Nodal sets and geometric control Steve Zelditch Northwestern University Nodal sets and geometric control Joint work with John Toth Quantissima in the Serenissima, Steve Zelditch 2017 Northwestern University Joint work with John Toth Quantissima in the Serenissima, 2017 August 10, 2017
Nodal intersection problems Nodal sets Let ( M m , g ) be a compact real analytic Riemannian manifold without boundary of and geometric control dimension m . We denote by { ϕ j } ∞ j =0 an orthonormal basis of Laplace Steve Zelditch eigenfunctions, Northwestern University − ∆ ϕ j = λ 2 Joint work j ϕ j , � ϕ j , ϕ k � = δ jk , with John Toth � where λ 0 = 0 < λ 1 ≤ λ 2 ≤ · · · and where � u , v � = M uvdV g ( dV g being the Quantissima in the volume form). Serenissima, 2017 We denote the nodal set of an eigenfunction ϕ λ of eigenvalue − λ 2 by N ϕ λ = { x ∈ M : ϕ λ ( x ) = 0 } . Sharp upper bounds for H m − 1 ( N ϕ λ ) were proved by Donnelly-Fefferman in the 80’s in the real analytic case: c λ ≤ H m − 1 ( N ϕ λ ) ≤ C λ.
Restrictions of eigenfunctions to a submanifold Nodal sets and geometric control Steve Zelditch Northwestern University Joint work Let H ⊂ M be a real analytic submanifold. Much work has gone into the study of with John Toth the restrictions ϕ j | H , its norms and its zeros. Quantissima in the Serenissima, Let S = { j k } ∞ k =1 be a subsequence (indices of) eigenvalues. (We also let S denote 2017 { λ j k } or the sequence { ϕ j k } of eigenfunctions from the given orthonormal basis.) Question: For curves or hypersurfaces, estimate the Hausdorff measure of N ϕ λ ∩ H = nodal set of ϕ j k | H .
Extreme cases Nodal sets and geometric control The answer depends on dim H at least and we mainly consider dim H = 1 (curve) Steve Zelditch or dim H = m − 1 (hypersurface). Northwestern University Joint work with John If H = Fix ( σ ) is the fixed point set of an isometric involution σ : M → M , then H Toth can be a hypersurface (e.g. x n → − x n on S n − 1 or on R n ) or of lower dimension Quantissima in the Z on C m is totally real R m ). (e.g. the fixed point set of Z → ¯ Serenissima, 2017 Odd eigenfunctions vanish on Fix ( σ ). I.e. ‘half’ of all eigenfunctions vanish on this set. Bourgain-Rudnick: Characterize H ⊂ M such that there exists some infinite sequence S such that ϕ j k | H = 0. We call such submanifolds ‘nodal’. We are able to answer the question for subsequences of positive number density.
‘ S -Good submanifolds’ Nodal sets and geometric control Steve Zelditch Northwestern University Joint work A less restrictive condition than nodal is ‘ S -bad’: It means that with John Toth sup x ∈ H | ϕ j k | H ≤ Ce − M λ jk for all M > 0. “Super-exponential decay’. Quantissima in the Serenissima, 2017 We say that H is S - ‘good’ if there exists M > 0 so that | ϕ j k | H ≥ Ce − M λ jk . sup x ∈ H
A good curve Nodal sets and geometric control Steve Zelditch Northwestern University Joint work with John Toth Quantissima in the H Serenissima, 2017 Figure 1: Nodal lines of a high energy state, λ ∼ 84, in the quarter stadium.
Main results in a nutshell Nodal sets and geometric We prove that the number n ( ϕ j k , C ) of nodal points on a connected irreducible control S -good real analytic curve C of a sequence S of Laplace eigenfunctions ϕ j of Steve Zelditch Northwestern eigenvalue − λ 2 j of a real analytic Riemannian manifold ( M , g ) of any dimension m University Joint work is bounded above as follows: with John Toth Quantissima in the n ( ϕ j k , C ) ≤ A g , C λ j k . Serenissima, 2017 Moreover, we prove that the codimension-two Hausdorff measure H m − 2 ( N ϕ λ ∩ H ) of nodal intersections with a connected, irreducible real analytic hypersurface H ⊂ M satisfies H m − 2 ( N ϕ λ ∩ H ) ≤ A g , H λ j k . We further give a geometric control condition on H which is sufficient that H be S -good for a density one subsequence of eigenfunctions.
Remembrance of things past Nodal sets and geometric control Steve Zelditch Northwestern Theorem University Joint work (Toth-Z,’09) Let Ω ⊂ R 2 be piecewise analytic and let n ∂ Ω ( λ j ) be the number of with John Toth components of the nodal set of the jth Neumann or Dirichlet eigenfunction which Quantissima in the intersect ∂ Ω . Then, n ∂ Ω ( λ j ) ≤ C Ω λ j . Serenissima, 2017 Theorem (Toth-Z ‘09) Suppose that Ω ⊂ R 2 is a C ∞ plane domain, and let C ⊂ Ω be a good interior real analytic curve. . Let n ( λ j , C ) = # N ϕ λ j ∩ C be the number of intersection points of the nodal set of the j-th Neumann (or Dirichlet) eigenfunction with C. Then there exists A C , Ω > 0 such that n ( λ j , C ) ≤ A C , Ω λ j .
New results Nodal sets and geometric control Steve Zelditch Northwestern University Joint work The first set of new results generalize the plane domain theorems to real anaytic with John Toth Riemannian manifolds of any dimension. One then must consider what dimension Quantissima in the the submanifold C should have. The new results work in all co-dimensions but we Serenissima, 2017 only state the results for curves and for hypersurfaces. The new results also assume ∂ M = ∅ . The counting techniques are based on analytic continuation of the wave kernel, which so far have not been generalized to the boundary case.
Results assuming goodness Nodal sets and geometric control Theorem Steve Zelditch Northwestern Suppose that ( M m , g ) is a real analytic Riemannian manifold of dimension m University Joint work without boundary and that C ⊂ M is connected, irreducible real analytic curve. If C with John Toth is S -good, then there exists a constant A S , g so that Quantissima in the Serenissima, 2017 n ( ϕ j , C ) := # {C ∩ N ϕ j } ≤ A S , g λ j , j ∈ S . Theorem Let ( M m , g ) be a real analytic Riemannian manifold of dimension m and let H ⊂ M be a connected, irreducible, S -good real analytic hyperurface. Then, there exists a constant C > 0 depending only on ( M , g , H ) so that H m − 2 ( N ϕ jk ∩ H ) ≤ C λ j k , ( j k ∈ S ) .
Goodness? Nodal sets and geometric control Steve Zelditch Northwestern University Joint work with John Why irreducible? Suppose C 1 is a good curve and C 2 is bad, e.g. the fixed point Toth set of an isometric involution. Then C 1 ∪ C 2 is good but the counting results do Quantissima in the not work. Serenissima, 2017 We now give sufficient geometric control conditions for ‘goodness’. The definition of S -good makes sense for any connected, irreducible analytic submanifold H ⊂ M , not only curves.
Notation and assumptions Nodal sets and geometric control Given a submanifold H ⊂ M , we denote the restriction operator to H by Steve Zelditch Northwestern γ H f = f | H . To simplify notation, we also write γ H f = f H . The criterion that a University Joint work pair ( H , S ) be good is stated in terms of the associated sequence with John Toth Quantissima u j := 1 in the log | ϕ j | 2 (1) Serenissima, λ j 2017 of normalized logarithms, and in particular their restrictions j := γ H u j := 1 u H log | ϕ H j | 2 (2) λ j to H . We only consider the goodness of connected, irreducible, real analytic submanifolds.
Definition of Good Nodal sets and geometric control Steve Zelditch Northwestern Definition: Given a subsequence S := { ϕ j k } , we say that a connected, irreducible University Joint work real analytic submanifold H ⊂ M is S -good, or that ( H , S ) is a good pair, if the with John Toth sequence (2) with j k ∈ S does not tend to −∞ uniformly on compact subsets of Quantissima in the H, i.e. there exists a constant M S > 0 so that Serenissima, 2017 u H sup j ≥ − M S , ∀ j ∈ S . H If H is S -good when S is the entire orthonormal basis sequence, we say that H is completely good. If S has density one we say that H is almost completely good. The opposite of a good pair ( H , S ) is a bad pair.
Equivalent notions of good Nodal sets and geometric The following are equivalent on a real analytic curve. control Steve Zelditch Northwestern University 1 Goodness in the sense of Definition 13, or equivalently in the sense that Joint work with John � ϕ j | H � L ∞ ( H ) ≥ e − a λ j . Toth Quantissima in the Serenissima, 2 Goodness in the sense � ϕ H j � L 2 ( H ) ≥ e − a λ j . 2017 1 3 Goodness in the sense that λ j log | ϕ j | H | → −∞ does not hold uniformly on the real H . λ j log | ϕ C 1 4 Goodness in the sense that j | H | → −∞ does not hold uniformly on the complex H .
Geometric control conditionsfor Goodness Nodal sets and geometric control Steve Zelditch Northwestern University Joint work with John Toth Quantissima The criteria consists of two conditions on H : in the Serenissima, (i) asymmetry with respect to geodesic flow, and 2017 (ii) a full measure flowout condition.
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