Peter D. Hislop, Carl V. Lutzer
Spectral asymptotics of the Dirichlet-to-Neumann map on multiply connected domains in R^d
(105K, LaTex 2e)

ABSTRACT.  We study the spectral asymptotics of the Dirichlet-to-Neumann operator 
$\Lambda_\gamma$ on a multiply-connected, 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, positive-definite, symmetric matrix-valued 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 high-energy spectral 
asymptotics of $\Lambda_\gamma$.