Post on 09-Feb-2022
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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 ?