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
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
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
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
+ Vc(q,
+ W (q,

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
unitary group
generating solutions of (1.4) is unitarily equivalent to an
’effective’ unitary group exp(−iHeff
t) associated with (1.2) up to errors of order
uniformly for bounded initial energies. In Section 2.2 we provide the asymptotic
expansion of Heff
up to terms of order
i.e. we compute Heff,0, Heff,1 and Heff,2
in Heff = Heff,0 + εHeff,1 +
Furthermore, in Section 2.3 and 2.4 we explain how to obtain quasimodes of

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

and apply our formulas to quantum
wave guides, i.e. the special case of curves in
As corollaries we obtain results
generalizing in some respects those by Friedlander and Solomyak obtained in [16]
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