Semiclassical Approximations for Hamiltonians with Operator-Valued Symbols.

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 $${\varepsilon \ll 1}$$ controls the separation of time scales and the limit $${\varepsilon\to 0}$$ corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time $${\varepsilon\to 0}$$ is the semiclassical limit for the slow degrees of freedom. In this paper we show that the $${\varepsilon}$$ -dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn (Phys Rev A (3) 44(8):5239-5256, ), coming from an $${\varepsilon}$$ -dependent classical Hamilton function and an $${\varepsilon}$$ -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 $${\varepsilon^2}$$ . In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics (Xiao et al. in Rev Mod Phys 82(3):1959-2007, ). 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. [ABSTRACT FROM AUTHOR]