Lp Norms of Eigenfunctions on Regular Graphs and on the Sphere

Abstract

We prove upper bounds on the L p norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the L p norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite collection of algebraic rotations of the 2-sphere. Under mild conditions, such joint eigenfunctions are shown to satisfy for large p the same bounds as those known for Laplace eigenfunctions on a surface of non-positive curvature.

Delocalization of Schrödinger Eigenfunctions

“…A careful reading also reveals that the proof shows some logarithmic upper bound on the L 1 -norm of eigenfunctions: k k 1 = O((log N ) 1/4 ). Very recently, Brooks and Le Masson have announced an improvement of the power 1/4 under a stronger assumption than (11): see Brooks and Le Masson [2017]. Anantharaman and Le Masson [2015], a general statement of "quantum ergodicity" was obtained for the first time in the large scale limit, namely for the discrete Laplacian on large regular graphs.…”

$$L^p$$ Norms and Support of Eigenfunctions on Graphs

“…This condition was used in [21]. While for a given graph, it may occur that such n > ℓ G , note that we always have n ≤ log D−1 |G|, cf.…”

“…• The fact that eigenfunction delocalization depends on the behavior of the Green function was not apparent in [21], but is actually well-known, see [24,14,17,9] to name a few references. For example, note that if ψ j is an eigenfunction of H G on G with corresponding eigenvalue λ j , then for any η > 0,…”

“…There are notable differences here however. In [21,25], the spectral cluster is essentially a function of the Laplacian. Here M n,λ is not normal, and in fact defined on a different Hilbert space : ℓ 2 (B) instead of ℓ 2 (V ).…”

“…Lower bounds on the support were provided in [20], where it is shown more generally that the eigenfunctions cannot concentrate on small sets. The recent paper [21] provides norm estimates which read as ψ λ p 1 (log q |G|) 1/2 for all p > 2, for generic regular graphs. All the preceding results apply to deterministic graphs.…”