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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 17/2000

M. I. BELISHEV
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ON RELATIONS BETWEEN SPECTRAL
AND DYNAMICAL INVERSE DATA

This preprint was accepted December 25, 2000

Contact:
` M. Gordin `

ABSTRACT:
As is known, boundary spectral data of the compact Riemannian
manifold $\Omega$ (spectrum of Laplacian with zero Dirichlet
boundary condition plus traces of normal derivatives of
eigenfunctions at $\partial\Omega$) determine its boundary
dynamical data (dynamical Dirichlet-to-Neumann map) $R^{2T}$ for
all $T>0$. In the paper the procedures recovering spectral data
of the submanifold~$\Omega^T\!=\!\{x\in\Om\mid\dist(x,\cd\Om)\!<\!T\}$
via given $R^{2T}$ with any
prescribed $T>0$ and continuing $R^{2T}$ from
$\partial\Omega\times(0,2T)$ onto $\partial\Omega\times(0,\infty)$
are proposed. The procedures do not invoke solving
the inverse problems; main fragment is constructing (via
$R^{2T}$) and use of a model of dynamical system associated with
$\Om^T$.

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