Effective Hamiltonians for constrained quantum systems / Jakob Wachsmuth, Stefan Teufel.

Format:

Book

Author/Creator:

Wachsmuth, Jakob, 1981- author.

Teufel, Stefan, 1970- author.

Series:

Memoirs of the American Mathematical Society ; Volume 230, Number 1083.

Memoirs of the American Mathematical Society, 1947-6221 ; Volume 230, Number 1083

Language:

English

Subjects (All):

Hamiltonian operator.

Quantum theory.

Eigenvalues.

Mechanics.

Physical Description:

1 online resource (96 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2013.

Language Note:

English

Summary:

The authors consider the time-dependent Schrodinger 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 Schrodinger 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 $\varepsilon DEGREES3t$ 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 $\varepsilon DEGREES3$ with those of the full Hamiltonian under reasonab

Contents:

""Contents""; ""Chapter 1. Introduction""; ""1.1. The model""; ""1.2. Comparison with existing results""; ""Chapter 2. Main results""; ""2.1. Effective dynamics on the constraint manifold""; ""2.2. The effective Hamiltonian""; ""2.3. Approximation of eigenvalues""; ""2.4. Application to quantum wave guides""; ""Chapter 3. Proof of the main results""; ""3.1. Proof of adiabatic decoupling""; ""3.2. Pullback of the results to the ambient space""; ""3.3. Derivation of the effective Hamiltonian""; ""3.4. Proof of the approximation of eigenvalues""; ""Chapter 4. The whole story""

""4.1. Elliptic estimates for the Sasaki metric""""4.2. Expansion of the Hamiltonian""; ""4.3. Construction of the superadiabatic subspace""; ""Appendix A. Geometric definitions and conventions""; ""A.1. Manifolds of bounded geometry""; ""A.2. The geometry of submanifolds""; ""Bibliography""

Notes:

"Volume 230, Number 1083 (fifth of 5 numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-1673-5