Quantum Ergodicity and Averaging Operators on the Sphere

Abstract

Abstract. We prove quantum ergodicity for certain orthonormal bases of L 2 (S 2 ), consisting of joint eigenfunctions of the Laplacian on S 2 and the discrete averaging operator over a finite set of rotations, generating a free group. If in addition the rotations are algebraic we give a quantified version of this result. The methods used also give a new, simplified proof of quantum ergodicity for large regular graphs.

Recent results of quantum ergodicity on graphs and further investigation

“…The proof given here bears similarities with a proof by Brooks, Lindenstrauss and Le Masson [25] in the special case where R = 0, i.e. K is a function on the vertex set V N .…”

“…The main idea there is the following : it is true that 1 λ r j A r G is difficult to analyze, but perhaps P r (A G ) is more approachable for a good polynomial P r . The authors of [25] choose the Chebyshev polynomial P r (cos θ) = cos(rθ). They start by observing that for any θ ∈ [0, π] and n ≥ 10, one has | 1 n n r=1 cos(rθ) 2 | ≥ 0.3.…”

“…Our aim in this paper is to present a comprehensive outline of the main ideas behind some recent proofs of quantum ergodicity on large graphs [7,10] or [25]. We hope this will be a helpful guide for an audience with different backgrounds : spectral geometry, quantum chaos, graph theory and Anderson localization.…”

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.…”

“…Let (u ) 0< ≤1 be a normalized sequence in L 2 (M) verifying the oscillation assumptions (6) and (7). For a given scale of times τ := (τ ) Thus, µ can be extended to a continuous linear form on L 1 (R, C 0 0 (T * M)), where C 0 0 (T * M) denotes the set of continuous functions vanishing at infinity.…”