We study the Birkhoff coordinates (Cartesian action angle coordinates) of the Toda lattice with periodic boundary condition in the limit where the number N of the particles tends to infinity. We prove that the transformation introducing such coordinates maps analytically a complex ball of radius R/Nα (in discrete Sobolev-analytic norms) into a ball of radius R′/Nα (with R,R′>0 independent of N) if and only if α≥2. Then we consider the problem of equipartition of energy in the spirit of Fermi-Pasta-Ulam. We deduce that corresponding to initial data of size R/N2, 0<R≪1, and with only the first Fourier mode excited, the energy remains forever in a packet of Fourier modes exponentially decreasing with the wave number. Finally we consider the original FPU model and prove that energy remains localized in a similar packet of Fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet. The proof of the theorem on Birkhoff coordinates is based on a new quantitative version of a Vey type theorem by Kuksin and Perelman which could be interesting in itself.

1. Birkhoff coordinates for the Toda Lattice in the
limit of infinitely many particles with an
application to FPU
D. Bambusi, A. Maspero
September 4, 2014

2. Introduction Main results Ingredients of the proof
Overview
Toda’s chain with N particles:
H(p, q) =
1
2
N−1
j=0
p2
j +
N−1
j=0
eqj −qj+1
periodic boundary conditions: pj+N = pN , qj+N = qj
phase space: p, q ∈ RN
:
N−1
j=0 pj =
N−1
j=0 qj = 0 .
symplectic form: j dpj ∧ dqj
Definition:
Let H(p, q) be a Hamiltonian. Birkhoff coordinates are canonical
coordinates (x, y) such that the Hamiltonian in the new coordinates is a
function only of the actions
H(x, y) = H (Ij ) , Ij =
x2
j + y2
j
2
Φ : (p, q) → (x, y) is called Birkhoff map
Aim: study the domain of analyticity of BC when N → ∞

3. Introduction Main results Ingredients of the proof
Overview
Toda’s chain with N particles:
H(p, q) =
1
2
N−1
j=0
p2
j +
N−1
j=0
eqj −qj+1
periodic boundary conditions: pj+N = pN , qj+N = qj
phase space: p, q ∈ RN
:
N−1
j=0 pj =
N−1
j=0 qj = 0 .
symplectic form: j dpj ∧ dqj
Definition:
Let H(p, q) be a Hamiltonian. Birkhoff coordinates are canonical
coordinates (x, y) such that the Hamiltonian in the new coordinates is a
function only of the actions
H(x, y) = H (Ij ) , Ij =
x2
j + y2
j
2
Φ : (p, q) → (x, y) is called Birkhoff map
Aim: study the domain of analyticity of BC when N → ∞

4. Introduction Main results Ingredients of the proof
Motivation
Dynamic of chains with many particles: metastability in FPU
HFPU (p, q) =
N−1
j=0
p2
j
2
+
(qj − qj+1)2
2
+
(qj − qj+1)3
6
+ β
(qj − qj+1)4
24
Long time behaviour of (specific) energies of normal modes:
Ek :=
1
N
|ˆpk |2
+ ω k
N
2
|ˆqk |2
2
, 1 ≤ k ≤ N − 1 ,
and of their time average
Ek (t) :=
1
t
t
0
Ek (s)ds
FPU: initial data with E1(0) = 0 and Ek (0) = 0 ∀k = 1, N − 1.

5. Introduction Main results Ingredients of the proof
Ek (t) present a recurrent behaviour, Ek (t) quickly relax to a sequence
¯Ek exponentially decreasing with k.
- Benettin, Ponno ’11: numerical analysis. Phenomenon persists in
the thermodynamic limit if initial phases are randomly chosen.
Question: Stability of metastable packet in Toda?

6. Introduction Main results Ingredients of the proof
Related results
In the regime
Energy
N
∼
1
N4
- Bambusi, Ponno ’06: analytical analysis. Persistence of the
metastable packet in FPU for long times using KdV approximation.
- Bambusi, Kappeler, Paul ’13: spectral data Toda converges to
spectral data of KdV

7. Introduction Main results Ingredients of the proof
Birkhoff map for Toda’s chain
Linear Birkhoff variables Xk , Yk in which the quadratic part of HToda is
H0 =
N−1
k=1
ωk
X2
k + Y 2
k
2
, ωk := 2 sin
kπ
2N
(1)
Theorem (Henrici, Kappeler ’08)
For any integer N ≥ 2 there exists a global real analytic symplectic
diffeomorphism ΦN : RN−1
× RN−1
→ RN−1
× RN−1
, (X, Y ) = ΦN (x, y)
with the following properties:
(i) The Hamiltonian HToda ◦ ΦN is a function of the actions Ik :=
x2
k +y2
k
2
only, i.e. (xk , yk ) are Birkhoff variables for the Toda Lattice.
(ii) The differential of ΦN at the origin is the identity: dΦN (0, 0) = 1.
Main goal: Analyticity properties of the map ΦN as N → ∞.
ΦN is globally analytic but it might (and actually does!) develop
singularities as N → ∞.

8. Introduction Main results Ingredients of the proof
Topology: Sobolev space of sequences
For any s ≥ 0, σ ≥ 0 define
(X, Y )
2
s,σ :=
1
N
N−1
k=1
[k]
2s
N e2σ[k]N
ωk
|Xk |2
+ |Yk |2
2
,
where [k]N := min(|k|, |N − k|).
Let Bs,σ
(R) the ball or center 0 and radius R in this topology.
Remarks:
- (X, Y )
2
0,0 ∼
Energy
N
- σ > 0: exponentially localized states

9. Introduction Main results Ingredients of the proof
Main theorem
Theorem (Bambusi, M.)
For any s ≥ 1, σ ≥ 0 there exist strictly positive constants Rs,σ,Rs,σ,
such that for any N 1, the map ΦN is real analytic as a map from
Bs,σ
(Rs,σ/N2
) to Bs,σ
(Rs,σ/N2
). The same holds for the inverse map
Φ−1
N possibly with a different Rs,σ.
Remark:
- We work in the regime
Energy
N
∼ 1
N4
- Stability of metastable packett of Toda for all times

10. Introduction Main results Ingredients of the proof
Application to FPU
Theorem
Consider the FPU system. Fix s ≥ 1 and σ ≥ 0; then there exist
constants R0, C2, T, such that the following holds true. Consider a real
initial datum such that
E1(0) ≤
R
N4
, R ≤ R0 Ek (0) = 0 , k = 1 .
Then, along the corresponding solution, one has
Ek (t) ≤
R
N4
e−2σk
, ∀ 1 ≤ k ≤ N/2 ,
for
|t| ≤
TN4
R2
·
1
|β − 1| + C2R 1
N2
.

11. Introduction Main results Ingredients of the proof
Keys ingredients in the proof
Construction of ΦN :
Vey type theorem for infinite dimensional systems (Kuksin, Perelman
’10);
perturbative analysis of spectral theory of Lax operator of Toda;

12. Introduction Main results Ingredients of the proof
Kuksin-Perelman theorem ’10
2
w := (ξj )j≥1, ξj ∈ C : j w2
j |ξj |2
< ∞ , as a real Hilbert space.
Let ξ, η = Re j ξj ηj , ω0(ξ, η) = iξ, η .
Theorem
Assume that there exists a real analytic map Ψ : Bw
(ρ) → 2
w , ρ > 0,
with
(i) dΨ(0) = 1
(ii) Ψ = (ψj )j≥1 fulfills: |ψj |2
, j ≥ 1, are in involution;
(iii) analyticity of Cauchy majorants of Ψ and dΨ∗
Then there exists Φ = (φj )j≥1, Φ : Bw
(ρ ) → 2
w , such that (i), (ii), (iii)
hold and
1 the foliation in a neighborhood of the origin in 2
w defined by
|ψj |
2
= cj , ∀j is the same foliation as |φj |
2
= cj , ∀j
2 Φ∗
ω0 = ω0; namely {φj }j are canonical coordinates.

13. Introduction Main results Ingredients of the proof
Lax matrices
Flaschka coordinates: bj = −pj , aj = e
1
2 (qj −qj+1)
− 1, not canonical.
L(b, a) =











b0 1 + a0 0 . . . 1 + aN−1
a0 b1 1 + a1
...
.
.
.
0 1 + a1 b2 . . . 0
.
.
.
...
...
... 1 + aN−2
1 + aN−1 . . .
... 1 + aN−2 bN−1











,
The equations of motion are equivalent to: d
dt L = [B, L]
Eigenvalues of L are conserved quantities λj (b(t), a(t)) = λj (b0, a0)
Poisson commute {λj , λi } = 0

14. Introduction Main results Ingredients of the proof
Jacobi matrix spectrum and spectral gaps
Theorem
For (b, a) ∈ RN
× RN
>−1, the spectrum of L±
is a sequence of eigenvalues
satisfying the relation
λ0 < λ1 ≤ λ2 < λ3 ≤ λ4 < . . . < λ2N−3 ≤ λ2N−2 < λ2N−1.
If N is even λ0, λ3, λ4, . . . , λ2N−1 are eigenvalues of L+
, while the others
are eigenvalues of L−
. Viceversa if N is odd.
Definition
The j-th spectral gap is γj := λ2j − λ2j−1
Fact:
1 Battig, Gr´ebert, Guillot, Kappeler ’93 the sequence of gaps determine all the spectrum;
2 {γ2
j (b, a)}j are set of complete integrals, real analytic and in involution

15. Introduction Main results Ingredients of the proof
Perturbative analysis
L(b, a) =









0 1 0 . . . 1
1 0 1
...
.
.
.
0 1 0 . . . 0
.
.
.
...
...
... 1
1 . . . . . . 1 0









L0
+









b0 a0 0 . . . a2N−1
a0 b1 a1
...
.
.
.
0 a1 b2 . . . 0
.
.
.
...
...
... a2N−2
a2N−1 . . . . . . a2N−2 b2N−1









Lp:perturbation
Eigenvalues L0: λ0
0 = −2; λ0
2l−1 = λ0
2l = −2 cos lπ
N ; λ0
2N−1 = 2
Distance of unperturbed eigenvalues:
λ0
2l − λ0
2j ∼
l2
− j2
N2
Aim: reconstruct the spectral data of L from the spectral data of L0.

16. Introduction Main results Ingredients of the proof
Spectral perturbation
(1) Spectral projectors on the eigenspaces of L(b, a)
Ej (b, a) subspace associated to λ2j−1(b, a), λ2j (b, a).
Pj (b, a) spectral projection on Ej (b, a):
Pj (b, a) = −
1
2πi Γj
(L(b, a) − λ)
−1
dλ,
(2) Transformation operator
Uj (b, a) = 1 − (Pj (b, a) − Pj0)
− 1
2 Pj (b, a)
maps isometrically the unperturbed eigenspace into the perturbed one
ImUj (b, a) = Ej (b, a) , Uj (b, a)f C2N = f C2N if f ∈ Ej0
(3) define
fj (b, a) = Uj (b, a)fj0
zj (b, a) = 2
N
ω j
N
−1/2
L(b, a) − λ
0
2j fj (b, a), fj (b, a)
(4) prove that zj j
’s (as functions of (X, Y )) satisfy Kuksin-Perelman assumptions. In particular for b, a real
one has
|zj (b, a)|
2
= 2
N
ω j
N
−1
γ
2
j (b, a)
Apply Kuksin-Perelman with weights w2
j = N4
[j]2s
N e2σ[j]N 2
N
ω j
N
.

17. Introduction Main results Ingredients of the proof
Theorem (Bambusi, M.)
Assume that for some s ≥ 1, σ ≥ 0 there exist strictly positive R, R and
α ≥ 0, Ns,σ ∈ N, s.t., for any N ≥ Ns,σ, the map ΦN is analytic in the
complex ball Bs,σ
(R/Nα
) and fulfills
ΦN Bs,σ R
Nα
⊂ Bs,σ R
Nα
, (2)
then one has α ≥ 2.
Theorem (Bambusi, M.)
For any s ≥ 1, σ ≥ 0 there exist strictly positive R, C, Ns,σ ∈ N, such
that, for any N ≥ Ns,σ, α ∈ R, the quadratic form d2
ΦN (0, 0) fulfills
sup
v∈Bs,σ
R ( R
Nα )
d2
ΦN (0, 0)(v, v) s,σ
≥ CR2
N2−2α
. (3)
As N → ∞, the real diffeomorphism ΦN develops a singularity at zero in
the second derivative.

18. Introduction Main results Ingredients of the proof
Proof of the inverse theorem
(1) Taylor expansion of ΦN at the origin:
ΦN = 1 + QΦN + O( ξ, η 3
s,σ) ,
QΦN is a quadratic polynomial and Hamiltonian vector field:
QΦn = XχΦN
χΦN
cubic complex valued polynomial.
(2) Taylor expansion of HToda ◦ ΦN at the origin
HToda ◦ ΦN = H0 + H0, χΦN
+ H1 + h.o.t.
ΦN Birkhoff map ⇒ HToda ◦ ΦN in BNF ⇒ only terms of even degree.
H0, χΦN
+ H1 = 0
If χΦN
= |K|+|L|=3 χK,L ξK ηL, then χK,L =
HK,L
−iω·(K−L)
(3) Compute explicitly! For ¯v = ((ξ1, 0, 0, ..., 0), (¯ξ1, 0, 0, ..., 0))
2ω1 − ω2 ∼
1
N3
=⇒ QφN (¯v)
s,σ
≥ N2
¯v s,σ