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 xk, k=1,...,n, and jumps Ak=A(xk +0)/A(xk-0). We show that if the discontinuity points x1,...,xn are noncommensurable, i.e., none of their linear combinations with integer coefficients vanishes; then the spectral function of the operator determines all discontinuity points xk and jumps Ak uniquely. We give an algorithm for finding xk and Ak in finitely many steps.