1. INTRODUCTION 3

work considering constraint manifolds with general geometries in quantum mechan-

ics from this point of view. In particular, we believe that our effective equations

have not been derived or guessed before and are new not only as a mathematical

but also as a physics result. In the mathematics literature we are aware of two

predecessor works: in [45] the problem was solved for constraint manifolds C which

are d-dimensional subspaces of Rd+k, while Dell’Antonio and Tenuta [11] consid-

ered the leading order behavior of semiclassical Gaussian wave packets for general

geometries.

Another result about submanifolds of any dimension is due to Wittich [48], who

considers the heat equation on thin tubes of manifolds. Finally, there are related

results in the wide literature on thin tubes of quantum graphs. A good starting

point for it is [18] by Grieser, where mathematical techniques used in this context

are reviewed. Both works and the papers cited there, properly translated, deal with

the case of small tangential energies.

We now give a non-technical sketch of the structure of our result. The detailed

statements given in Section 2 require some preparation.

We implement the limit of strong confinement by mapping the problem to the

normal bundle NC of C and then scaling one part of the potential in the normal

direction by

ε−1.

With decreasing ε the normal derivatives of the potential and thus

the constraining forces increase. In order to obtain a non-trivial scaling behavior of

the equation, the Laplacian is multiplied with a prefactor

ε2.

The reasoning behind

this scaling, which is the same as in [17, 32], is explained in Section 1.2. With q

denoting coordinates on C and ν denoting normal coordinates our starting equation

on NC has, still somewhat formally, the form

i∂tψε

=

−ε2ΔNCψε

+ Vc(q,

ε−1ν)ψε

+ W (q,

ν)ψε

=:

Hεψε

(1.4)

for

ψε|t=0

∈

L2(NC).

Here ΔNC is the Laplace-Beltrami operator on NC, where

the metric on NC is obtained by pulling back the metric on a tubular neighborhood

of C in A to a tubular neighborhood of the zero section in NC and then suitably

extending it to all of NC. We study the asymptotic behavior of (1.4) as ε goes to zero

uniformly for initial data with energies of order one. This means that initial data are

allowed to oscillate on a scale of order ε not only in the normal direction, but also in

the tangential direction, i.e. that tangential kinetic energies are of the same order as

the normal energies. More precisely, we assume that ε∇hψ0 ε 2 = ψ0 ε | − ε2Δhψ0ε

is of order one, in contrast to the earlier works [17, 32], where it was assumed to

be of order ε2. Here ∇h is a suitable horizontal derivative to be introduced in

Definition 1.1.

Our final result is basically an effective equation of the form (1.2). It is presented

in two steps. In Section 2.1 it is stated that on certain subspaces of

L2(NC)

the

unitary group

exp(−iHεt)

generating solutions of (1.4) is unitarily equivalent to an

’effective’ unitary group exp(−iHeff

ε

t) associated with (1.2) up to errors of order

ε3|t|

uniformly for bounded initial energies. In Section 2.2 we provide the asymptotic

expansion of Heff

ε

up to terms of order

ε2,

i.e. we compute Heff,0, Heff,1 and Heff,2

in Heff = Heff,0 + εHeff,1 +

ε2Heff,2

+

O(ε3).

Furthermore, in Section 2.3 and 2.4 we explain how to obtain quasimodes of

Hε

from the eigenfunctions of Heff,0 + εHeff,1 +

ε2Heff,2

and quasimodes of Heff,0 +

εHeff,1

+ε2Heff,2

from the eigenfunctions of

Hε

and apply our formulas to quantum

wave guides, i.e. the special case of curves in

R3.

As corollaries we obtain results

generalizing in some respects those by Friedlander and Solomyak obtained in [16]