Spectre du Laplacien, graphes et topologie de Fell

We find sharp upper bounds for the multiplicities and the numerical values of all the distinct eigenvalues on a surface of revolution diffeomorphic to the sphere.

Sharp Bounds for Eigenvalues and Multiplicities on Surfaces of Revolution

/pdf/sur-la-multiplicite-des-valeurs-propres-du-laplacien-5glmbfb5c5.pdf

Sur la multiplicité des valeurs propres du laplacien

We prove two explicit bounds for the multiplicities of Steklov eigenvalues $\sigma_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given smooth Riemannian surface with boundary, the multiplicities of Steklov eigenvalues $\sigma_k$ are uniformly bounded in $k$.

Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

The recent progress in the study of multiplicities of the eigenvalues of the Laplacian has brought through new ideas, in particular the link between the operator theory on graphs and operator theory on manifolds. This has lead to a complete solution of the original problem in dimension greater than 2. The case of Dimension 2 is still open, but the conjecture mentioned above is an aim to reach.

On the multiplicy of the eigenvalues of the Laplacian

We produce examples of Riemann surfaces whose n-first eigenvalues of the laplacian are of large multiplicity On donne des exemples de surfaces de Riemann dont les n premieres valeurs propres du laplacien sont de multiplicite elevee

A propos de la multiplicité de la première valeur propre du laplacien d'une surface de Riemann

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A propos de la multiplicité de la première valeur propre du laplacien d'une surface de Riemann