Focal points and sup-norms of eigenfunctions

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1 Focal points and sup-norms of eigenfunctions Chris Sogge (Johns Hopkins University) Joint work with Steve Zelditch (Northwestern University)) Chris Sogge Focal points and sup-norms of eigenfunctions 1 /

2 Background Compact real analytic manifold (M, g) of dimension n. Eigenfunctions: e j (x) = λ j e j (x), e j dv = 1 Give fundamental modes of vibration: u j (t, x) = cos tλ j e j (x). Know e j L (M) Cλ n 1 j Question: When can you improve this estimate for eigenfunctions (or, more generally, quasi-modes)? Since u j (t, x) provide high-frequency solutions of wave equations, ( t )u j = 0, expect answer to depend on long-term dynamics of geodesic flow (known to be linked to propagation of singularities for t ) Chris Sogge Focal points and sup-norms of eigenfunctions /

3 Terminology to be used Say that e j L (M) = O(λ n 1 j ), if, as above e j L (M) Cλ n 1 j. We are interested in when we have improvements, e j L (M) = o(λ n 1 j ), meaning that lim sup λ j λ n 1 j e j L (M) = 0. If we cannot do this, we say that e j L (M) = Ω(λ n 1 j ), meaning that lim sup λ j λ n 1 j e j L (M) > 0. (We ll have to capture Ω(λ n 1 j ) sup-norm behavior using quasi-modes, though, in which case we ll also be able to use liminf over natural sequence µ k of frequencies arising from the geometry) Chris Sogge Focal points and sup-norms of eigenfunctions 3 /

4 Loops through a point Let Φ t (x, ξ) = (x(t), ξ(t)), with (x(0), ξ(0)) = (x, ξ) S x M, denote unit-speed geodesic flow in cosphere bundle. Thus, γ = {x(t)} is geodesic in M starting at x. Let L x0 = {ξ S x 0 M : Φ t (x 0, ξ) = (x 0, ξ(t)), some t > 0} denote initial directions of loops through x 0. Figures from A. Besse, Manifolds all of whose geodesics are closed, (Springer, 1978). Chris Sogge Focal points and sup-norms of eigenfunctions 4 /

5 Why real analytic? In the real analytic case, can show that the set of looping directions satisfies either L x0 = 0 or L x0 = S x 0 M AND there is an l > 0 so that Φ l (x 0, ξ) = (x 0, η(ξ)). (i.e., all geodesics starting at x 0 loop back in time l). Say that x 0 a self-focal point [sfp] in second case Chris Sogge Focal points and sup-norms of eigenfunctions 5 /

6 A Few Examples T n, then L x = 0 for all x. (No spf s) S n, have Φ π (x, ξ) = (x, ξ), i.e., periodic flow (Everything a sfp) Surfaces of revolution. Periodic at poles Triaxial ellipsoid: x + z = 1, 0 < a < b < c < 1. Loops of b c equal length thru 4 umbilic points a + y Latter has much different dynamics: Call these 4 points dissipative. Chris Sogge Focal points and sup-norms of eigenfunctions 6 /

7 Dissipative self-focal points x 0 M If x 0 is a spf, there s a minimum l > 0 (first return time) so that Φ l (x 0, ξ) = (x 0, η(ξ)) ξ Sx 0 M. Call η : Sx 0 M Sx 0 M first return map Defines unitary operator (Perron-Frobenius operator) U = U x0 : L (S x 0 M) L (S x 0 M) f Uf (ξ) = f (η(ξ)) J(ξ), J = Jacobian. Define: sfp x 0 M dissipative if U has no invariant L -functions Chris Sogge Focal points and sup-norms of eigenfunctions 7 /

8 von Neumann Ergodic Theorem and self-focal points Assume x is a self-focal point. Then if 1 1 on S x M and η ν = η η η (ν-times), if J ν (ξ) is Jacobian and dµ is Liouville measure on S x M, by von Neumann, as T, T 1 T ν=1 S x M J ν (ξ) dµ(ξ) = T 1 T U ν 1, 1 Π(1), 1, where Π : L (S x M) L (S x M) is projection onto U-invariant functions. Conclude (as U 0) T 1 T ν=1 S x M J ν (ξ) dµ(ξ) ν=1 { 0, if x dissipative c x > 0, if x not dissipative Chris Sogge Focal points and sup-norms of eigenfunctions 8 /

9 Necessary and sufficient conditions for improved sup-norms Define projection onto δ-bands of spectrum (0 < δ 1) by χ [λ,λ+δ] f = λ j [λ,λ+δ] E j f, where E j f (x) = f, e j e j (x) is proj onto j-th eigenspace. Know χ [λ,λ+1] L (M) L (M) = O(λ n 1 ). Local estimate always sharp. Only possible improvements if use shrinking bands of intervals (i.e., δ = δ(λ) 0 as λ ). Problem: n 1 χ [λ,λ+o(1)] L L = o(λ )? Means: ε > 0, δ(ε) > 0 and Λ ε < so that χ [λ,λ+δ(ε)] f ελ n 1 f, λ Λ ε. (1) Chris Sogge Focal points and sup-norms of eigenfunctions 9 /

10 Main result Theorem Have improved sup-norm bounds, (1), if and only if every self-focal point is dissipative. Since χ [λ,λ+δ] e j = e j if λ j [λ, λ + δ], conclude that we have the following result for eigenfunctions: Corollary If all self-focal points are dissipative then e j = o(λ n 1 j ). Chris Sogge Focal points and sup-norms of eigenfunctions 10 /

11 Some earlier results S-Zelditch (00) Have (1) if L x = 0 for all x M. S-Toth Zelditch (011). Same if R x = 0, where R x S x M denotes the set of recurrent directions for the flow. Specifically initial unit directions ξ such that, given ε > 0, there is a t 0 so that Φ t0 (x, ξ) S x M (loops back) and Φ t0 (x, ξ) (x, ξ) < ε. Note that if C x S x M is the set of initial directions for periodic geodesics (i.e. smoothly closed loops) through x, then have C x R x L x and so L x = 0 = R x = 0 = C x = 0. By the Poincaré recurrence theorem if x is a sfp and R x = 0 then the Perron-Frobenius operator U x cannot have 0 f L (S x M) with U x f = f. For then f dµ a finite invariant measure in the class of dµ. Whence, by Poincaré, a.e. point with respect to f dµ would be recurrent and so R x > 0. Chris Sogge Focal points and sup-norms of eigenfunctions 11 /

12 Some earlier results, continued Recall Weyl formula: N(λ) = c M λ n + O(λ n 1 ), if N(λ) = #λ j λ. By the Duistermaat-Guillemin Theorem (1975), if the set C S M of all periodic points for the geodesic flow is of measure zero, then the error term in the Weyl formula is o(λ n 1 ). Our results imply that if the error term in Weyl law is Ω(λ n 1 ) then not only must one have C > 0, but also there must be a non-dissipative sfp. For many years thought that C x = 0 for all x M, should be sufficient for the o(λ n 1 ) sup-norm bounds (1). WRONG (?) On the other hand, as we ll see below, if there is a non-dissipative sfp x 0 M then the recipe for producing quasi-modes that are large there is related to another theorem in D-G ( 75). Chris Sogge Focal points and sup-norms of eigenfunctions 1 /

13 Equivalent quasi-mode formulation of Main Theorem Functions φ λk C (M) are quasi-modes of order o(λ) ( fat quasi-modes ) if φ λk dv = 1 and ( + λ k )φ λ k = o(λ k ). () M (N.B. as λ j λ k = λ j λ k (λ j + λ k ) measures L -concentration in o(1)-sized intervals) Need an extra condition besides () when n 4. Corollary Have φ λk = o(λ n 1 k ) for every sequence λ k if and only if every self-focal point is dissipative. Chris Sogge Focal points and sup-norms of eigenfunctions 13 /

14 Ω(λ n 1 ) bounds at a non-dissipative sfp x 0 M If x 0 is a non-dissipative sfp, let µ k = π l (k + β/4), k = 1,,..., (3) where β is number of conjugate points (w/ multiplicity) along a unit speed geodesic {γ(t) : 0 t < l}, where l is first return time and γ(0) = γ(l) = x 0 Then if T k (slow enough) the coherent state at x 0 φ µk (y) = µ n 1 k ρ(t k (µ k λ j )) e j (x 0 )e j (y) j=0 satisfies (), and has φ µk (x 0 ) µ n 1 k, assuming that 0 ρ S(R) has ρ(t) 1, t 1, ˆρ 0, and supp ˆρ [ 1, 1]. Chris Sogge Focal points and sup-norms of eigenfunctions 14 /

15 Related earlier result of Duistermaat-Guillemin (1975) Theorem (Duistermaat-Guillemin) If geodesic flow on M is periodic with minimal period l > 0 and µ k as in (3) and intervals I k are given by I k = [µ k Ck 1, µ k + Ck 1 ], with C > 1 sufficiently large but fixed, have #{λ j λ : λ j k=1 I k} 1, as λ. #{λ j λ} Chris Sogge Focal points and sup-norms of eigenfunctions 15 /

16 Ideas in proofs Want to show that if all sfp s are dissipative then, given ε > 0, if T (ε), Λ(ε) large n 1 χ [λ,λ+t 1 (ε)] L L ελ, λ Λ(ε), or this cannot happen if there are non-dissipative sftp s. Concentrate for now on former. By orthogonality, χ [λ,λ+t 1 (ε)] L L = sup x M λ j [λ,λ+t 1 (ε)] ( ej (x) ), and so task is equivalent to showing that ( ej (x) ) ε λ n 1, λ 1 (4) λ j [λ,λ+t 1 (ε)] By earlier results of S-Zelditch, know have uniform such bounds near any point x 0 M with L x0 = 0. So need same near given dissipative sfp. Chris Sogge Focal points and sup-norms of eigenfunctions 16 /

17 o-bounds near dissipative self-focal point x 0 M Use above bump function 0 ρ S satisfying ρ(t) 1, t [ 1, 1] and supp ˆρ [ 1, 1] and ˆρ 0. Have λ j [λ,λ+t 1 (ε)] ( ej (x) ) ρ(t (λ λ j )) (e j (x)), j=0 T = T (ε), and so would have (4) near our diss sfp x 0 if could show there that ρ(t (λ λ j )) (e j (x)) ε λ n 1 (5) j=0 Use ρ(t (λ λ j )) (e j (x)) = 1 ˆρ(t/T )e itλ ( e itλ j (e j (x)) ) dt πt j=0 j = 1 T ˆρ(t/T )e itλ ( e it ) (x, x) dt πt T Chris Sogge Focal points and sup-norms of eigenfunctions 17 /

18 FIOs and propagation of singularities Near any t 0 (mod C ) can write exp( it )(x, y) as finite sum j = 1,..., N(t 0 ) of terms (π) n R n e is j (t,y,ξ) ix ξ a j (t, x, y, ξ) dξ, where a S 0, a(lν, x, x, ξ) = e iνβπ/ J ν (ξ), ν Z, t S j = p(y, x S j ) Factors i νβ called Maslov factors Our assumptions (basically) also imply that exp( it )(y, y) is smooth if y close to sfp x and t supp ˆρ( /T ) is not close to {lν : ν = 0, ±1, ±,... } [ T, T ] Chris Sogge Focal points and sup-norms of eigenfunctions 18 /

19 Punch line via stationary phase Using above facts write ξ = rω, ω Sx M and use stationary phase in t, r variables. Miracle: Modulo lower order terms in λ, LHS of (5) is ρ(t (λ λj ))(e j (x)) = λ n 1 ( e ilνλ e iνβπ/)ˆρ(lν) ( 1 J T ν (ω)dµ(ω) ) (6) Sx M ν T If x = x 0 dissipative sfp then (mod l.o.t. s) this is Cλ n 1 T 1 U ν 1, 1, ν T and coefficient is small if T large by the von Neumann ergodic Theorem If x non-diss sfp and λ = µ k as in (3) then blue factor in (6) is one and ˆρ 0. Thus, if 0 < c 0 = lim T T 1 ν T Uν 1, 1, the LHS of (6) c 0 µ n 1 k for every fixed T > 1 if λ 1 Chris Sogge Focal points and sup-norms of eigenfunctions 19 /

20 Remarks and questions Did not use real analyticity in essential way. For instance, get )-sized quasi-modes at any x M if Φ l (x, ξ) Sx M for some l > 0, f L, Uf = f, f 0, and the measure of initial directions of loops through x shorter than l has measure zero. Ω(µ n 1 k As in the real analytic case above, the q.m. s satisfy φ µk (x) = Ω(µ n 1 k ) and M φ µk dv = 1 and ( + µ k )φ µ k = o(µ k ). Question: Can you reach a stronger conclusion by replacing the red term by O(1)? (More traditional definition of q.m. s.) If the Perron-Frobenius operator Uf (ξ) = J(ξ) f (η(ξ)) had a smooth invariant function, would be able to solve a transport equation and do this. In our results, the conclusion or assumption just concerns L -invariant functions (used to obtain o-bounds via von Neumann when all sfp s diss). Chris Sogge Focal points and sup-norms of eigenfunctions 0 /

21 Remarks and questions, continued By interpolation with CS 88 L p estimates: Know that if all sfp s are dissipative then every quasi-mode as in () must satisfy φ λ L p (M) = o(λ n( 1 1 p ) 1 ), p > (n + 1) n 1 (7) What about endpoint p = (n+1) n 1? Can show that all sfp s being dissipative is a necessary condition for (7) when p = (n+1) n 1. Much recent work (Bourgain, CS, CS-Zelditch, Blair-CS) on obtaining improved L p estimates when < p < (n+1) n 1. Relevant thing for this range: concentration along periodic geodesics (as opposed to concentration at points). Different proofs for this range. Boils down to improving geodesic restriction estimates of Burq-Gérard-Tzvetkov (n = ) and geodesic shrinking tube estimates of X. Chen-CS and M. Blair-CS. Chris Sogge Focal points and sup-norms of eigenfunctions 1 /

22 Some applications Bourgain-Shao-CS-Yao: Improved L p estimates for large p lead to improvements of earlier resolvent estimates of Shen and dos Santos Ferreira-Kenig-Salo. Possible for torus (big improvement) and nonpositive curvature (log-improvements) Improvements for < p < (n+1) n 1 lead to improved known results for nodal sets/domains: Colding-Minicozzi, CS-Zelditch, Blair-CS; Gosh-Reznikov-Sarnak... Chris Sogge Focal points and sup-norms of eigenfunctions /

Focal points and sup-norms of eigenfunctions

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