Effective Dynamics for Constrained Quantum Systems

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 ε 1, the solutions concentrate in an ε-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 ε3|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. MSC 2010: 81Q15; 35Q41, 58J37, 81Q70. ∗Supported by the DFG within the SFB Transregio 71. University of Tübingen, Mathematical Institute, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email: [email protected] & [email protected].