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We consider the Gel'fand inverse problem and continue studies of Mandache (Inverse Problems 17: 1435–1444, 2001). We show that the Mandache-type instability remains valid even in the case of Dirichlet-to-Neumann map given on the energy intervals. These instability results show, in particular, that the logarithmic stability estimates of Alessandrini (Appl. Anal. 27: 153–172, 1988), Novikov and Santacesaria (J. Inverse Ill-Posed Probl., 2010) and especially of Novikov (2010) are optimal (up to the value of the exponent).
Keywords.: Hyperbolic equations; Gel'fand inverse problem; Dirichlet-to-Neumann map; stability and instability estimates
Published Online: 2011-08-05
Published in Print: 2011-August
© de Gruyter 2011