A gentle introduction to Quantum Ergodicity

A gentle introduction to Quantum Ergodicity Tuomas Sahlsten University of Bristol, UK 7 March 2017 Arithmetic study group, Durham, UK Joint work with Etienne Le Masson (Bristol) Supported by the EU: Horizon 2020 (MSCA-IF grant 655310) and 7th framework (ERC grant 306494)

Laplacian Let M be a compact Riemannian manifold.

Laplacian Let M be a compact Riemannian manifold. The Laplacian f(x) of f : M C at x M is ˆ f(x) rate at which f deviates from f(x) as r 0 S(x,r) Here S(x,r) f is the average value of f on the sphere S(x, r) M.

Eigenfunctions of the Laplacian For a compact M, there exists a sequence 0 λ 1 λ 2 + and functions ψ j : M C, j N, satisfying ψ j (x) = λ j ψ j (x), x M. Maps ψ j are called eigenfunctions of for eigenvalues λ j.

Eigenfunctions of the Laplacian: ψ j = λ j ψ j Eigenfunctions ψ j are building blocks for harmonic analysis on M: - they can be chosen to form an orthonormal basis of L 2 (M): ˆ { 1, i = j; ψ i, ψ j := ψ i (x)ψ j (x) dx = δ ij = 0, i j. f = j f, ψ j ψ j f L 2 (M)

Eigenfunctions of the Laplacian: ψ j = λ j ψ j...they describe the geometry of M. - Kac s inverse problem: Identify M from just λ j :s. Hearing the shape of a drum -...and provide approximations of the manifold M: Sturm-Liouville approximations Picture (c) G. Rong et al. (Visual Comput. 2008)

Eigenfunctions of the Laplacian: ψ j = λ j ψ j...arise in analytic number theory in the study of L-functions (for Hecke-Maass forms ) L(z) = n=1 a n n z, z C, Re(z) > 1. ( ˆ Watson s formula ) 2 f(x) ψ j (x) 2 π 4 L j,f ( 1 dx = 2 ) 216 L j (1) 2 L f (1)

Eigenfunctions of the Laplacian: ψ j = λ j ψ j...and, are (stationary) quantum states in quantum mechanics ψ j 2 for two j:s on a heart shaped domain M R 2 (with Dirichlet data on M) Picture (c) T. Kriecherbauer, J. Marklof, A. Soshnikov (PNAS 2001)

Use dynamics to study them Geodesic flow (g t ) t 0 is a natural dynamical system on T M (cotangent bundle).

Use dynamics to study them Geodesic flow (g t ) t 0 is a natural dynamical system on T M (cotangent bundle). It describes the dynamics of traveling along geodesics in M:

Use dynamics to study them Geodesic flow (g t ) t 0 is a natural dynamical system on T M (cotangent bundle). It describes the dynamics of traveling along geodesics in M: Typical geodesics of interest: - Closed geodesics, i.e. periodic orbits; - Equidistributed geodesics, i.e. orbits that visit everywhere uniformly: for any open A T M. {t [0, T ] : g t (x, θ) A} lim = Vol(A) T T

Ergodic geodesic flow Geodesic flow is ergodic if almost all geodesics are equidistributed. 1...i.e. for a.e. initial point x M and initial direction θ the orbit (g t (x, θ)) t 0 is equidistributed. 1 To be honest, one needs also all sets of positive measure (Birkhoff ergodic thm).

Ergodic geodesic flow: examples Hyperbolic surfaces: i.e. M with constant negative curvature 1. - They can be obtained as quotients M = Γ \ H from actions Γ H on the hyperbolic plane H, where Γ < SL 2 (R) is a discrete subgroup.

Geodesic flow and Laplacian Take an orthonormal basis of eigenfunctions of Laplacian on M : ψj = λj ψj, 0 λ1 λ2..., hψi, ψj i = δij.

Geodesic flow and Laplacian Take an orthonormal basis of eigenfunctions of Laplacian on M : ψj = λj ψj, 0 λ1 λ2..., hψi, ψj i = δij. Numerics: If the geodesic flow on M is ergodic, then ψj 2 should flatten as j. ( i.e. in high energy/frequencies )

Geodesic flow and Laplacian: Quantum Ergodicity ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. Theorem (Shnirelman-Zelditch-Colin de Verdière, 1970s-1980s) If the geodesic flow on T M is ergodic, then lim ψ j 2 = 1 (weakly) j j S for some density 1 subsequence S N. Alexander Shnirelman (Montreal) Steve Zelditch (Northwestern) Yves Colin de Verdière (Grenoble) Density 1 needed by Hassell s constructions (Annals of Maths. 2008).

Improving Quantum Ergodicity? On hyperbolic surfaces the geodesic flow is more chaotic For example, the dynamics is exponentially mixing

Quantum Unique Ergodicity Conjecture ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. For hyperbolic surfaces, it is conjectured that one could improve QE: QUE Conjecture (Rudnick-Sarnak 1990s) If M is a hyperbolic surface, then the whole sequence converges: lim ψ j 2 = 1 j (weakly).

Quantum Unique Ergodicity Conjecture ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. Still open... only partial results available.

Quantum Unique Ergodicity Conjecture ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. Still open... only partial results available. In 2001, Thomas Watson approached QUE for arithmetic surfaces: Theorem (Watson s PhD thesis 2001) QUE conjecture is true when M = Γ \ H is compact, where Γ < SL 2 (Z) is a congruence subgroup;

Quantum Unique Ergodicity Conjecture ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. In 2006, using tools from ergodic theory, Lindenstrauss removed the Generalised Riemann Hypothesis Theorem (Lindenstrauss 2006) QUE conjecture is true when M = Γ/H is compact, where Γ < SL 2 (Z) is a congruence subgroup; ψ j are also eigenfunctions for Hecke operators. Elon Lindenstrauss

Quantum Unique Ergodicity Conjecture ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. Anantharaman s result (Annals of Maths. 2008) is the best in general: Theorem (Anantharaman 2008) If M is hyperbolic, any weak limit measure µ of the sequence ψ 1 2, ψ 2 2, ψ 3 2,... must have Hausdorff dimension dim H µ := inf{dim H A : µ(a) > 0} > 1. Nalini Anantharaman

Open problems/research directions ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. Improving Hausdorff dimension bounds (full dimension implies QUE!)

Open problems/research directions ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. Improving Hausdorff dimension bounds (full dimension implies QUE!) Multiplicities for eigenfunctions on hyperbolic surfaces? (If 1, then one can drop Hecke from Lindenstrauss s resolution) Ergodicity/mixing of the weak limits of ψ j 2 lifted to T M (only invariance w.r.t. geodesic flow is known)

Quantum ergodicity for large surfaces

Quantum ergodicity for large surfaces ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. - Quantum ergodicity for large eigenvalues (i.e. large frequencies): Equidistribution of eigenfunctions for large frequencies and a fixed surface. - Quantum ergodicity for large surfaces: Equidistribution of eigenfunctions for fixed frequencies and large surfaces.

Quantum ergodicity for large surfaces Premise: Fix any bounded interval I (0, + ) ( interval of frequencies )

Quantum ergodicity for large surfaces Premise: Fix any bounded interval I (0, + ) ( interval of frequencies ) Consider just those ψ j with λ j I.

Quantum ergodicity for large surfaces Premise: Fix any bounded interval I (0, + ) ( interval of frequencies ) Consider just those ψ j with λ j I. Now vary the geometric parameters of M (e.g. volume, genus, injectivity radius etc.) such that M becomes large (while keeping I R fixed!)

Example: injectivity radius Let M be a hyperbolic surface, i.e. constant negative curvature 1.

Example: injectivity radius Let M be a hyperbolic surface, i.e. constant negative curvature 1. Pointwise injectivity radius of M at x: InjRad(M, x) is the largest R > 0 such that B(x, R) M is isometric to a ball in H.

Example: injectivity radius ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij, I (1/4, ). Increase InjRad(M) = More eigenvalues λ j I.

Quantum ergodicity for large surfaces ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij, I (1/4, ). Do most ψ j 2, λ j I, equidistribute when InjRad(M) is large? Open Problem (Colin de Verdière) Fix a bounded interval I [0, ). Is it true that as InjRad(M)? 1 {λ j I} j:λ j I dist( ψ j 2, 1) 0 (assuming uniform spectral gap : λ 1 = λ 1 (M) is uniformly bounded from below)

Back to the 1970s ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. The classical QE theorem can be reformulated in a similar manner: Theorem (Shnirelman-Zelditch-Colin de Verdière 1970s-1980s) If the geodesic flow on M is ergodic, then as λ. 1 {λ j λ} j:λ j λ dist( ψ j 2, 1) 0 Alexander Shnirelman (Montreal) Steve Zelditch (Northwestern) Yves Colin de Verdière (Grenoble)

Back to the 1970s ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. The classical QE theorem can be reformulated in a similar manner: Theorem (Shnirelman-Zelditch-Colin de Verdière 1970s-1980s) If the geodesic flow on M is ergodic, then lim ψ j 2 = 1 (weakly) j j S for some density 1 subsequence S N. Alexander Shnirelman (Montreal) Steve Zelditch (Northwestern) Yves Colin de Verdière (Grenoble)

Previous work Modular forms for large arithmetic surfaces: - P. Nelson: Equidistribution of cusp forms in the level aspect. Duke Math. J., 160(3):467 501, 2011. - P. Nelson, A. Pitale, A. Saha: Bounds for Rankin-Selberg integrals and quantum unique ergodicity for powerful levels. J. Amer. Math. Soc., 27(1):147 191, 2014. Eigenfunctions of the Laplacian on large graphs: - N. Anantharaman, E. Le Masson: Quantum ergodicity on large regular graphs. Duke Math. J., 164(4):723 765, 2015. - S. Brooks, E. Le Masson, E. Lindenstrauss: Quantum ergodicity and averaging operators on the sphere, IMRN, to appear, 2015...but for eigenfunctions on manifolds (and Maass forms) nothing is known.

Resolution Suppose M is any compact hyperbolic surface. Theorem (Le Masson-S. 2016) Fix a compact interval I (1/4, + ) and let a L (M). Then 1 {λ j I} ˆ λ j I ˆ a(x) ψ j (x) 2 dx a(x) dx 2 a 2 C I ϱ(λ 1 ) InjRad(M), where ϱ(λ 1 ) is a function of the first eigenvalue λ 1 of on M. Colin de Verdière s problem holds for compact hyperbolic surfaces. Etienne Le Masson

Examples Surfaces which satisfy the assumptions of Colin de Verdière s problem: Compact arithmetic surfaces M = Γ \ H of congruence type when the level tends to infinity Abert, Bergeron, Biringer, Gelander, Nikolov, Raimbault, Samet: On the growth of L 2 -invariants for sequences of lattices in Lie groups. Ann. of Math. (2016), to appear Large random hyperbolic surfaces for a natural model satisfy the conditions almost surely R. Brooks, E. Makover: Random Construction of Riemann Surfaces, J. Differential Geom. 68(1), 2004.

Proof Weyl law from Benjamini-Schramm Wave Propagation via renormalised Disc Averages instead of microlocal analysis The Seven Samurais Abert, Bergeron, Biringer, Gelander, Nikolov, Raimbault, Samet On the growth of L2-invariants for sequences of lattices in Lie groups Annals of Mathematics (2016), to appear Spectral theory of radial integral operators S. Brooks E. Le Masson E. Lindenstrauss Hilbert-Schmidt norm bounds and Ergodic theory for general averaging sets A. Selberg S. Zelditch A. Gorodnik A. Nevo

Prospects Relation to large eigenvalue limits?

Prospects Relation to large eigenvalue limits? Phase space version (i.e. lift ψ j 2 to T M)

Prospects Relation to large eigenvalue limits? Phase space version (i.e. lift ψ j 2 to T M) Quantum Unique Ergodicity: arithmetic QUE, entropy results

Prospects Relation to large eigenvalue limits? Phase space version (i.e. lift ψ j 2 to T M) Quantum Unique Ergodicity: arithmetic QUE, entropy results More general manifolds, variable curvature, Maass forms