Effective Hamiltonians for Constrained Quantum Systems

The authors consider the time-dependent Schroedinger equation on a Riemannian manifold $\mathcal{A}$ with a potential that localizes a certain subspace of states close to a fixed submanifold $\mathcal{C}$. When the authors 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 wave guides and for the motion of nuclei in electronic potential surfaces in quantum molecular dynamics. The authors derive an effective Schroedinger 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, the authors 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.