The Modulus of Continuity¶for Γ0(m)\? Semi-Classical Limits

Abstract:

We study the behavior of a large-eigenvalue limit of eigenfunctions for the hyperbolic Laplacian for the modular quotient SL(2;ℤ)\?. Féjer summation and results of S. Zelditch are used to show that the microlocal lifts of eigenfunctions have large-eigenvalue limit a geodesic flow invariant measure for the modular unit cotangent bundle. The limit is studied for Hecke–Maass forms, joint eigenfunctions of the Hecke operators and the hyperbolic Laplacian. The first modulus of continuity result is presented for the limit. The singular concentration set of the limit cannot be a compact union of closed geodesics and measured geodesic laminations.