Quantum Ergodicity and L p Norms of Restrictions of Eigenfunctions

Abstract

We prove an analogue of Sogge's local L p estimates [So16] for L p norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq-Gérard-Tzvetkov [BuGeTz07], Hu [Hu09], and Chen-Sogge [ChSo14]. The improvements are logarithmic on negatively curved manifolds (without boundary) and by o(1) for manifolds (with or without boundary) with ergodic geodesic flows. In the case of ergodic billiards with piecewise smooth boundary, we get o(1) improvements on L ∞ estimates of Cauchy data away from a shrinking neighborhood of the corners, and as a result using the methods of [GhReSa13, JuZe16a, JuZe16b], we get that the number of nodal domains of 2-dimensional ergodic billiards tends to infinity as λ → ∞. These results work only for a full density subsequence of any given orthonormal basis of eigenfunctions.We also present an extension of the L p estimates of [BuGeTz07, Hu09, ChSo14] for the restrictions of Dirichlet and Neumann eigenfunctions to compact submanifolds of the interior of manifolds with piecewise smooth boundary. This part does not assume ergodicity on the manifolds. 1 See [LuSa95, Yo16, HeRi16b, LeRu16] for parallel results in the arithmetic setting.

Eigenfunction scarring and improvements in L∞ bounds

“…However, in Theorem 2, we make no analyticity or dynamical assumptions on (M, g) whatsoever, only an assumption on the particular defect measure associated with the eigenfunction sequence. Recently, Hezari [Hez16] and Sogge [Sog16] gave independent proofs of Theorem 2.…”

Improved critical eigenfunction restriction estimates on Riemannian manifolds with constant negative curvature

“…If r ≥ CcoshT , then by (4.2)-(4.5), E ≥ (A − 4re s+t ) 2 ≥ Cr 2 e 2s+2t ≥ Cr 4 (coshT ) −2 , |D α E| ≤ C α r 4 (coshT ) 8 , |D α G| ≤ C α r 3 (coshT ) 5 .…”

“…|D α φ st | ≤ C α r(rcoshT ) 2 (r 4 (coshT ) −2 ) 3/2+|α| r 3 (coshT ) 5 (r 4 (coshT ) 8 ) |α| ≤ C α e (10|α|+10)T , If r ≤ CcoshT , then by (4.2)-(4.5), E ≥ (A − 4re s+t ) 2 ≥ Cr 2 e 2s+2t ≥ C(coshT ) −6 , |D α E| ≤ C α (coshT ) 12 , |D α G| ≤ C α (coshT ) 8 .…”

“…Sogge and Zelditch [17] and Chen and Sogge [6] showed that one can improve (1.2) for 2 ≤ p ≤ 4, in the sense that In the 3-dimensional case, under the assumption of nonpositive curvature, Chen [5] also proved a (log λ) − 1 2 gain for all p > 2. With the assumption of constant negative curvature, Chen and Sogge [6] showed that (1.8) sup Moreover, Hezari and Rivière [9] and Hezari [8] used quantum ergodic methods to get logarithmic improvements at critical exponents in the cases above on negatively curved manifolds for a density one subsequence.…”

Local and global analysis of nodal sets

“…The first result on counting nodal domains by counting intersections with a curve was proved by Ghosh-Reznikov-Sarnak for M = H 2 /SL(2, Z). Theorem 5.1 has been extended to a rather general class of billiard tables with ergodic billiard flow by H. Hezari in [He16b].…”