The behaviour of eigenstates of arithmetic hyperbolic manifolds. (English) Zbl 0836.58043

Summary: We study some problems arising from the theory of quantum chaos, in the context of arithmetic hyperbolic manifolds. We show that there is no strong localization (“scarring”) onto totally geodesic submanifolds. Arithmetic examples are given, which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom.


11F72 Spectral theory; trace formulas (e.g., that of Selberg)
58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
81Q50 Quantum chaos
11F32 Modular correspondences, etc.
11F37 Forms of half-integer weight; nonholomorphic modular forms
Full Text: DOI


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