Quantum ergodicity and symmetry reduction

The equivariant spectral function of an invariant elliptic operator. L -bounds, caustics, and concentration of eigenfunctions

“…Thus, spherical harmonics with fixed m concentrate on the poles as k becomes large. This fact is in accordance with the probability of finding a classical particle of zero angular momentum near singular orbits and the shape of the corresponding equivariant quantum limits, see [18,Section 9.2]. Furthermore, if c denotes a closed geodesic on S 2 we obtain for the restriction of Y k,m to c the L ∞ -bounds…”

“…In particular, this gives a new interpretation of the classical bounds for spherical harmonics in terms of caustics of the equivariant spectral function, generalizing them to eigenfunctions on arbitrary compact manifolds with symmetries. The concentration of eigenfunctions along singular orbits was already observed in [18] for Schrödinger operators in the context of equivariant quantum ergodicity under the additional assumption that the reduced Hamiltonian flow is ergodic. Our results can be viewed as part of the more general problem of studying the eigenfunctions of a commuting family of differential operators on a general compact manifold that are independent in some sense [21].…”

“…[π γtriv |Gx : 1] =ˆG x γ triv (g)dG x (g) = |Gx| l=1 |G x |, dG x being the counting measure. For the volume factor, see[18, Remark 6.2].…”

Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

“…In particular, in this case, the normalized Riemannian volume is the only accumulation point of the sequence of eigenfunctions under consideration. Note that quantum ergodicity properties were also proved recently for sequences of eigenfunctions on S d satisfying certain symmetry assumptions [36,7]. Yet, there are in fact some situations for which one has N (∞) = N g , which is in a certain sense the opposite situation to quantum ergodicity.…”

Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

“…But since there are only finitely many torsion points on a fundamental domain of an arithmetic quotient[38, Theorem 4.15] we have K prin = {1}, and consequently [π σ|Kprin : 1] = 1. In addition, by singular cotangent bundle reduction[28, Remark 3.4] we have Ω/K ≃ T * (M/K) as orbifolds, so that as in (4.12) one deduces ˆM vol [(Ω ∩ S * x (M ))/K] dx = (d − dim K) ˆM/K vol [B * x•K (M/K)] d(x • K) = (d − dim K) ̟ d−dim K vol (M/K).…”