Semiclassical approximations for Hamiltonians with operator-valued symbols

Abstract

We consider the semiclassical limit of quantum systems with a Hamiltonian given by the Weyl quantization of an operator valued symbol. Systems composed of slow and fast degrees of freedom are of this form. Typically a small dimensionless parameter $\epsilon \ll 1$ controls the separation of time scales and the limit $\epsilon \to 0$ corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time $\epsilon \to 0$ is the semiclassical limit for the slow degrees of freedom. In this paper we show that the $\epsilon$-dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn, coming from an $\epsi$-dependent classical Hamilton function and an $\epsilon$-dependent symplectic form, has a concrete mathematical and physical meaning: Based on this flow we prove a formula for equilibrium expectations, an Egorov theorem and transport of Wigner functions, thereby approximating properties of the quantum system up to errors of order ${\epsilon }^2$. In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics. Hence we discuss in some detail the application of the general results to the Hofstadter model, which describes a two-dimensional gas of non-interacting electrons in a constant magnetic field in the tight-binding approximation.

Final version to appear in Commun. Math. Phys. Results have been strengthened with only minor changes to the proofs. A section on the Hofstadter model as an application of the general theory was added and the previous section on other applications was removed

References in corpus (4)

• Berry Phase Effects on Electronic Properties
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• Applications of Magnetic PsiDO Techniques to Space-adiabatic Perturbation Theory
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