CONVERGENCE OF A

QUANTUM NORMAL FORM

AND AN EXACT

QUANTIZATION FORMULA

Sandro Graffi

Universita di Bologna, Italy

Thierry Paul

Ecole Polytechnique, France

Hamiltonian family:

Hε(ξ, x) = L(ξ) + εV(x, ξ), ε ∈ R (1)

Hε has a normal form if ∀N ∈ N there is a

canonical map Cε,N such that:

(Hε Cε,N)(ξ, x) = L(ξ) +N∑

k=1

Bk(ξ)εk

+εN+1RN+1,ε(ξ, x)

Quantum counterpart: Hε = L+εV operator

of Weyl symbol Hε.

H admits a uniform QNF if ∀N ∈ N there is

UN,ε(~) = UN,ε(~)∗ such that

UN,ε(~) = eiWN,ε(~)/~, WN,ε(~) =N∑

k=1

Wk(~)εk,

UN,ε(~)HεU∗N,ε(~) = L +

N∑k=1

Bk(L, ~)εk

+εN+1RN+1,ε(~).

Bk(ξ; 0) = Bk(ξ); Wk(ξ, x,0) =Wk(ξ, x),

RN+1,ε(x, ξ; 0) = RN+1,ε(x, ξ)

If the QNF converges uniformly with respect

to ~ (notion to be made precise) then

W∞,ε(ξ, x; ~) := 〈ξ, x〉+∞∑

k=1

Wk(ξ, x; ~)εk

B∞,ε(ξ, ~) := L(ξ) +∞∑

k=1

Bk(ξ; ~)εk (2)

eiW∞,ε(~)/~Hεe−iW∞,ε(~)/~ = B∞,ε(L; ~).

Therefore, if the eigenvalues of L are L(n~):

(A1) Exact quantization formula

λn,ε(~) = B∞,ε(n~, ~), |ε| < ε∗ (3)

for the eigenvalues of Hε.

(A2) Classical NF convergent, |ε| < ε∗.

Problem: find explicit conditions on L and

V ensuring uniform convergence of the QNF.

Model: L := Lω(ξ) = ω1ξ1+ . . .+ωlξl. Phase

space: Rl × Tl, 2l-cylinder.

Quantum counterpart:

Lω := −i~(ω1∂x1 + . . . + ωl∂xl) on L2(Tl), ei-

genvalues Lω(n~). Let:

(t, x; ~) 7→ F(t, x; ~) ∈ C∞(R× Tl × [0,1];C),

F(t, x; ~) =∑

q∈Zl

Fq(t, ~)ei〈q,x〉.

Define Fω(ξ, x, ~) ∈ C∞(Rl × Tl × [0,1];C):

Fω = F(Lω(ξ), x, ~) =∑

q∈Zl

Fω,q(ξ, ~)ei〈q,x〉 (4)

Fq(ξ, ~) =1√2π

∫RFq(p, ~)e−ipξ dp, (5)

Fω,q(ξ, ~) =1√2π

∫RFq(p, ~)e−ipLω(ξ) dp (6)

Let ρ > 0, σ > 0, t 7→ F(t; ~) smooth. Denote:

‖F(~)‖σ :=∫R|F(p, ~)|eσ|p| dp, ‖F‖σ = max

[0,1]‖F(~)‖σ

J (σ; ~) = (ξ, ~) 7→ F(Lω(ξ); ~) | ‖F(~)‖σ < +∞;

J (σ) := (ξ, ~) 7→ F(Lω(ξ), ~) | ‖F‖σ < +∞

Consider (t, x, ~) 7→ F(t, x; ~) smooth. Set:

‖F(~)‖ρ,σ =∑

q∈Zl

eρ|q|‖Fq(~)‖σ;

‖F‖ρ,σ =∑

q∈Zl

max[0,1]

eρ|q|‖Fq(~)‖σ

J (σ, ρ; ~) := F(Lω(ξ); ~) | ‖F(~)‖ρ,σ < +∞;

J (ρ, σ) := F(Lω(ξ); ~) | ‖F‖ρ,σ < +∞ (7)

F (~) = quantization of F. We will see:

‖F (~)‖ := ‖F (~)‖L2→L2 ≤ ‖F(~)‖ρ,σ,

‖F‖ := ‖F‖L2→L2 := max~∈[0,1]

‖F (~)‖ ≤ ‖F‖ρ,σ

Theorem 1 Assume:

1. ∃ γ > 0, τ > l − 1 such that

|〈ω, ν〉|−1 ≤ γ|ν|τ , ν ∈ Zl, ν 6= 0. (8)

2. ‖V‖ρ,σ < +∞ for some ρ > 0, σ > 0.

Then Hε = Lω+εVω admits a QNF such that:

(i) It converges in the ‖ · ‖ρ/2,σ/2 norm, with

uniform convergence radius larger than

ε∗ :=[5‖V ‖ρ,σe8+2τ+γ+ζ

]−1,

ζ :=∞∑

k=1

ln k2−k

(ii) Let B∞(ξ, ε, ~) be defined by (2). Then

for any |ε| < ε∗ ∃B∞(t, ε, ~) : R × [0,1] → C

with ‖B∞(t, ε, ~)‖σ/2 < +∞ such that

B∞,ω(ξ, ε, ~) := B∞(t, ε, ~)|t=Lω(ξ); |ε| < ε∗.

(iii) Set ε∗0(ρ, σ) := ε∗(ρ, σ) and: ∀ r ∈ N,

D∗r(ρ, σ) := ε ∈ C | |ε| < ε∗r(ρ, σ),

D∗r(ρ, σ) := ε ∈ C | |ε| < ε∗r(ρ, σ)

ε∗r(ρ, σ) :=[5‖V‖ρ,σe8+2τ+γe3r+ζ

]−1.

Then there exist Cr(ρ, σ, ε∗) > 0 such that,

for ε ∈ D∗r(ρ, σ):

r∑γ=0

max~∈[0,1]

‖∂γ~B∞,ω(ξ; ε, ~)‖σ/2 ≤ Cr, r = 0,1, . . .(9)

(iv) Denote λn,ε(~) : n ∈ Zl the eigenvalues

of Hε. Let |ε| < ε∗r, r = 0,1, . . .. Then ~ 7→

Bs(t, ~) ∈ Cr([0,1];Cω(t ∈ C | |=t| < σ/2)

and

λn,ε(~) = B∞,ω(n~, ε, ~) (10)

= Lω(n~) +∞∑

s=1

Bs(Lω(n~), ~)εs

Remarks

[(1)] (10) first example of exact quantization

formula, apart exactly solvable models.

[(2)] The EBK quantization formula:

λEBKn,ε (~) := 〈ω, n〉~+

∞∑k=1

Bk(n~)εk = B∞,ε(n~)

reproduces Spec(Hε) up to order ~.

[(3)] Apart Cherry’s case (S.G., C.Villegas-

Blas, CMP 2008) no convergence criterion

for the QNF, not even for the classical NF.

Thm 1: a convergence criterion for the NF.

Lω(ξ) harmonic-oscillator Hamiltonian in R2l,

ξi > 0. Assuming (8), holomorphy and:

Bs(ξ) = Fs(Lω(ξ)), s = 0,1, . . . (11)

Russmann (1967): Birkhoff NF converges.

No examples. Here we construct Fs(t; ~) s.t.:

Bs(ξ; ~) = Fs(Lω(ξ); ~), s = 0,1, . . . (12)

No application to Russmann’s case: the map

T (ξ, x) = (p, q) : R2l \ 0,0 ↔ R2l \ 0,0

pi = −√

ξi sinxi, qi =√

ξi cosxi, i = 1, . . . , l

is canonical but does not preserve holomor-

phy at the origin. On the other hand:

(Hε T )(p, q) =l∑

s=1

ωs(p2s + q2s ) + ε(V T )(p, q)

:= H0(p, q) + εH1(p, q)

(V T )(p, q) =1

2

∑k∈Zl

(RVk H0)(p, q)×

×

p1 + iq1√p21 + q21

k1

· · ·

pl + iql√p2l + q2l

kl

+

+∑

k∈Zl

(=Vk H0)(p, q)×

×

p1 − iq1√p21 + q21

k1

· · ·

pl − iql√p2l + q2l

kl

Therefore:

Corollary 1 The Birkhoff NF of the Hamilto-

nian Hε(p, q) = H0(p, q)+ εH1(p, q) is conver-

gent on any compact of R2l \ 0,0, |ε| < ε∗.

Sketch of proof of statements (i) and (ii).

Weyl quantization of the cylinder

Dual of Rl×Tl: Rl×Zl. Relevant Heisenberg

group: Hl(Rl×Zl×R), the subgroup of Hl(Rl×

Rl × R) with law:

(u, t) · (v, s) = (u + v, t + s +1

2Ω(u, v))

Ω(u, v) := 〈u1, v2〉 − 〈v1, u2〉

u := (p, q), p ∈ Rl, q ∈ Zl, t ∈ R

Unitary representations of Hl(Zl × Rl × R)

U~(p, q, t)f(x) := ei~t+i〈q,x〉+~〈p.q〉/2f([x + ~p])

∀ ~ 6= 0, ∀ (p, q, t) ∈ Hl; fulfill the Weyl CCR

U~(u)U~(v) = ei~Ω(u,v)U~(u + v)

U~ = Schrodinger representation of the Weyl

CCR, unique up to unitary equivalences.

Quantization of A(x, ξ, ~) : Tl×Rl× [0,1]→ C

A(ξ, x, ~) =∫Rl

∑q∈Zl

Aq(p; ~)ei(〈p.ξ〉+〈q,x〉) dp

Definition 1 Weyl quantization of A(ξ, x; ~):

(A(~)f)(x) :=∫Rl

∑q∈Zl

Aq(p; ~)U~(p, q)f(x) dp

Weyl quantization of F(ξ, x; ~) ∈ J (ρ, σ)

(F (~)f)(x) :=∫R

∑q∈Zl

Fq(p; ~)Uh(ωp, q)f(x) dp

Uh(ωp, q)f(x) := ei〈q,x〉+i~p〈ω,q〉f(x + ~ωp), p ∈ R

Properties Set:

Fq(~) :=∫RFq(p; ~)Uh(ωp, q) dp, q ∈ Zl

Then:

‖F (~)‖L2→L2 ≤∑

q∈Zl

‖Fq(~)‖L2→L2 ≤ ‖F‖L1 ≤ ‖F‖ρ,σ

‖F‖L1 :=∑

q∈Zl

max0≤~≤1

∫R|Fq(p, ~)| dp

We write, with abuse of notation:

‖F (~)‖ = ‖F(~)‖ρ,σ, ‖F‖ = ‖F‖ρ,σ

Action of the Weyl quantization and matrix

elements

u =∑

m∈Zl

umem ∈ L2(Tl), em = (2π)−l/2ei〈m,x〉

(F (~)u)(x) =∑q∈Zl

ei〈q,x〉 ∑m∈Zl

Fq(~〈ω, (m + q/2)〉, ~)umei〈m,x〉

‖F (~)u‖Hk(Tl) ≤ C(k, s)‖u‖Hs(Tl)

〈em+s, Fq(~)em〉 = δq,sFq(Lω(m + s/2)~, ~)

〈em+s, F (~)em〉 = Fs(Lω(m~ + s~/2), ~)

〈em, F (~)en〉 = Fm−n(〈ω, (m + n)〉~/2, ~)

Compositions, Moyal brackets, and uniform

estimates Given F(~),G(~) ∈ J (ρ, σ), define

their twisted convolutions: w := (p, q)

Ωω(w′ − w, w′) = Ωω(w′, w) :=

(p′ − p)〈ω, q′〉 − p′〈(q′ − q), ω〉 = p′〈q, ω〉 − p〈q′, ω〉

Cq(p;ω; ~) := (13)

1

~∑

q′∈Zl

∫RFq′−q(p

′ − p, ~)Gq′(p′, ~) sin[~Ωω(w′, w)/2] dp′

C(x, ξ; ~) :=∑

q∈Zl

∫RCq(p, ω; ~)eipLω(ξ)+i〈q,x〉 dp

Proposition 1

[F (~), G(~)]i~

=∫R

∑q∈Zl

Cq(p; ~)U~(ωp, q) dp

Symbol of [F (~), G(~)]/i~: C(x,Lω(ξ); ~), Mo-

yal bracket of F ,G. At lowest order:

F ,GM(x,Lω(ξ); ~) = F ,G(x,Lω(ξ)) + O(~)

=⇒ F ,Lω(ξ)M = F ,Lω(ξ)

by the linearity of Lω.

Crucial property of observables F ∈ J (ρ, σ):

stability of the dependence on Lω under com-

positions.

Estimates uniform in ~.

Proposition 2 Let F ,G ∈ J (ρ, σ). Then:

1

~‖[F (~), G(~)]‖L2→L2 ≤ (14)

‖F ,GM‖ρ−d,σ−δ ≤1

d2δ2‖F‖ρ,σ · ‖G‖ρ,σ

Sketch of the proof.

1

~| sin[~Ωω(w′, w)/2]| ≤ |w′ ∧ w|

and therefore, by (13)

‖F(~),G(~)M‖ρ−d,σ−δ ≤∑

r,r′∈Zl

e(ρ−d)|r| ×

max0≤~≤1

∫R2|Fr′(y

′; ~)Gr′−r(y′ − y, ~)| · |w′ ∧ w|eσ|y| dy′dy

Change variables:

‖F(~),G(~)M‖ρ−d,σ−δ ≤∑

k,j∈Zl

|k|e(ρ−d)|k||j|e(ρ−d)|j| ×

×∫R2|Fk(u, ~)Gj(v, ~)| · |u||v|e(σ−δ)(|u|+|v)| dudv

≤1

d2δ2

∑k,j∈Zl

eρ(|k|+|j|)∫R2|Fk(u, ~)Gj(v, ~)|eσ(|u|+|v)| =

=‖F(~)‖ρ,σ‖G(~)‖ρ,σ

d2δ2=⇒ ‖F ,GM‖ρ−d,σ−δ

= max0≤~≤1

‖F(~),G(~)M‖ρ−d,σ−δ ≤‖F‖ρ,σ‖G‖ρ,σ

d2δ2.

Corollary 2 Let F ,G ∈ J (ρ, σ). Then:

1

k!‖F , F , . . . , F ,GM . . .M‖ρ−d,σ−δ

≤ ([dδ]−2‖F‖ρ,σ)k‖G‖ρ,σ (15)

‖[F, Lω]/i~‖ρ− d, σ = ‖F ,LωM‖ρ−d,σ ≤l ω

d‖F‖ρ,σ, ω := max

1≤k≤l|ωk| (16)

KAM in the quantum context: first step.

Look for U0,ε = eiεW0/~ : L2(Tl) ↔ L2(Tl),

W0 = W ∗0, such that

S0,ε := U0,ε(Lω + εV0)U∗0,ε = F1,ε(Lω) + ε2V1,ε,

V0 := V, F1,ε(Lω) = Lω + εN0(Lω).

First order expansion← homological equation

[Lω, W0]

i~+ V = N0

V1,ε = the second order remainder. Iterating:

U`,ε := eiε2`W`/~; S`,ε := U`.ε(F`,ε(Lω) + ε2

`V`,ε)U

∗`,ε

= F`+1,ε(Lω) + ε2`+1

V`+1(ε),

[F`,ε(Lω), W`,ε]

i~+ V`,ε = N`,ε

F`,ε(Lω), N`,ε(Lω), V`,ε(Lω) the symbols.

Problem Solve for W and N the equation:

[F(Lω), W ]

i~+ V = N(Lω) (17)

in terms of (Lω,W,V,N ) ∈ J (ρ, σ) to esti-

mate unif in ~, with remainder. In symbols:

F(Lω(ξ), ~),W(x, ξ; ~)M + (18)

V(x, Lω(ξ); ~) = N (Lω(ξ), ~)

Here F(Lω),WM 6= F(Lω),W!

Assumptions on F

(to accomodate F = x + εN (x, ~)):

1. x 7→ F(x; ~) ∈ C∞(R× [0,1];R);

2. inf(x,~)∈R×[0,1]

Fx(x; ~) > 0; lim|x|→∞

|F(x, ~)||x|

=

C > 0 uniformly with respect to ~ ∈ [0,1].

3. supq∈Zl‖Kq(~)‖σ < +∞, where

Kq(u; ~) =

〈ω, q〉~/[F(u− 〈ω, q〉~/2; ~)−F(u + 〈ω, q〉~/2; ~)]

Theorem 2

‖W‖ρ−d,σ ≤ γ

(τ

d

)τ‖V‖ρ,σ sup

q∈Zl‖Kq‖σ

2. N ∈ J(ρ, σ); ‖N‖ρ,σ ≤ ‖V ‖ρ,σ; N = V.

Proof: Construct the symbols. V ∈ J (ρ, σ):

∑q∈Zl

supRl×[0,1]

|Vq(Lω(ξ); ~)|e(ρ−d)|q| < K(d) < +∞

=⇒∑

q∈Zl

supRl×[0,1]

∣∣∣∣∣Vq(Lω(ξ); ~)〈ω, q〉

Kq(Lω(ξ); ~)∣∣∣∣∣

≤ C∑

q∈Zl

supRl×[0,1]

∣∣∣∣∣Vq(Lω(ξ); ~)〈ω, q〉

∣∣∣∣∣ ≤C∑

q∈Zl

supRl×[0,1]

|Vq(Lω(ξ); ~)|eρ|q| e−d|q|

|〈ω, q〉|

≤ CK(0)(

τ

d

)τ

by the small denominator estimate

supq∈Zl

e−d|q|

|〈ω, q〉|≤(

τ

d

)τ. (19)

implied by (8). Thus define W as follows:

W(Lω(ξ), x; ~) = −i∑

06=q∈Zl

Wq(Lω(ξ), ~)ei〈q,x〉,

Wq(Lω(ξ); ~) :=Vq(Lω(ξ); ~)〈ω, q〉

Kq(Lω(ξ); ~)

N (Lω(ξ); ~) := V(Lω(ξ); ~) (20)

Now:

〈em,[F(Lω), W ]

i~en〉+ 〈em, V en〉 = 〈em, N(Lω)en〉δmn

〈em, Wem+q〉 =i~〈em, V em+q〉

F(〈ω, m〉~)−F(〈ω, (m + q)〉~), q 6= 0,

〈em, Wem〉 = 0, 〈em, Nem〉 = 〈em, V em〉, 〈em, Nem+q〉 = 0, q 6= 0

By the matrix elements formulae:

〈em, Wem+q〉 =Wq(〈ω, (m + q/2)〉~; ~);

〈em, Nem〉 = N (〈ω, m〉~) = V(Lω(〈ω, m〉~); ~)

This identifies the operators W and N .

As ~→ 0, m→∞, m~→ ξ, |q| 6= 0 bounded:

Kq(〈ω, (m + q/2)〉~; ~)→ F ′(Lω(ξ)),

〈em, Wem+q〉 →Vq(ξ)

F ′(Lω(ξ))〈ω, q〉

generating function of the canonical map.

Second step: estimate the ρ− d, σ − δ norms

of N , W in terms of ‖V‖ρ,σ. By (19) we can

define the symbol Y(Lω(ξ), x; ~):

Y(Lω(ξ), x; ~) :=∑

q∈Zl

Yq(Lω(ξ), ~)ei〈q,x〉;

Yq(Lω(ξ; ~) =Vq(Lω(ξ), ~)〈ω, q〉

=⇒

W =∑

q∈Zl

Yq(Lω(ξ), ~)Kq(Lω(ξ), ~)ei〈q,x〉 (21)

Proposition 3

‖N‖ρ,σ ≤ ‖V ‖σ ≤ ‖V‖ρ,σ (22)

‖Y‖ρ−d,σ ≤ γ

(τ

d

)τ‖V‖ρ,σ (23)

‖W‖ρ−d,σ ≤ (24)

γ

(τ

d

)τ‖V‖ρ,σ sup ‖Kq(~)‖σ

(22) obvious from (20). To see (23): by (19)

‖Y(Lω(ξ), x; ~)‖ρ−d,σ =∑q∈Zl

e(ρ−d)|q|

|〈ω, q〉|max

0≤~≤1

∫R|Vq(p; ~)|e|σ|p| dp

≤ γ

(τ

d

)τ ∑q∈Zl

eρ|q| max0≤~≤1

∫R|Vq(p; ~)|e|σ|p| dp

= γ

(τ

d

)τ‖V‖ρ,σ =⇒ (23).

To see (24), convolution estimate by (21):

‖W‖ρ−d,σ =∑

q∈Zl

e(ρ−d)|ν| ×

× max~∈[0,1]

∫R

(∣∣∣∣∫R Yq(y′, ~)Kq(y

′ − y, ~) dy′∣∣∣∣) eσ|y| dy

≤ γ

(τ

d

)τ ∑q∈Zl

eρ|q| ×

× max~∈[0,1]

∫R

∫R|Vq(y

′, ~)Kq(y′ − y, ~)|eσ|y| dy′dy

‖W‖ρ−d,σ ≤ γ

(τ

d

)τ ∑q∈Zl

eρ|q| max~∈[0,1]

‖Vq(~)‖σ‖Kq(~)‖σ

≤ γ

(τ

d

)τsup

(q,~)∈Zl×[0,1]‖Kq(~)‖σ

∑q∈Zl

eρ|q| max~∈[0,1]

‖Vq(~)‖σ

= γ

(τ

d

)τsup

(q,~)∈Zl×[0,1]‖Kq(~)‖σ‖V‖ρ,σ

which is (24). The proposition is proved.

Start of the KAM iteration:

Theorem 3

eiεW/~(F(Lω) + εV )e−iεW/~ = (25)

(F + εN)(Lω) + ε2V1,ε

‖V1,ε‖ρ−2d,σ−2δ ≤ (26)

γ

(dδ)2

(τ

d

)τ‖V ‖2ρ,σKσ

2 + εγ(dδ)2

(τd

)τ‖V ‖ρ,σKσ

1− εγ(dδ)2

(τd

)τ‖V ‖ρ,σKσ

Kσ := sup(q,~)∈Zl×[0,1]

‖Kq(~)‖σ (27)

Preliminary result:

Lemma 1 Define:

Aε(~) := eiεW/~Ae−iεW/~. (28)

Then, ∀0 < d1 < ρ, ∀0 < δ1 < σ:

‖Aε(~)‖ρ−d1,σ−δ1 ≤‖A‖ρ,σ

1− ε‖W‖ρ,σ/(d1δ1)2.

(29)

Proof: commutator expansion.

Proof of Theorem 3 By construction:

ε 7→ Sε := eiεW/~(F(Lω) + εV )e−iεW/~(30)

Expansion at ε = 0 with 2- order remainder:

Sεu = F(Lω)u + εN(Lω)u + ep2V1,εu, u ∈ H1(Tl)

V1,ε =1

2

∫ ε

0(ε− t)eitW/~ER1,te

−itW/~ dt

R1,t := ([N, W ] + [W, V ])/~ + t[W, [W, V ]]/~2

Obviously :

‖V1,ε‖ ≤ |ε|2 max0≤|t|≤|ε|

‖S′′(t)‖ (31)

Symbol of R1,ε:

R1.ε(Lω(ξ), x; ~) =

N ,WM + V,WM + εW, W,VMM

Individual estimates:

‖[N, W ]/i~‖ρ−d,σ−δ ≤ ‖N ,WM‖ρ−d,σ−δ ≤1

(dδ)2‖W‖ρ,σ‖N‖ρ,σ ≤

γ(d/τ)τ

(dδ)2‖V‖ρ,σKσ

‖[V, W ]/i~‖ρ−d,σ−δ ≤ ‖V,WM‖ρ−d,σ−δ

≤1

(dδ)2‖V‖ρ,σ‖W‖ρ,σ ≤

γ(d/τ)τ

(dδ)2‖V‖ρ,σKσ

‖[W, [W, V ]]/(i~)2‖ρ−d,σ−δ ≤

‖W, W,VMM‖ρ−d,σ−δ ≤‖W‖2ρ,σ

(dδ)4‖V‖ρ,σ

≤γ(d/τ)2τ

(dδ)4‖V‖ρ,σK2

σ

We can now apply Lemma 1, which yields:

‖eiεW/~[N, W ]e−iεW/~/i~‖ρ−d−d1,σ−δ−δ1 ≤γ(d/τ)τ

(dδ)2‖V‖ρ,σKσ

1− ε‖W‖ρ,σ/(d1δ1)2

‖eiεW/~[V, W ]e−iεW/~/i~‖ρ−d−d1,σ−δ−δ1 ≤γ(d/τ)τ

(dδ)2‖V‖ρ,σKσ

1− ε‖W‖ρ,σ/(d1δ1)2

‖eiεW/~[W, [W, V ]]e−iεW/~/(i~)2‖ρ−d−d1,σ−δ−δ1 ≤

γ(d/τ)2τ

(dδ)4‖V‖ρ,σK2

σ

1− ε‖W‖ρ,σ/(d1δ1)2

Summing up:

‖V1,ε‖ρ−d−d1,σ−δ−δ1 ≤γ(d/τ)τ

(dδ)2‖V‖ρ,σKσ

1− ε‖V‖ρ,σ/(d1δ1)2[2 + |ε|

γ(d/τ)τ

(dδ)2‖K‖σ]

With d1 = d, δ1 = δ this is (26). Theorem 3

is proved.

Estimate ‖K‖σ for our particular F.

Theorem 4 Set:

Fε(u; ~) = u + εNε(u; ~) (32)

with max|ε|≤L

‖Nε‖σ < +∞, L > 0. Then:

supq‖Kq,ε‖σ−δ ≤

[1−|ε|4δ‖Nε‖σ

]−1

(33)

Proof By definition:

Kq,ε(u; ~) = [1 + εQq,ε(u, ~)]−1 (34)

Qq,ε(u, ~) :=

[Nε(u + 〈ω, q〉~/2; ~)−Nε(u− 〈ω, q〉~/2; ~)]/〈ω, q〉~

Now prove the estimate:

‖Qq,ε(u, ~)‖σ−δ ≤1

4δ‖Nε‖σ

Since:

Nε(u + 〈ω, q〉~/2; ~)−Nε(u− 〈ω, q〉~/2; ~)

=∫RNε(p; ~)[eip〈ω,〉~/2 − e−ip〈ω,〉~/2] eiup dp

supη∈R

[|η|−1|eipη − e−ipη|] ≤|p|4

, supp∈R

e−δ|p||p| =1

δ, δ > 0

we get

‖Qq(u, ~; ε)‖σ−δ ≤1

4

∫R|Nε(p; ~)|e(σ−δ)|p||p| dp

≤1

4δ

∫R|Nε(p; ~)|eσ|p| dp =

1

4δ‖Nε‖σ

‖(Qq,ε)n‖σ = ‖Q∗nq,ε‖σ ≤ (4δ)−n‖Nε‖nσ

Majorization independent of q. By (34):

supq‖Kq,ε‖σ−δ ≤

∞∑n=0

(|ε|4δ

)n

‖Nε‖nσ =1

1− |ε|4δ‖Nε‖σ

and this concludes the proof of the Theorem.

Before proceeding to the iteration: relation

between eiεW/i~ and the canonical map φεW(0)

generated by the flow of W(ξ, x; ~)|~=0 :=

W(0) at time ε.

Variant of the semiclassical Egorov theorem:

Theorem 5

Sε := eiεW~ (Lω + A)e−iεW

~ = Lω + A′

where:

‖A′‖ρ−2d,σ−δ ≤1

1− |ε|‖W‖ρ,σ

(dδ)2

[‖A‖ρ,σ+l |ω|d‖ε|‖W‖ρ,σ], ω := max

1≤k≤lωk

Moreover:

Lω +A′ = (Lω +A) ΦεW0

+ O(~)

ΦεW0

= time ε flow of W0,ε :=Wε(ξ, x; ~)|~=0.

Proof Commutator expansion

Sε = Lω +∞∑

k=1

(iε)k

~kk![W, [W, . . . , [W,Lω] . . .]

+∞∑

k=0

(iε)k

~kk![W, [W, . . . , [W, A] . . .]

S(x, ξ; ~, ε) = Lω(ξ) +∞∑

k=1

εk

k!W, W, . . . , W,Lω . . .M

+∞∑

k=0

εk

k!W, W, . . . , W,AM . . .M

because W,LωM = W,Lω. Hence:

‖∞∑

k=1

εk

k!W, W, . . . , W,Lω . . .M‖ρ−2d,σ−δ ≤

≤l |ω|d

∞∑k=1

(|ε|‖W‖ρ,σ

(dδ)2

)k

‖∞∑

k=0

εk

k!W, W, . . . , W,AM . . .M‖ρ−d,σ−δ ≤

‖A‖ρ,σ

∞∑k=0

(|ε|‖W‖ρ,σ

(dδ)2

)k

Therefore,:

A′ :=∞∑

k=1

(iε)k

~kk![W, [W, . . . , [W,Lω] . . .]

+∞∑

k=0

(iε)k

~kk![W, [W, . . . , [W, A] . . .]

and remark that ‖ · ‖ρ−2d ≤ ‖ · ‖ρ−d. Then:

‖A′‖ρ−2d,σ−δ ≤1

1− |ε|‖W‖ρ,σ

(dδ)2

[‖A‖ρ,σ +l |ω|d‖ε|‖W‖ρ,σ]

This proves assertions (1) and (2). Now:

S0ε (x, ξ; ~)|~=0 = Lω +A′ε(ξ, x; ~)|~=0 =∞∑

k=0

(ε)k

k!W0, . . . , W0,L+A . . . = e

εLW0(L+A)

Taylor’s expansion

eεLW0(Lω +A) = (Lω +A) φε

W0(x, ξ)

and this concludes the proof. Explicitly:

W0 =1

F ′(Lω(ξ))

∑q∈Z`

Vq(ξ)

〈ω, ν〉ei〈q,x〉

eεLW0(F(Lω) + εA) = F(Lω) + εN0,ε(Lω) + O(ε2)

KAM in the quantum context: iteration

At the `-th iteration:

S`,ε := eiε`W`/~ · · · eiε2W1/~eiεW0/~ ×

×(F(Lω) + εV )e−iεW0/~e−iε2W1/~ · · · e−iε`W`/~

= eiε`W`/~(F`,ε(Lω) + ε2`V`)e

−iε`W`/~

= F`+1,ε(Lω) + ε`+1V`+1,ε,

F`,ε(Lω) = F(Lω) +∑

k=1

εkNk(Lω),

[F`(Lω), W`]/i~ + V` = N`(Lω, ε)

V`+1,ε :=∞∑

k=0

(iε`)k

~kk![W`, [W`, . . . , [W`, R`] . . .]

R`+1,ε := [N`, W`]/~ + [W`, V`]/~ + ε`[V`, [W`, W`]]/~2

κ` := d`κ`, γ` :=γ

κ2`

(τ

d`

)τ

; η` =1

γ`, ε` = ε2

`

ρ` = ρ`−1 − 2d`, σ` = σ`−1 − 2δ`; ρ0 := ρ σ0 := σ.

By Theorems 3 and 4 we can write:

‖V`,ε‖ρ`,σ` ≤ γ`‖V`−1,ε‖2ρ`−1,σ`−1× (35)

×K`−1,εσ`−1

2 + |ε`|γ`‖V`−1.ε‖ρ`−1,σ`−1K`−1,εσ`−1

1− |ε`|γ`‖V`−1,ε‖ρ`−1,σ`−1K`−1.εσ`−1

K`−1,εσ`−1−δ`

:= sup(q,~)∈Zl×[0,1]

‖K`−1q,ε (~)‖σ`−1−δ`

(36)

F`−1,ε(u; ~) = u + εN0,ε(u; ~) + . . .(37)

+ε`−1N`−1,ε(u; ~)

Iterating the argument of Theorem 4:

Lemma 2 Let F(u; ~) have the form (37)

with sup|ε|<L

‖Nj,ε‖σj < +∞. Let:

P0(σ0;N0) := sup|ε|<L

[1− |ε0|‖N0,ε‖σ0/4δ0

]−1;

Pj(σ0, . . . , σj;N0, . . . ,Nj) :=

sup|ε|<L

[1− |εj|Pj−1‖Nj,ε‖σj/4δj

]−1, 1 ≤ j ≤ `− 1

∀0 < δj < σj : j = 0, . . . , `− 1. Then:

K`−1σ`−1−δ`

≤`−1∏j=0

Pj(σ0, . . . , σj;N0, . . . ,Nj)

(38)

Monotonic dependence of d`, δ` on `. Now

σ` + δ` = σ`−1 − δ`−1 + δ` < σ`−1, ‖ · ‖σ ≤ ‖ · ‖τ ⇒

‖Vj‖σ`+δ`= ‖Vj‖σ`−1−δ`

≤ ‖Vj‖σ`−1 ≤ ‖Vj‖σj

j = 0, . . . , `− 1 =⇒ (Nj = Vj)

P`−1(σ0, . . . , σ`−1;V 0, . . . , V `−1) ≤[1− |ε`−1|P`−2‖V `−1‖σ`−1/4δ`−1

]−1

K`−1σ`−1−δ`

≤`−1∏j=0

[1− |εj|Pj−1‖V j‖σj/4δj

]−1, j = 0, . . . , `− 1.(39)

Thus finally

‖V`(ε)‖ρ`,σ` ≤ γ`

‖V`−1‖2ρ`−1,σ`−1∏`−1j=0

[1− |εj|Pj−1‖V j‖σj/4δj

] ××

2 + |ε`|γ`Z`

1− |ε`|γ`Z`(40)

Z` :=‖V`−1‖ρ`−1,σ`−1

[∏`−1

j=0

[1− |εj|Pj−1‖V j‖σj/4δj

]Set now:

Γ` = ‖V`‖ρ`,σ`, P0(ε0,Γ0) = [1− |ε0|Γ0/4δ0]−1 (41)

Pj(Γ0, . . . ,Γj) =[1− |εj|Pj−1Γj/4δj

]−1(42)

(40) becomes:

Γ` ≤Γ2

`−1

η`−1

1− `−1∏j=0

|εj|Pj−1Γj/4δj

−1

×

2 + |ε`|Γ`−1

[1−

∏`−1j=0 |εj|Pj−1Γj/4δj

]−1

1− |ε`|Γ`−1

[1−

∏`−1j=0 |εj|Pj−1Γj/4δj

]−1 (43)

Theorem 6 Solution of the inequalities (43):

Γ` ≤ (5‖V ‖ρ,σe8+2τ+γ+D)2`, D =

∞∑k=1

2−k ln k

True if ` = 1. Assume, ∀1 ≤ j ≤ ` − 1, ∃0 <

K < 1 such that

(H`−1)

`−1∏j=1

|εj|Pj−1Γj/4δj ≤1

2

1 ≤ j ≤ `− 1

|εj|Γj/

1− `−1∏j=1

|εj|Pj−1Γj/4δj

≤ K < 1

Set

H := 22 + K

1−K. (44)

Then clearly:

Γ` ≤ HΓ2

`

η`−1=⇒ Γ` ≤

(HΓ0)2`∏l

k=1 η2k`−k

(45)

Choice of δ` and d` free provided:

∞∑`=1

δ` < σ,∞∑

`=1

d` < ρ =⇒

δ` =σ

4(` + 1)2, d` =

ρ

4(` + 1)2=⇒ (46)

δ−∞∑

`=0

δ` = δ−π2

24>

σ

2; d−

∞∑`=0

d` = ρ−π2

24>

ρ

2.

η` =d2` δ2`γ

(τ

d`

)−τ

≤ γ−1`−(8+2τ)

1∏`k=1 η2k

`−k

≤ (e8+2τ+γ+D)2`

By (45) with Γ0 = ‖V ‖ρ,σ:

Γ` ≤ (H‖V ‖ρ,σe8+2τ+γ+D)2`= (H‖V ‖ρ,σµ)2

`;

µ := e8+2τ+γ+D; ε`Γ` ≤ (H‖V ‖ρ,σµε)2`

(H`−1) =⇒ (H`). If K = 1/2 =⇒ H = 5:

ε`Γ` ≤ (5αεµ)2`, α := ‖V ‖ρ,σ (47)

First condition. By (47)

5αεµ < min[1

2,1

2σ

]⇒ P`−1 ≤ . . . ≤ P0 ≤

1

2⇒

`−1∏j=1

εjPj−1Γj/4δj ≤ P `−10 σ−l+1(l!)2 ×

×`−1∏j=1

εjΓj ≤ (σ)−l+1(l!)2(5αεµ)2`−2 <

1

2⇒

∏j=1

εjPj−1Γj/4δj ≤1

2(48)

because

∏j=1

εjPj−1Γj/4δj

≤ (σ/2)−l+1(l!)2(5αεµ)2`−2 · ε`P`−1Γ`/4δ`

≤1

2ε`P`−1Γ`/4δ`

≤1

2

(l + 1)2(5αεµ)2`/σ

1− l2(5αεµ)2`−1/2σ

≤1

2· 2(l + 1)2(5αεµ)2

`/σ <

1

2, ` > 2

This verifies the first condition (H`). For the

second, by (47) and (48):

εjΓj

1−∏`−1

j=1 εjPj−1Γj/4δj

≤(5αεµ)2

j

1− 1/2≤

1

2=⇒

εjΓj

1−∏`

j=1 εj|Pj−1Γj/4δj≤

(5αεµ)2j

1− 1/2≤

1

2, j = 2, . . . , `.

This concludes the sketch of the proof of the

Theorem.

Final estimates of W`, N`, R`.

Inserting everything in (24) we get:

ε`‖W`,ε‖ρ`+1,σ` ≤ 2γ

(τ

d`

)−τ

ε`‖V`,ε‖ρ`.σ` ≤

2γ

(4

τ

)τ(` + 1)2τ(5αεµ)2

`(49)

N`,ε = V`,ε,entails:

ε`‖N`,ε‖ρ`,σ` = ε`‖V`,ε‖ρ`,σ` ≤ (5αεµ)2`

(50)

Finally, inserting everyting in (26):

‖V`+1,ε‖ρ`+1,σ`+1 ≤ γ`+12‖V ‖2ρ`,σ`

2 + 2εγ`‖V ‖ρ`,σ`

1− 2εγ`‖V ‖ρ`,σ`

≤

2γττ43τ(` + 1)6τ(5αεµ)2`+12 + 2εττ43τ`6τ(5αεµ)2

`

1− 2εττ43τ`6τ(5αεµ)2`

Lemma 3 ~ fixed, |ε| < ε∗ := 1/5αµ. Set:

Un,ε(~) := eiεnWn/~ · · · eiεW1/~ (51)

Then ∃ U∞,ε(~) unitary in L2(Tl) such that

limn→∞ ‖Un,ε(~)− U∞,ε(~)‖L2→L2 = 0

Existence and analyticity of the limit of the

KAM iteration.

Theorem 7 For k = 1,2, . . . let:

Dk,ε(~, ω) := Uk,ε(~)(Lω + εV )U−1k,ε (~) (52)

Dk,ε(Lω(ξ), ~) := Lω +k∑

`=1

N`,ε(Lω(ξ); ~)ε2`−1

(53)

ε∗ :=1

5‖V ‖ρ,σe8+2τ+γ+D, D :=

∞∑k=1

ln k2−k (54)

Then for |ε| < ε∗:

1. Dk.ε(~) has a norm limit D∞,ε(~) as k →

∞, uniformly in ~ ∈ [0,1].

2. D∞,ε(~) = U∞,ε(~)(Lω + εV )U∞,−ε(~);

3. D∞,ε(~) is self-adjoint with p.p. spec-

trum,diagonal on the eigenvector basis of Lω.

3. limk→∞

Dk,ε(Lω(ξ), ~) = D∞,ε(Lω(ξ), ~) in the

‖ · ‖σ/2 norm.

4. D∞,ε(Lω(ξ), ~) is the symbol of D∞,ε(~).

λn,ε(~) = D∞,ε(Lω(n~), ~)

1. We have, by construction:

Dk,ε(~) = Lω +∑`=0

N`,ε(L; ~)ε2`+ ε2

`+1V`+1,ε

‖ε2`+1

V`+1,ε‖L2→L2 ≤

2γττ43τ(` + 1)6τ(5αεµ)2`+12 + 2εττ43τ`6τ(5αεµ)2

`

1− 2εττ43τ`6τ(5αεµ)2`→ 0, `→∞

because 5αεµ < 1 if ε < ε∗. Denote now:

Nk,ε(ω, ~) := Lω +k∑

`=0

N`,ε(L; ~)ε2`

2. Consequence of the norm convergence.

3. Follows by the estimate (50).

4. Symbol of Nk,ε = Nk,ε,. [Nk,ε, L] = 0, the

eigenvalues of Nk,ε are Nk,ε(Lω(n~); ~). As-

sertion again by the convergence statement.

Proof of Theorem 1

Operator family ε 7→ D∞,ε(~) bounded holo-

morphic for |ε| < ε∗, uniformly in ~ ∈ [0,1].

Hence

D∞,ε(~) = Lω +∞∑

k=1

Bk(L, ~)εk, |ε| < ε∗

which is the uniformly convergent quantum

normal form.