Post on 19-Feb-2022
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.