Semiclassical Limits of Eigenfunctions on Flat n-Dimensional Tori

Abstract

Abstract. We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an n-dimensional flat torus T n , and the Fourier transform of squares of the eigenfunctions |ϕ λ | 2 of the Laplacian have uniform l n bounds that do not depend on the eigenvalue λ. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on T n+2 . We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an n-dimensional sphere S n (λ) of radius √ λ, and we use it in the proof.

Uniform estimates for the solutions of the Schrödinger equation on the torus and regularity of semiclassical measures

“…where the constant K d is independent of the eigenvalue λ. This result was established in [6,12] for d = 2 and in [8,11,1] for d ≥ 3. Theorem 1.1 shows that as a function of space and time, e it∆ u is more regular than being merely a function in C T; L 2 T d , but this regularity is only expressed in terms of the summability properties of the Fourier coefficients of its modulus square.…”

“…Consider the Schrödinger flow e it∆ on the arithmetic flat d-dimensional torus T d := R d /2πZ d . That is, given any u ∈ L 2 T d , the function ψ(x, t) := e it∆ u(x) is the solution to: (1) i∂ t ψ(x, t) + ∆ψ(x, t) = 0, ψ(·, 0) = u.…”

Semiclassical completely integrable systems: long-time dynamics and observability via two-microlocal Wigner measures

“…Finally, we point out that in this case the regularity of semiclassical measures can be precised. The elements in M (∞) are trigonometric polynomials when d = 2, as shown in [23]; and in general they are more regular than merely absolutely continuous, see [1,23,36]. The same phenomenon occurs with those elements in M (1/h) that are obtained through sequences whose corresponding semiclassical measures do not charge {ξ = 0}, see [2].…”

“…They are H-invariant and the convex hull of the set of uniform orbit measures is dense in the set of H-invariant measures. Considering initial data that are linear combinations of wave packets of the form (60), we see that the convex hull of uniform orbit measures is contained in M(τ ), and since the latter is closed, it contains all measures invariant by φ s as stated in Theorem 1.3 (1).…”

“…At that stage of the paper, the proofs of Theorem 1.10 and Theorem 1.3(2) when τ h ∼ 1/h are then achieved. Examples are developed in Section 5 in order to prove Theorem 1.3 (1). Finally, the results concerning hierarchy of time-scales are proved in Section 6 (and lead to Theorem 1.3 for τ h ≫ 1/h), whereas the proof of Theorem 1.16 is given in Section 7.…”