 09201 Jakob Wachsmuth, Stefan Teufel
 Effective Hamiltonians for Constrained Quantum Systems
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Nov 9, 09

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Abstract. We consider the timedependent Schr\"odinger 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 $\veps$neighborhood of $\mathcal{C}$. This situation
occurs for example in quantum wave guides and for the motion of nuclei
in electronic potential surfaces in quantum molecular dynamics. We derive an
effective Schr\"odinger 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 $\varepsilon^3t$ at time~$t$. Furthermore, we prove that the eigenvalues of the corresponding effective Hamiltonian
below a certain energy coincide up to errors of order $\varepsilon^3$ with
those of the full Hamiltonian under reasonable conditions.
Our results hold in the situation where tangential and normal energies are
of the same order, and where exchange between these energies occurs. In earlier results tangential energies were assumed to be small compared to normal energies, and rather restrictive assumptions were
needed, to ensure that the separation of energies is maintained during the
time evolution. Most importantly, we can allow for constraining
potentials that change their shape along the submanifold, which is the
typical situation in the applications mentioned above.
Since we consider a very general situation, our effective Hamiltonian contains many nontrivial terms of different origin. In particular,
the geometry of the normal bundle of $\mathcal{C}$ and a generalized Berry connection on
an eigenspace bundle over $\mathcal{C}$
play a crucial role. In order to explain the meaning and the relevance of
some of the terms in the effective Hamiltonian, we analyze in some detail
the application to quantum wave guides, where $\mathcal{C}$ is a curve
in $\mathcal{A}=\mathbb{R}^3$.
This allows us to generalize two recent results
on spectra of such wave guides.
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