1.1. THE MODEL 11
σ(Hf)
q
f(q) E
I(q)
cgap
Figure 3. Ef (q) has to be separated by a local gap that is uniform
in q.
ϕf (q, ·)ψ(q, ·)
Hf (q)
ϕf (q, ν). This defines a unitary mapping between the corre
sponding subspace P0H and
L2(C,dμ):
Definition 1.5. Let the eigenspace bundle EP0 corresponding to a simple con
straint energy band Ef admit a normalized section ϕf ∈ Γ(EP0 ). The partial isom
etry U0 : H →
L2(C,dμ)
is defined by (U0ψ)(q) := ϕf (q, ·)ψ(q, ·)
Hf (q)
. Then
U0
∗U0
= P0 and U0U0
∗
= 1 with U0
∗
given by (U0
∗ψ)(q,
ν) = ϕf (q, ν)ψ(q).
So any ψ ∈ P0H has the product structure ψ = (U0ψ)ϕf . Since V0 and there
fore ϕf depends on q, such a product will, in general, not be invariant under the time
evolution. However, it will turn out to be at least approximately invariant. For short
times this follows from the fact that the commutator [Hε,P0] = [−ε2Δh,P0]+O(ε)
is of order ε. For long times this is a consequence of adiabatic decoupling.
On the macroscopic scale the corresponding eigenfunction Dεϕf is more and more
localized close to the submanifold: most of its mass is contained in the εtube
around C and it decays like
e−Λ0ν/ε.
This is visualized in Figure 4.
V0
0 0
O(1)
ν
φ (q)
V0
O( ε)
(q)
ν
Dεφ
*
 

ν/ε) (q, ν) (q,
f
f
Figure 4. On the macroscopic level ϕf is localized on a scale of
oder ε.
Our goal is to obtain an effective equation of motion on the submanifold for
states that are localized close to the submanifold in that sense. More precisely, for
each subspace P0H corresponding to a constraint energy band Ef we will derive