CHAPTER 1
Introduction
Although the mathematical structure of the linear Schr¨ odinger equation
(1.1) i∂tψ = −Δψ + V ψ =: , ψ|t=0
L2(A,dτ)
is quite simple, in many cases the high dimension of the underlying configuration
space A makes even a numerical solution impossible. Therefore it is important to
identify situations where the dimension can be reduced by approximating the solu-
tions of the original equation (1.1) on the high dimensional configuration space A
by solutions of an effective equation
(1.2) i∂tφ = Heff φ , φ|t=0
L2(C,dμ)
on a lower dimensional configuration space C.
The physically most straightforward situation where such a dimensional reduction
is possible are constrained mechanical systems. In these systems strong forces
effectively constrain the system to remain in the vicinity of a submanifold C of the
configuration space A.
For classical Hamiltonian systems on a Riemannian manifold (A,G) there is a
straightforward mathematical reduction procedure. One just restricts the Hamilton
function to T ∗C by embedding T ∗C into T ∗A via the metric G and then studies the
induced dynamics on T ∗C. For quantum systems Dirac [12] proposed to quantize
the restricted classical Hamiltonian system on the submanifold following an ’intrin-
sic’ quantization procedure. However, for curved submanifolds C there is no unique
quantization procedure. One natural guess would be an effective Hamiltonian Heff
in (1.2) of the form
(1.3) Heff = −ΔC + V |C ,
where ΔC is the Laplace-Beltrami operator on C with respect to the induced metric
and V |C is the restriction of the potential V : A R to C.
However, to justify or invalidate the above procedures from first principles, one
needs to model the constraining forces within the dynamics (1.1) on the full space A.
This is done by adding a localizing part to the potential V . Then one analyzes the
behavior of solutions of (1.1) in the asymptotic limit where the constraining forces
become very strong and tries to extract a limiting equation on C. This limit of strong
confining forces has been studied in classical mechanics and in quantum mechanics
many times in the literature. The classical case was first investigated by Rubin and
Ungar [40], who found that in the limiting dynamics an extra potential appears
that accounts for the energy contained in the normal oscillations. Today there is
a wide literature on the subject. We mention the monograph by Bornemann [4]
for a result based on weak convergence and a survey of older results, as well as the
book of Hairer, Lubich and Wanner [19], Section XIV.3, for an approach based on
classical adiabatic invariants.
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