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Article Dans Une Revue Communications in Mathematical Physics Année : 2011

Spectral simplicity and asymptotic separation of variables

Résumé

We describe a method for comparing the real analytic eigenbranches of two families of quadratic forms that degenerate as t tends to zero. One of the families is assumed to be amenable to `separation of variables' and the other one not. With certain additional assumptions, we show that if the families are asymptotic at first order as t tends to 0, then the generic spectral simplicity of the separable family implies that the eigenbranches of the second family are also generically one-dimensional. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian is one-dimensional. We also show that for all but countably many t, the geodesic triangle in the hyperbolic plane with interior angles 0, t, and t, has simple spectrum.

Dates et versions

hal-00445680 , version 1 (11-01-2010)

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Luc Hillairet, Chris Judge. Spectral simplicity and asymptotic separation of variables. Communications in Mathematical Physics, 2011, 302 (2), pp.291-344. ⟨10.1007/s00220-010-1185-6⟩. ⟨hal-00445680⟩
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