A refinement of strong multiplicity one for spectra of hyperbolic manifolds

Kelmer, Dubi

doi:10.1090/S0002-9947-2014-06102-3

A refinement of strong multiplicity one for spectra of hyperbolic manifolds
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Trans. Amer. Math. Soc. 366 (2014), 5925-5961 Request permission

Abstract:

Let $\mathcal {M}_1$ and $\mathcal {M}_2$ denote two compact hyperbolic manifolds. Assume that the multiplicities of eigenvalues of the Laplacian acting on $L^2(\mathcal {M}_1)$ and $L^2(\mathcal {M}_2)$ (respectively, multiplicities of lengths of closed geodesics in $\mathcal {M}_1$ and $\mathcal {M}_2$) are the same, except for a possibly infinite exceptional set of eigenvalues (respectively lengths). We define a notion of density for the exceptional set and show that if it is below a certain threshold, the two manifolds must be isospectral.

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