4 1. INTRODUCTION
and by Bouchitt´ e et al. in [6]. In addition, we discuss how twisted closed wave
guides display phase shifts somewhat similar to the Aharanov-Bohm effect but
without magnetic fields.
The crucial step in the proof is the construction of closed infinite dimensional sub-
spaces of
L2(NC)
which are invariant under the dynamics (1.4) up to small errors
and which can be mapped unitarily to L2(C), where the effective dynamics takes
place. To construct these ’almost invariant subspaces’, we define at each point q C
a Hamiltonian operator Hf (q) corresponding to the dynamics in the fibre NqC. If
it has a simple eigenvalue band Ef (q) that depends smoothly on q and is isolated
from the rest of the spectrum for all q, then the corresponding eigenspaces define
a smooth line bundle over C. Its L2-sections define a closed subspace of L2(NC),
which after a modification of order ε becomes the almost invariant subspace as-
sociated with the eigenvalue band Ef (q). In the end, to each isolated eigenvalue
band Ef (q) there is an associated line bundle over C, an associated almost invariant
subspace and an associated effective Hamiltonian Heff ε .
We now come to the form of the effective Hamiltonian associated with a simple
band Ef (q). For Heff,0 we obtain, as expected, the Laplace-Beltrami operator of the
submanifold as kinetic energy term and the eigenvalue band Ef (q) as an effective
potential,
Heff,0 =
−ε2ΔC
+ Ef .
We note that (Vc+W )|C is contained in Ef . This is the quantum version of the result
of Rubin and Ungar [40] for classical mechanics. However, the time scale for which
the solutions of (1.4) propagate along finite distances are times t of order
ε−1.
On
this longer time scale the first order correction εHeff,1 to the effective Hamiltonian
has effects of order one and must be included in the effective dynamics. We do not
give the details of Heff,1 here and just mention that at next to leading order the
kinetic energy term, i.e. the Laplace-Beltrami operator, must be modified in two
ways. First, the metric on C needs to be changed by terms of order ε depending on
exterior curvature, whenever the center of mass of the normal eigenfunctions does
not lie exactly on the submanifold C. Furthermore, the connection on the trivial
line bundle over C (where the wave function φ takes its values) must be changed
from the trivial one to a non-trivial one, the so-called generalized Berry connection.
For the variation of the eigenfunctions associated with the eigenvalue band Ef (q)
along the submanifold induces a non-trivial connection on the associated eigenspace
bundle. This was already discussed by Mitchell in the case that the potential (and
thus the eigenfunctions) only twists.
When Ef is constant as in the earlier works, there is no non-trivial potential term
up to first order and so the second order corrections in Heff,2 become relevant.
They are quite numerous. In addition to terms similar to those at first order, we
find generalizations of the Born-Huang potential and the off-band coupling both
known from the Born-Oppenheimer setting, and an extra potential depending on
the inner and the exterior curvature of C, whose occurence had originally lead to
Marcus’ reply to Dirac’s proposal. Finally, when the ambient space is not flat, there
is another extra potential already obtained by Mitchell.
We note that in the earlier works it was assumed that
−ε2ΔC
is of order
ε2
and thus of the same size as the terms in Heff,2. That is why the extra potential
depending on curvature appeared at leading order in these works, while it appears
only in Heff,2 for us. And this is also the reason that assumptions were necessary,
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