Stability estimate for a multidimensional inverse spectral problem with partial spectral data

In Bellassoued, Choulli and Yamamoto (2009) [4], we proved a log–log type stability estimate for a multidimensional inverse spectral problem with partial spectral data for a Schrödinger operator, provided that the potential is known in a small neighbourhood of the boundary of the domain. In the present paper we discuss the same inverse problem. We show a log type stability estimate under an additional condition on potentials in terms of their X-ray transform. In proving our result, we follow the same method as in Alessandrini and Sylvester (1990) [1], and Bellassoued, Choulli and Yamamoto (2009) [4]. That is we relate the stability estimate for our inverse spectral problem to a stability estimate for an inverse problem consisting in the determination of the potential in a wave equation from a local Dirichlet to Neumann map (DN map in short).