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