Effective Hamiltonians for Constrained Quantum Systems

Jakob Wachsmuth, Stefan Teufel

Memoirs of the AMS 1083 (2013).
[arXive:0907.0351]

Zusammenfassung

We consider the time dependent Schrödinger equation on a Riemannian manifold A with a potential that localizes a certain class of states close to a fixed submanifold C, the constraint manifold. When we scale the potential in the directions normal to C by a parameter epsi<<1, the solutions concentrate  in an epsi-neighborhood of the submanifold. We derive an effective Schrödinger equation on the submanifold  C and show that its solutions, suitably lifted to A, approximate the solutions of the original equation on A up to errors of order epsi^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 insure 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.