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Effective Hamiltonians for Constrained Quantum Systems

About this Title

Jakob Wachsmuth, Department of Technology Development and Design, University of Bremen, Enrique-Schmidt-Str. 7, 28359 Bremen, Germany. and Stefan Teufel, Mathematics Institute, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 230, Number 1083
ISBNs: 978-0-8218-9489-7 (print); 978-1-4704-1673-7 (online)
DOI: https://doi.org/10.1090/memo/1083
Published electronically: December 24, 2013
Keywords: Schrödinger equation on manifolds, constraints, effective dynamics, adiabatic limit
MSC: Primary 81Q15, Secodary, 35Q41, 58J37, 81Q70

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Table of Contents

Chapters

• 1. Introduction
• 2. Main results
• 3. Proof of the main results
• 4. The whole story
• A. Geometric definitions and conventions

Abstract

We consider the time-dependent 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 we 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. We 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 $\varepsilon ^3|t|$ at time $t$. Furthermore, we 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.

Our results hold in the situation where tangential and normal energies are of the same order, and where exchange between these 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 allow for constraining potentials that change their shape along the submanifold, which is the typical situation in the applications mentioned above.

Since we consider a very general situation, our effective Hamiltonian contains many non-trivial terms of different origin. In particular, the geometry of the normal bundle of $\mathcal {C}$ and a generalized Berry connection on an eigenspace bundle over $\mathcal {C}$ play a crucial role. In order to explain the meaning and the relevance of some of the terms in the effective Hamiltonian, we analyze in some detail the application to quantum wave guides, where $\mathcal {C}$ is a curve in $\mathcal {A}=\mathbb {R}^3$. This allows us to generalize two recent results on spectra of such wave guides.