Quantum Unique Ergodicity Introduction 1 Planar Exercise Lior - PowerPoint PPT Presentation

QUE Lior Silberman The University of British Columbia Quantum Unique Ergodicity Introduction 1 Planar Exercise Lior Silberman 1 2 Classical and quantum mechanics The University of British Columbia 3 Arithmetic eigenfunctions 4 Without arithmetic April 30, 2020 5 Scarring for quasimodes 1 lior@math.ubc.ca ; https://www.math.ubc.ca/~lior/

Scarring QUE [Heller 1984] Lior Silberman The University of British Columbia Introduction 1 Planar Exercise 2 Classical and quantum mechanics 3 Arithmetic eigenfunctions 4 Without arithmetic 5 Scarring for quasimodes

Other examples QUE Lior Silberman The University of British Columbia Introduction 1 Planar Exercise 2 Classical and quantum mechanics 3 Arithmetic eigenfunctions 4 Without arithmetic 5 Scarring for quasimodes (Images: Bäcker, Stromberg)

Quantum Unique Ergodicity QUE Problem : What happens as λ → ∞ ? What is a “feature”? Lior Silberman The University Pointwise How big does � u λ � ∞ get as λ → ∞ ? of British Columbia | u λ | 2 f d vol as λ → ∞ ? ´ Weakly What happens to Introduction 1 Planar Theorem (Schnirel’man–Zelditch–Colin de Verdière) Exercise 2 Classical and If the billiard dynamics is chaotic (ergodic) then for almost all quantum | u λ | 2 f d vol → mechanics 1 eigenfunctions ´ ´ f d vol vol 3 Arithmetic eigenfunctions Conjecture (Rudnick–Sarnak) 4 Without arithmetic 5 Scarring for On a manifold of negative sectional curvature, replace “almost quasimodes all” with “all”. Hassell 2008: For stadium billiard, can’t remove “almost”.

Plan QUE Lior Silberman The University of British Columbia Introduction 1 Bounds on eigenfunctions on the tree and in the plane 1 Planar 2 “Classical” and “quantum” mechanics Exercise 2 Classical and 3 “Arithmetic” QUE quantum mechanics 4 Without arithmetic 3 Arithmetic eigenfunctions 5 Negative results for approximate eigenfunctions 4 Without arithmetic 5 Scarring for quasimodes

A pointwise bound QUE Theorem (Hörmander bound) Lior Silberman n − 1 4 � u λ � 2 . � u λ � ∞ ≤ C λ The University of British Columbia Proof (in spirit). Introduction 1 Planar Use u λ as the initial condition for an evolution equation, e.g. Exercise 2 Classical and h ∂ quantum i ¯ ∂ t ψ ( t , x ) = − ∆ x ψ ( t , x ) . mechanics 3 Arithmetic eigenfunctions 4 Without ψ ( t , x ) = e − i λ t u λ ( x ) is a solution. arithmetic 5 Scarring for But solutions tend to follow classical trajectories. quasimodes So ψ ( t , x ) looks like u λ “averaged” over a region near x , and can relate ψ ( t , x ) to � u λ � 2 .

Some physics QUE Lior Silberman The University of British Columbia Introduction 1 Planar Exercise 2 Classical and quantum mechanics 3 Arithmetic eigenfunctions 4 Without arithmetic 5 Scarring for quasimodes

The Space of Lattices QUE Lior Silberman The University of British Move to curved geometry and periodic boundary conditions. Columbia P n = Introduction { symmetric, positive-definite n -matrices X , det ( X ) = 1 } 1 Planar Exercise SL n ( R ) acts by g · X := gXg t , preserving metric: 2 Classical and quantum i = 1 | log µ i | 2 � 1 / 2 � ∑ n mechanics dist ( Id , X ) = , µ i = eigenvalues . 3 Arithmetic For n = 2, P n is the hyperbolic plane. eigenfunctions 4 Without Study the quotient L n = SL n ( Z ) \P n arithmetic = isometry classes of unimodular lattices in R n . 5 Scarring for quasimodes

Arithmetic QUE QUE Lior Silberman Domain has number-theoretic symmetries , manifest as The University Hecke operators (T p f = ∑ y ∼ x f ( y ) ) of British Columbia Introduction T p ∆ = ∆ T p , T p T q = T q T p 1 Planar Exercise Study limits of joint eigenfunctions. Start with n = 2: 2 Classical and quantum Rudnick–Sarnak 1994: limits don’t scar on closed mechanics geodesics. 3 Arithmetic eigenfunctions Iwaniec–Sarnak 1995: savings on Hörmander bound 4 Without small balls have small mass arithmetic 5 Scarring for Bourgain–Lindenstrauss 2003: limits have positive entropy quasimodes small dynamical balls have small mass Lindenstrauss 2006: from this get equidistribution.

Higher-rank QUE QUE Lior Silberman The University What about n ≥ 3 ? of British Columbia No longer negatively curved – extend Rudnick–Sarnak Introduction conjecture 1 Planar S–Venkatesh 2007: limits respect Weyl chamber flow Exercise 2 Classical and S–Venkatesh: (non-degenerate) limits are uniformly quantum mechanics distributed if n is prime (division algebra quotient). 3 Arithmetic eigenfunctions QUE Results proceed by 4 Without Lift to the bundle where classical flow lives. arithmetic 5 Scarring for Bound mass of dynamical balls (“positive entropy”) quasimodes Apply measure-classification results to identify the limit.

QUE on general manifolds QUE Lior Silberman The University of British Columbia In µ ( B ( C , ε )) ≪ ε h , h measures the complexity of µ . Introduction 1 Planar Related to the metric entropy h ( µ ) . Exercise Anantharaman ~2003: On a manifold of negative 2 Classical and quantum curvature, every quantum limit has positive entropy. mechanics 3 Arithmetic Anatharaman + others: quantitative improvements eigenfunctions Idea: “quantum partition” 4 Without arithmetic 5 Scarring for quasimodes

Applied to the space of lattices QUE Lior Silberman L n not negatively curved (has flats). The University of British Nevertheless limits have positive entropy: Columbia Microlocal calculus adapted to locally symmetric spaces. Introduction Entropy contribution from “rapidly expanding” directions. 1 Planar Exercise Measure-classification 2 Classical and Restriction on possible ergodic components. quantum mechanics Use quantitative entropy bound. 3 Arithmetic eigenfunctions Theorem (Anantharaman–S) 4 Without arithmetic Let X = Γ \P 3 be compact. Then every quantum limit on X is 5 Scarring for quasimodes at least 1 4 Haar measure.

New uncertainty principle QUE Lior Silberman The University Density is now known for n = 2: of British Columbia Theorem (Dyatlov–Jin 2018) Introduction 1 Planar Every quantum limit on a compact hyperbolic surface has full Exercise 2 Classical and support. quantum mechanics 3 Arithmetic eigenfunctions Theorem (Dyatlov–Jin–Nonnenmacher 2019) 4 Without arithmetic The same on a compact surface with Anosov geodesic flow. 5 Scarring for quasimodes

Approximate eigenfunctions QUE Lior Silberman The University of British Method of Anantharaman applies to approximate Columbia eigenfunctions. Introduction √ 1 Planar λ Exercise � ∆ u λ + λ u λ � ≤ C 2 Classical and log λ quantum mechanics Entropy depends on C . 3 Arithmetic eigenfunctions 4 Without Problem arithmetic 5 Scarring for What are the possible limits of these “log-scale quasimodes”? quasimodes

Scarring of quasimodes QUE Problem Lior Silberman The University On a manifold M, construct log-scale quasimodes which of British Columbia concentrate on singular measures √ Introduction λ 1 Planar � ∆ u λ + λ u λ � ≤ C Exercise log λ 2 Classical and quantum ˆ ˆ mechanics | u λ | 2 f d vol = lim f d µ 3 Arithmetic λ → ∞ eigenfunctions 4 Without arithmetic Brooks 2015: M = hyperbolic surface , µ = geodesic . 5 Scarring for quasimodes Uses the geometry explicitely (Eisenstein packets) Eswarathasan–Nonnenmacher 2016: M =any surface , µ = hyperbolic geodesic .

High dimensions QUE Theorem (Eswarathasan–S 2017) Lior Silberman The University Let M be a hyperbolic manifold, and let N ⊂ M be a compact of British totally geodesic submanifold. Then there is a sequence of Columbia log-scale quasimodes uniformly concentrating on N. Introduction 1 Planar Exercise Includes the case N = closed geodesic . 2 Classical and Actually, any quantum limit on N achievable. quantum mechanics 3 Arithmetic Corollary eigenfunctions (M compact) every invariant measure on M is a limit of 4 Without arithmetic log-scale quasimodes. 5 Scarring for quasimodes Proof. In a hyperbolic system, closed orbits are dense in the space of invariant measures.