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 $x_k$, $k=1,\dots,n$, and jumps $A_k=A(x_k+0)/A(x_k-0)$. We show that if the discontinuity points $x_1,\dots,x_n$ 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.