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Hyperbolic triangles without embedded eigenvalues. (English) Zbl 1407.58011

Any noncompact complete hyperbolic surface of finite area has the continuous spectrum \([1/4, +\infty)\). On the other hand, the existence of square integrable eigenfunctions is a very subtle problem. A conjecture due to Phillips and Sarnak says that a generic such hyperbolic surface should not have any square integrable functions whose eigenvalues lie in the continuous spectrum. Some previous work on this problem are conditional. In this paper, the authors study this conjecture for a model surface: hyperbolic triangles with one cusp, i.e., whether there exists square integrable eigenfunctions with respect to the Neumann boundary condition. They show that a generic such triangle does not have such eigenfunctions. The result is unconditional. To prove this result, the author study the behavior of the real-analytic eigenvalue branches of a degenerating family of triangles. A careful analysis of spectral projections near the crossings of these eigenvalue branches with the eigenvalue branches of a model operator is crucial.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P99 Spectral theory and eigenvalue problems for partial differential equations
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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