouville Operators with Discontinuous Coefficients

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 x₁,...,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.