09-102 Jakob Wachsmuth, Stefan Teufel

Effective Dynamics for Constrained Quantum Systems (695K, PDF) Jul 2, 09

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Abstract. We consider the time dependent Schr\"odinger equation on a Riemannian manifold $\mathcal{A}$ with a potential that localizes a certain class of states close to a fixed submanifold $\mathcal{C}$, the constraint manifold. When we scale the potential in the directions normal to $\mathcal{C}$ by a parameter $\varepsilpon\ll 1$, the solutions concentrate in an $\varepsilpon$-neighborhood of the submanifold. 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 $\varepsilon^3|t|$ at time~$t$. Our result holds in the situation where tangential and normal energies are of the same order, and where exchange between normal and tangential 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 now allow for constraining potentials that change their shape along the submanifold, which is the typical situation in applications like molecular dynamics and quantum waveguides.

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