2 research outputs found
The Selberg trace formula for Dirac operators
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
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 controls the separation of time
scales and the limit corresponds to an adiabatic limit, in
which the slow and fast degrees of freedom decouple. At the same time
is the semiclassical limit for the slow degrees of freedom.
In this paper we show that the -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 -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 . 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.Comment: 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 remove