1.1. THE MODEL 9
Instead of (1.7) we now consider a Schr¨ odinger equation on the normal bundle,
thought of as a Riemannian manifold (NC, g). There we can immediately implement
the idea of squeezing the potential in the normal directions: Let
V
ε(q,
ν) = Vc(q,
ε−1ν)
+ W (q, ν)
for fixed real-valued potentials Vc,W Cb
∞(C,Cb ∞(NqC)).
Here we have split up
any Q NC as (q, ν) where q C is the base point and ν is a vector in the fiber
NqC at q. We allow for an ’external potential’ W which does not contribute to the
confinement and is not scaled. Then ε 1 corresponds to the regime of strong
confining forces. The setting is sketched in Figure 2.
Figure 2. The width of is ε but it varies on a scale of order
one along C.
So we will investigate the Schr¨ odinger equation
(1.14) i∂tψ =
Hεψ
:=
−ε2ΔNCψ
+ V
ε
ψ , ψ|t=0 = ψ0
ε
H ,
where ΔNC is the Laplace-Beltrami operator on (NC, g), i.e. the operator associ-
ated with
NC
g(dψ, dψ)dμ. The operator

will be called the Hamiltonian.
We note that

is real, i.e. it maps real-valued functions to real-valued func-
tions. Furthermore, it is bounded from below because we assumed Vc and W to be
bounded. In Section 1.3 of [42]

is shown to be selfadjoint on its maximal domain
D(Hε)
for any complete Riemannian manifold M, thus in particular for (NC, g).
Let W 2,2(NC, g) be the second Sobolev space, i.e. the set of all L2-functions with
square-integrable covariant derivatives up to second order. We emphasize that,
in general, W 2,2(NC, g) D(Hε) but W 2,2(NC, g) = D(Hε) for a manifold of
unbounded geometry.
We only need one additional assumption on the potential, that ensures localiza-
tion in normal direction. Before we state it, we clarify the structure of separation
between vertical and horizontal dynamics:
After a unitary transformation can at leading order be split up into an oper-
ator which acts on the fibers only and a horizontal operator. That unitary trans-
formation is given by multiplication with the square root of the relative density
ρ :=

dμ⊗dν
of our starting measure and the product measure on NC that was in-
troduced above. We recall from (1.12) that this density is bounded and strictly
positive. After the transformation it is helpful to rescale the normal directions.
Definition 1.2. Set H :=
L2(NC,dμ
dν) and ρ :=

dμ⊗dν
.
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