2 1. INTRODUCTION
For the quantum mechanical case Marcus [29] and later on Jensen and Koppe [23]
pointed out that the limiting dynamics depends, in addition, also on the embed-
ding of the submanifold C into the ambient space A. In the sequel Da Costa [9]
deduced a geometrical condition (often called the no-twist condition) ensuring that
the effective dynamics does not depend on the localizing potential. This condition
is equivalent to the flatness of the normal bundle of C. It fails to hold for a generic
submanifold of dimension and codimension both strictly greater than one, which is
a typical situation when applying these ideas to molecular dynamics.
Thus the hope to obtain a generic ’intrinsic’ effective dynamics as in (1.3), i.e. a
Hamiltonian that depends only on the intrinsic geometry of C and the restriciton of
the potential V to C, is unfounded. In both, classical and quantum mechanics, the
limiting dynamics on the constraint manifold depends, in general, on the detailed
nature of the constraining forces, on the embedding of C into A and on the initial
data of (1.1). In this work we present and prove a general result concerning the
precise form of the limiting dynamics (1.2) on C starting from (1.1) on the ambient
space A with a strongly confining potential V . However, as we explain next, our
result generalizes existing results in the mathematical and physical literature not
only on a technical level, but improves the range of applicability in a deeper sense.
Da Costa’s statement (like the more refined results by Froese-Herbst [17], Maraner
[27] and Mitchell [32], which we discuss in Subsection 1.2) requires that the con-
straining potential is the same at each point on the submanifold. The reason behind
this assumption is that the energy stored in the normal modes diverges in the limit
of strong confinement. As in the classical result by Rubin and Ungar, variations in
the constraining potential lead to exchange of energy between normal and tangen-
tial modes, and thus also the energy in the tangential direction grows in the limit of
strong confinement. However, the problem can be treated with the methods used
in [9, 17,27,32] only for solutions with bounded kinetic energies in the tangential
directions. Therefore the transfer of energy between normal and tangential modes
was excluded in those articles by the assumption that the confining potential has
the same shape in the normal direction at any point of the submanifold. In many
important applications this assumption is violated, for example for the reaction
paths of molecular reactions. The reaction valleys vary in shape depending on the
configuration of the nuclei. In the same applications also the typical normal and
tangential energies are of the same order.
Therefore the most important new aspect of our result is that we allow for con-
fining potentials that vary in shape and for solutions with normal and tangential
energies of the same order. As a consequence, our limiting dynamics on the con-
straint manifold has a richer structure than earlier results and resembles, at leading
order, the results from classical mechanics. In the limit of small tangential energies
we recover the limiting dynamics by Mitchell [32].
The key observation for our analysis is that the problem is an adiabatic limit and
has, at least locally, a structure similar to the Born-Oppenheimer approximation in
molecular dynamics. In particular, we transfer ideas from adiabatic perturbation
theory, which were developed by Nenciu-Martinez-Sordoni and Panati-Spohn-Teufel
in [30, 31, 34–36, 43, 45], to a non-flat geometry. We note that the adiabatic na-
ture of the problem was observed many times before in the physics literature, e.g.
in the context of adiabatic quantum wave guides [7], but we are not aware of any
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