6 1. INTRODUCTION

under the assumption that the potential VA

ε

localizes at least a certain subspace of

states in an ε-tube of C with ε δ. The localization will be realized by simply

imposing that the potential is squeezed by

ε−1

in the directions normal to the

submanifold and not by assuming VA

ε

to become large away from C, which makes

the proof of localization more diﬃcult. To ensure proper scaling behavior, we have

multiplied the Laplacian in (1.7) by ε2. The physical meaning of this is explained

at the end of the next subsection. Here we only emphasize that an analogous

scaling was used implicitly or explicitly in all other previous works on the problem

of constraints in quantum mechanics. The crucial difference in our work is, as

explained before, that we allow for ε-dependent initial data ψ0 ε with tangential

kinetic energy of order one instead of order ε2.

In order to actually implement the scaling in the normal directions, we will now

construct a related problem on the normal bundle of C by mapping NC diffeomor-

phically to the tubular neighborhood B of C in a specific way and then choosing

a suitable metric g on NC (considered as a manifold). On the normal bundle the

scaling of the potential in the normal directions is straightforward. The theorem

we prove for the normal bundle will later be translated back to the original setting.

On a first reading it may be convenient to skip the technical construction of g and

of the horizontal and vertical derivatives

∇h

and

∇v

and to immediately jump to

the end of Definiton 1.1.

The mapping to the normal bundle is performed in the following way. There is

a natural diffeomorphism from the δ-tube B to the δ-neighborhood Bδ of the zero

section of the normal bundle NC. This diffeomorphism corresponds to choosing

coordinates on B that are geodesic in the directions normal to C. These coordinates

are called (generalized) Fermi coordinates. They will be examined in detail in

Section 4.2. In the following, we will always identify C with the zero section of the

normal bundle. Next we choose any diffeomorphism

˜

Φ ∈

C∞

(

R, (−δ, δ)

)

which is

the identity on (−δ/2,δ/2) and satisfies

(1.8) ∀ j ∈

N×

∃ Cj ∞ ∀ r ∈ R :

|˜

Φ

(j)

(r)| ≤ Cj (1 +

r2)−(j+1)/2

(see Figure 1). Now a diffeomorphism Φ ∈

C∞(NC,

B) is obtained by first apply-

ing

˜

Φ to the radial coordinate on each fibre NqC (which are all isomorphic to Rk)

and then using Fermi charts in the normal directions.

Figure 1.

˜

Φ converges to ±δ like 1/r.

The important step now is to choose a suitable metric and corresponding measure

on NC. On the one hand we want it to be the pullback

Φ∗G

of G on Bδ/2. On

the other hand, we require that the distance to C asymptotically behaves like the

radius in each fibre and that the associated volume measure on NC \ Bδ is dμ ⊗ dν,