Eigenvalue Asymptotics of Layered Media and Their Applications to the Inverse Problem

We compute the asymptotics of the eigenvalues of the classical Sturm–Liouville problem with a piecewise smooth coefficient q. This means that q and/or its derivatives can have jump discontinuities. The boundary conditions are arbitrary. Our results extend the classical work of H. Hochstadt (see [Comm. Pure Appl. Math., 14 (1961), pp. 749–764]) and some related formulas discovered by G. Borg (see [Acta Math., 78 (1946), pp. 1–96]). Then, we apply our results to the inverse problem of determining the interfaces in a layered medium from acoustic data, since the index of refraction of such a medium can be considered piecewise smooth.