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Contemporary Mathematics Finite Part of Spectrum and Isospectrality

Summary: Contemporary Mathematics
Finite Part of Spectrum and Isospectrality
Xianzhe Dai and Guofang Wei
Abstract. We study geometric conditions under which finitely many eigen-
values are sufficient to determine all spectral data. We also discuss briefly the
implication of this result to the local structure of moduli space of isopectral
1. Introduction
For a compact (smooth) manifold Mn
, a Riemannian metric gives rise to a
canonical differential operator, namely, the Laplacian (acting on smooth func-
tions). If M = , we put the Dirichlet boundary condition on the boundary. This
makes into a self-adjoint second order elliptic operator. Hence it has a discrete
spectrum all consisting of eigenvalues of finite multiplicity:
0 = 0 < 1 2 .
Here the eigenvalues are repeated according to their multiplicity.
Two Riemannian manifolds (M, g) and (M , g ) are said to be isospectral if
their corresponding eigenvalues are identical:
i(M, g) = i(M , g ), i = 1, 2, .
One of the main questions, popularized by Kac's question "can one hear the shape


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics