On the scarring of eigenstates for some arithmetic hyperbolic manifolds in dimension 2 and 3

Tristan Poullaouec (1)

1 IMJ - Institut de Mathématiques de Jussieu

In this paper, we deal with the conjecture of 'Quantum Unique Ergodicity'. Z. Rudnick and P. Sarnak showed that there is no 'strong scarring' on closed geodesics for arithmetic congruence surfaces derived from a quaternion division algebra. First, we extend this result to the congruence surface Gamma(2)\H^2 , where Gamma(2) is the kernel in SL(2,Z) of the projection modulo 2. Then, we extend it to a class of three-dimensional Riemannian manifolds X=Gamma\H^3 that are again derived from quaternion division algebras. We show that there is no 'strong scarring' on closed geodesics or on Gamma-closed imbedded totally geodesics surfaces of X.

Mots clés: Quantum Chaos; Laplace-Beltrami; Hyperbolic Manifolds; Modular Correspondences; Quaternion Algebras; Prime Numbers

Domaines: Physique mathématique [math-ph]; Théorie des nombres [math.NT]; Algèbres d'opérateurs [math.OA]

Tristan Poullaouec. On the scarring of eigenstates for some arithmetic hyperbolic manifolds in dimension 2 and 3. 2004. ⟨hal-00001491v2⟩