Abstract
This paper concerns the uniqueness of a coefficient inverse problem for a one-dimensional wave equation. We allow the unknown coefficient under consideration to be discontinuous. This type of inverse problem arises in seismology, corresponding to the case in which the medium to be detected has discontinuous interfaces. By virtue of spectra theory of the Sturm--Liouville problem and asymptotics of the Green function, we prove that the discontinuous coefficient can be uniquely recovered from the source function $f\in L^1 [0, +\infty)$ and the response function g. Furthermore, Lemma 6 in this paper corrects some mistakes in [T. Kimura and T. Suzuki, SIAM J. Appl. Math., 53 (1993), pp. 1747--1761] about the asymptotics of the Green function. This correction means that some related parts in the aforementioned article also need to be rectified correspondingly.
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