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Effective Hamiltonians for Constrained Quantum Systems
Jakob Wachsmuth, University of Bremen, Germany, and Stefan Teufel, University of Tübingen, Germany
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Memoirs of the American Mathematical Society
2014; 83 pp; softcover
Volume: 230
ISBN-10: 0-8218-9489-7
ISBN-13: 978-0-8218-9489-7
List Price: US$65 Individual Members: US$39
Institutional Members: US\$52
Order Code: MEMO/230/1083

The authors 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 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 $$\varepsilon^3|t|$$ 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.