Effective Hamiltonians for Constrained Quantum Systems

We consider the time dependent Schrödinger equation on a Riemannian
manifold $\mathcal{A}$ with a potential that localizes a certain
class of states close
to a fixed submanifold $\mathcal{C}$. When we scale the
potential in the directions normal to $\mathcal{C}$ by a parameter $
\varepsilon\ll 1$, the
solutions concentrate in an $\varepsilon$-neighborhood of $\mathcal{C}$.
This situation
occurs for example in quantum molecular dynamics for the motion of
nuclei
in electronic potential surfaces and in quantum wave guides. We derive
an
effective Schrödinger equation on the submanifold $\mathcal{C}$ and
show that its
solutions, suitably lifted to $\mathcal{A}$, approximate the solutions
of the original
equation on $\mathcal{A}$ up to errors of order $\varepsilon^3|t|$ at
time~$t$. Furthermore,
we prove that the eigenvalues of the corresponding effective Hamiltonian
below a certain fixed energy coincide up to errors of order $
\varepsilon^3$ with
those of the full Hamiltonian under reasonable conditions. This is
joint work with
Jakob Wachsmuth.