On the scarring of eigenstates in some arithmetic hyperbolic manifolds

Details On the scarring of eigenstates in some arithmetic hyperbolic manifolds On the scarring of eigenstates in some arithmetic hyperbolic manifolds

2004-04-27

arxiv.org/abs/math-ph/0404066v4

Physics

Mathematical Physics

Written in January-February 2003 and corrected in March 2004 and August 2005. 35 pages

Scientific paper

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. We extend this result 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.

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