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 ε ≪ 1 controls the separation of time scales and the limit ε → 0 corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time ε → 0 is the semiclassical limit for the slow degrees of freedom. In this paper we show that the ε-dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn [LF91], coming from an ε-dependent classical Hamilton function and an ε-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 ε 2 . In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics [XCN10]. 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.
We examine spectra of Dirac operators on compact hyperbolic surfaces. Particular attention is devoted to symmetry considerations, leading to non-trivial multiplicities of eigenvalues. The relation to spectra of Maaß-Laplace operators is also exploited. Our main result is a Selberg trace formula for Dirac operators on hyperbolic surfaces.
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