Geodesic
On relations between data of dynamical and spectral inverse problems
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 32, Tome 297 (2003), pp. 30-48
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As is well-known, the boundary spectral data of the compact Riemannian manifold determine its boundary dynamical data (the response operator of the wave equation) corresponding to the arbitrary time interval: the response operator is represented in the form of a series over the spectral data. The converse is true in the following sense: the response operator determines the manifold and, thus, its spectral data. For recovering the last, one can recover the manifold and then solve the (direct) boundary spectral problem. Nevertheless, such the way is not efficient and the question arises whether one can extract the spectral data from the response operator without solving the inverse problem (without recovering the manifold). The paper gives a positive answer and proposes a “direct” time-optimal procedure extracting the spectral data from the response operator. The procedure is based upon a variational principle. 
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M. I. Belishev. On relations between data of dynamical and spectral inverse problems. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 32, Tome 297 (2003), pp. 30-48. http://geodesic.mathdoc.fr/item/ZNSL_2003_297_a2/

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