10 1. INTRODUCTION

i) The unitary transform Mρ is defined by Mρ : H → H, ψ →

ρ−

1

2

ψ.

ii) The dilation operator Dε is defined by (Dεψ)(q, ν) := ε−k/2 ψ(q, ν/ε).

iii) The dilated Hamiltonian Hε and potential Vε are defined by

Hε := Dε

∗Mρ ∗HεMρDε,

Vε := Dε

∗Mρ ∗V εMρDε

= Vc + Dε

∗WDε

.

The index ε will consistently be placed down to denote dilated objects, while it will

placed up to denote objects in the original scale.

The leading order of Hε will turn out to be the sum of −Δv + Vc(q, ·) + W (q, 0)

and −ε2Δh (for details on Mρ and the expansion of Hε see Lemmas 3.1 & 3.7

below). When

−ε2Δh

acts on functions that are constant on each fibre, it is simply

the Laplace-Beltrami operator on C carrying an

ε2.

Hereby the analogy with the

Born-Oppenheimer setting is revealed where the kinetic energy of the nuclei carries

the small parameter given by the ratio of the electron mass and the nucleon mass

(see e.g. [36,46]).

We need that the family of q-dependent operators −Δv + Vc(q, ·) + W (q, 0) has

a family of exponentially decaying bound states in order to construct a subspace of

states that are localized close to the constraint manifold. The following definition

makes this precise. We note that the conditions are simpler to verify than one might

have thought in the manifold setting, since the space and the operators involved

are euclidean.

Definition 1.3. Let Hf (q) :=

L2(NqC,dν)

and V0(q, ν) := Vc(q, ν) + W (q, 0).

i) The selfadjoint operator (Hf (q),H2(NqC,dν)) defined by

(1.15) Hf (q) := −Δv + V0(q, .)

is called the fiber Hamiltonian. Its spectrum is denoted by σ

(

Hf (q)

)

.

ii) A function Ef : C → R is called an energy band, if Ef (q) ∈ σ

(

Hf (q)

)

for

all q ∈ C. Ef is called simple, if Ef (q) is a simple eigenvalue for all q ∈ C.

iii) An energy band Ef : C → R is called separated, if there are a constant

cgap 0 and two bounded continuous functions f± : C → R defining an

interval I(q) := [f−(q),f+(q)] such that

(1.16) {Ef (q)} = I(q) ∩ σ(Hf (q)) , inf

q∈C

dist

(

σ

(

Hf (q)

)

\{Ef (q)}, {Ef (q)}

)

= cgap.

iv) Set ν := 1 + |ν|2 = 1 + g(q,0)(ν, ν). A separated energy band Ef is

called a constraint energy band, if there is Λ0 0 such that the family of

spectral projections {P0(q)}q∈C with P0(q) ∈ L

(

Hf (q)

)

and Hf (q)P0(q) =

Ef (q)P0(q) satisfies

supq∈C

eΛ0 ν P0(q)eΛ0 ν

L(Hf (q))

∞.

Remark 1.4. Condition iii) is known to imply condition iv) in lots of cases

(see [21] for a review of known results), in particular for eigenvalues below the

continuous spectrum, which is the most important case in the applications. Besides,

condition iii) is a uniform but local condition (see Figure 3).

The family of spectral projections {P0(q)}q∈C associated with a simple energy

band Ef defines a smooth line bundle EP0 := {(q, ϕ) | q ∈ C, ϕ ∈ P0(q)Hf (q)}

over C. Smoothness of EP0 follows e.g. from Lemma 4.10. If this bundle has

a normalized section ϕf ∈ Γ(EP0 ), it holds for all q ∈ C that (P0ψ)(q, ν) =