Quantum Chaos, eigenvalue statistics (& Fibonacci)

Zeev Rudnick, Tel Aviv & IASJoint with V. Blomer, J. Bourgain & M. Radziwill

Hearing the shape of a drumM. Kac (1966): What can you tell about a “drum” from its spectrum?

“drum”= compact Riemannian surface planar domain with piecewise smooth boundary (“billiard”)

Eigenvalue problem: Δψ+ E ψ =0, (ψ =0 on bdry).

Δ= Laplace-Beltrami operator

●There is an orthonormal basis consisting of eigenfunctions.

● The spectrum of Δ is discrete and accumulates at infinity: 0< E1 ≤… ≤ En ≤ … →∞ .

What can be heardWe can hear the area of the drum:

Weyl’s law (1911):

{ } ∞→•≤ XXMareaXEn ,4

)(~#π

We can hear the length of the boundary of the drum and the connectivity h:

1 21

1 2

area( ) length( ) 1 ( )~ (1), 06

1/ 4 , 1/ 8

nE

n

M M h Me c o

c

c

c

t tt t

π π

∞−

=

∂ −+ +

= =

−∑

h=3

(smooth boundary)

Question: Can we hear the dynamics?

Geodesic flow

Billiard flow

Billiard dynamicsplanar billiards: motion in planar domain B (configuration space)

angle of reflection = angle of incidencePhase space (description of a particle):S*B=BxS1 = { position x of point & direction vector ξ of motion}

Regularity vs. chaosClassification of (conservative, Hamiltonian) dynamical systems, e.g. planar billiards, geodesic flows on surfaces

Chaotic:

•Typical orbits densely cover all of available phase space (ergodicity)

•Exponential divergence of nearby trajectories (hyperbolicity).

Regular (integrable):

•A full set of constants of motion

•dynamics confined to invariant tori in phase space.

•Linear separation of trajectories

Can we “hear” the difference between chaos and integrability ?

Level spacing distribution P(s) :=limiting distribution of the normalized gaps δn between adjacent levels

E1 E2E6E4E3

E5

spacing meannn

nEE −

= +1:δ

{ } ∫ →<≤ ∞→

x

Nn dssPxNnN 0

)(:#1 δ

Statistical models:

a) Uncorrelated levels: En independent, uniform in [0,1] (homogeneous Poisson process on the line with intensity 1). level spacing distribution is P(s)=exp(-s)

b) En = eigenvalues of a random NxN symmetric matrix (Gaussian Orthogonal Ensemble)H=HT , matrix elements=independent real Gaussians

P(s) was computed by Gaudin and Mehta (1960’s)

Δψ+ E ψ =0, (ψ =0 on bdry).

Hearing the dynamics: The universality conjectures for the level spacing distribution

Berry & Tabor 1977: For integrable dynamics expect* Poisson statistics

rectangular billiard, aspect ratio =√π/3

Bohigas, Giannoni & Schmit (1984): for chaotic dynamics expect* GOE statistics

*=Exclusions apply

Level spacing for regular graphs

D. Jakobson, S. Miller, I. Rivin and Z. Rudnick (1996) conjecture that the level spacing distribution of a random k-regular graph on N vertices tends to GOE for large N.

rectangular billiards

For the square (α=1), P(s) does not exist because of multiplicities in the spectrum (many ways of writing E=m2+n2).

However for Diophantine α, e.g. α =(1+√5)/2 , numerical evidence indicates Poissonian spacing distribution:

Theoretical evidence (pair correlation function): Sarnak (1996), Eskin-Margulis-Mozes (2005).

The spectrum of a rectangle of height 𝜋𝜋𝛼𝛼

and width π are the numbers 𝛼𝛼m2 + 𝑛𝑛2 (m,n>0 integers).

𝜋𝜋𝛼𝛼

π

Minimal gaps

The minimal gap between the first N levels (normalized to have unit mean spacing):

min 1( ) min{ :1 }j jN E E j Nδ += − ≤ ≤

For uncorrelated levels (Poisson sequence), the minimal gap is of size 𝛿𝛿min𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃(𝑁𝑁) ≈ 1𝑁𝑁

- exercise (the birthday problem)

For GOE, it is expected to be much larger, 𝛿𝛿min𝐺𝐺𝐺𝐺𝐺𝐺 𝑁𝑁 ≈ 1𝑁𝑁1/2

For CUE/GUE, the minimal gap is of size 𝛿𝛿min𝐺𝐺𝐺𝐺𝐺𝐺 𝑁𝑁 ≈ 1𝑁𝑁1/3 Vinson 2001, Ben Arous & Bourgade (2013)

Goal: 𝛿𝛿𝑚𝑚𝑃𝑃𝑃𝑃 𝑁𝑁 for the eigenvalues of the rectangular billiard

Minimal gaps for rectangular billiardsThe spectrum of a rectangle of height 𝜋𝜋

𝛼𝛼and width π are the numbers 𝛼𝛼m2 + 𝑛𝑛2 (m,n integers).

2 2 2 22 2 2 2

min ( ) min | ( ) ( ) |m n m n N

N m n m nαα α

δ α α′ ′+ ≠ + ≤

′ ′= + − +

Goals: Show that fora) quadratic irrationalities 2) “typical” α (in measure theoretic sense)

min 1 (1)

1( ) oNN

αδ −

-consistent with Poisson statistics !

Caveat emptor: the exponent is consistent with Poisson, but not the fine structure.

Small gaps and rational approximations2 2 2 2,m n m nλ α λ α′ ′ ′= + = +

2 2 2 2( ) ( )m m n n q ppq

λ λ α α′ ′ ′− = − − − = −

So 𝜆𝜆 − 𝜆𝜆′ small gives a good rational approximation p/q to α

Typically, the best possible approximations satisfy

- can be constructed from the continued fraction expansion of α.

2

1 1| | | |p q pq q q

α α− ≤ ⇔ − <

Quadratic irrationalities: All gaps are at least 1/N

2 2 2 2,m n m nλ α λ α′ ′ ′= + = +2 2 2 2( ) ( )m m n n q p

pqλ λ α α′ ′ ′− = − − − = −

Note: quadratic irrationalities (e.g. √𝐷𝐷) are “badly approximable”:( )| | , ,cp p qqqαα − ≥ ∈

2 2

1 1| , #{eigenvalue| s }Nm m N

λ λ λ′ ≥ =−

− ≥ ≤′

N.B. only a measure zero set of reals are badly approximable, so quadratic irrationalities are atypical

Badly approximable ↔ bounded partial quotients in continued fraction expansion.

Finding small gaps version 1Suppose we find a rational approximation p/q to α

2

1 1| | | |p q pq q q

α α− ≤ ⇔ − <

1, 1, 1Take 1, m q p n pm q n′ ′= − = + =+ −=

Define eigenvalues 2 2 2 21 2

2: , :m n m n qλ α λ α ′ ′= + = + ≈

2 2 2 21 2

1( ) ( )m m n n q pq

λ λ α α′ ′− = − − − ≈ − <

{eigenvalues }: #N λ λ≤ ≈=

1 2 1/2

1 1 1|max(

|, )q N

λ λλ λ

− ≤ ≈′

then

1/2min1( )N N

αδ

Want to beat this!

Finding small gaps version 2Suppose we find a rational approximation p/q to α

,' ,' ' , p p p qp q q q qp p q′ ′′ ′ ′′ ′ ′′≈ = ≈= ≈ ≈

2

1pq q

α − ≤

' '',Take ' '', '' '' , '' m q q n p p n p pm q q ′ ′= − == = + −+all of size about √q

Define eigenvalues 2 2 2 2

1 2: , :m n m n qλ α λ α ′ ′= + = + ≈2 2 2 2

1 21( ) ( )m m n n q q p p q pq

λ λ α α α′ ′ ′ ′′ ′ ′′− = − − − ≈ − = − <

{eigenvalues }: # qN q≤ ≈=

1 21 1 1|

max( , )|

q Nλ λ

λ λ− ≈ ≈

′

s.t. p and q are essentially squares

then

min1( )N N

αδ

Example: The golden ratio11 5 11 lim12 1 11

1

n

nn

FF

α +

→∞

+= = + =

++

+𝐹𝐹𝑃𝑃 = the Fibonacci sequence 2 1 0 1, 0, 1n n nF F FF F+ += + = =

Lets implement this strategy when α is the golden ratio

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fn 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987

Divisibility properties of rational approximants2| / | 1/p q qα − <

Rational approximations to the golden ratio 𝛼𝛼 = (1 + √5)/2, are obtained from the Fibonacci sequence 𝑝𝑝𝑃𝑃 = 𝐹𝐹𝑃𝑃+1, 𝑞𝑞𝑃𝑃 = 𝐹𝐹𝑃𝑃

Strong divisibility property (Lucas, 1876): gcd( , )gcd( , )m n m nF F F=

2 1 0 1, 0, 1n n nF F FF F+ += + = =

Goal: To find rational approximants with p, q essentially squares

𝐹𝐹2𝑃𝑃 is divisible by 𝐹𝐹𝑃𝑃 which is of size roughly 𝐹𝐹2𝑃𝑃, so 𝐹𝐹2𝑃𝑃 is essentially a square

N.B. For many quadratic irrationalities, a version of strong divisibility holds for both numerators and denominators of the convergents 𝑝𝑝/𝑞𝑞

In particular 𝐹𝐹𝑃𝑃 divides 𝐹𝐹𝑘𝑘𝑃𝑃

1 1( ( ) )2

n nnF α

α= − −

Claim: The even Fibbonacci numbers 𝐹𝐹2𝑃𝑃 are essentially squares !

Small gaps when α=quadratic irrationality

Theorem (2016): For 𝛼𝛼 = 𝐷𝐷 , or golden ratio, (& some other quadratic irrationalities), for all N

min 1 (1)

1( ) oNN

αδ −

-consistent with Poisson statistics !

2| / | 1/p q qα − <Since we found rational approximants with p, q essentially squares

Note: We do not know how to do all quadratic irrationalities, e.g. 𝛼𝛼 = 3 + √2

Typical α

For “typical” α (in the sense of measure theory), we can also exhibit Poissonian gaps

min 01 (1)

1( ) , ( )oN N NN

αδ α− ∀ >

Note: The Riemann Hypothesis implies: 𝜁𝜁 𝜎𝜎 + 𝑖𝑖𝑖𝑖 ≪ |𝑖𝑖|𝜀𝜀 , ∀𝜀𝜀 > 0 (12≤ 𝜎𝜎 ≤ 1, 𝑖𝑖 ≥ 2).

We do this by showing that almost all α have many rational approximations 2

1|| pq q δα −− <

with both p,q “essentially squares”.

A key input is a bound for the Riemann zeta function 𝜁𝜁 𝑠𝑠 near the line Re(s)=1

3/2(1 ) (1) 1( ) | | | 1, | | 2| ,2

A oit t tσζ σ σ− ++ ≤ ≤ ≥

Richert (1967): A=100,…., Heath Brown (2016) A=1/2

Open problems

$$ Show that there are arbitrarily large gaps in the sequence E=m2+√5n2 (implied by Poisson spacings!)

Can we show 𝛿𝛿𝑚𝑚𝑃𝑃𝑃𝑃(𝑁𝑁) < 1𝑁𝑁

for other quadratic irrationalities, e.g. 𝛼𝛼 = 3 + √2 ?

Algebraic irrationalities of higher degree, e.g. 3 2 ? or π? or e ?

Size of minimal gap in GOE?

Minimal gaps for other integrable examples?

Minimal gaps in a nonintegrable example ?