under the assumption that the potential VA
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
in the directions normal to the
submanifold and not by assuming VA
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
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 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
R, (−δ, δ)
which is
the identity on (−δ/2,δ/2) and satisfies
(1.8) j

Cj r R :

(r)| Cj (1 +
(see Figure 1). Now a diffeomorphism Φ
B) is obtained by first apply-
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
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 \ is dν,
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