Memoirs of the American Mathematical Society 2014; 83 pp; softcover Volume: 230 ISBN10: 0821894897 ISBN13: 9780821894897 List Price: US$65 Individual Members: US$39 Institutional Members: US$52 Order Code: MEMO/230/1083
 The authors consider the timedependent Schrödinger 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 Schrödinger 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. Table of Contents  Introduction
 Main results
 Proof of the main results
 The whole story
 Appendix A. Geometric definitions and conventions
 Bibliography
