 01326 Peter D. Hislop, Carl V. Lutzer
 Spectral asymptotics of the DirichlettoNeumann map on multiply connected domains in R^d
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Sep 12, 01

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Abstract. We study the spectral asymptotics of the DirichlettoNeumann operator
$\Lambda_\gamma$ on a multiplyconnected, bounded, domain in $\R^d$,
$d \geq 3$, associated
with the uniformly elliptic operator $L_\gamma =  \sum_{i,j=1}^d
\partial_i \gamma_{ij} \partial_j$, where $\gamma$
is a smooth, positivedefinite, symmetric matrixvalued function
on $\Omega$.
We prove that the operator is approximately diagonal
in the sense that $\Lambda_\gamma = D_\gamma + R_\gamma$, where $D_\gamma$
is a direct sum of operators, each of which acts on one boundary
component only, and $R_\gamma$ is a smoothing operator.
This representation follows from the
fact that the $\gamma$harmonic
extensions of eigenfunctions of $\Lambda_\gamma$ vanish
rapidly away from the boundary.
Using this representation, we study the inverse problem of
determining the number of holes in the body, that is,
the number of the connected
components of the boundary, by using the highenergy spectral
asymptotics of $\Lambda_\gamma$.
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