Generalized solutions and distributional shadows for Dirac equations

We discuss the application of recent results on generalized solutions to the Cauchy problem for hyperbolic systems to Dirac equations with external fields. In further analysis we focus on the question of existence of associated distributional limits and derive their explicit form in case of free Dirac fields with regularizations of initial values corresponding to point-like probability densities

Generalized Fourier Integral Operators on spaces of Colombeau type

Generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is based on a theory of generalized oscillatory integrals (OIs) whose phase functions as well as amplitudes may be generalized functions of Colombeau type. The mapping properties of these FIOs are studied as the composition with a generalized pseudodifferential operator. Finally, the microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets are investigated. This theory of generalized FIOs is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distributional data

Classes of generalized functions with finite type regularities

We introduce and analyze spaces and algebras of generalized functions which correspond to Hölder, Zygmund, and Sobolev spaces of functions. The main scope of the paper is the characterization of the regularity of distributions that are embedded into the corresponding space or algebra of generalized functions with finite type regularities

Dipole trap model for the metallic state in gated silicon-inversion layers

In order to investigate the metallic state in high-mobility Si-MOS structures, we have further developed and precised the dipole trap model which was originally proposed by B.L. Altshuler and D.L. Maslov [Phys. Rev. Lett.\ 82, 145 (1999)]. Our additional numerical treatment enables us to drop several approximations and to introduce a limited spatial depth of the trap states inside the oxide as well as to include a distribution of trap energies. It turns out that a pronounced metallic state can be caused by such trap states at appropriate energies whose behavior is in good agreement with experimental observations.Comment: 16 pages, 10 figures, submitte

Dynamic structure factor of the antiferromagnetic Kitaev model in large magnetic fields

We investigate the dynamic structure factor of the antiferromagnetic Kitaev honeycomb model in a magnetic field by applying perturbative continuous unitary transformations about the high-field limit. One- and two-quasi-particle properties of the dressed elementary spin flip excitations of the high-field polarized phase are calculated which account for most of the spectral weight in the dynamic structure factor. We discuss the evolution of spectral features in these quasi-particle sectors in terms of one-quasi-particle dispersions, two-quasi-particle continua, the formation of anti-bound states, and quasi-particle decay. In particular, a comparably strong spectral feature above the upper edge of the upmost two-quasi-particle continuum represents three anti-bound states which form due to nearest-neighbor density-density interactions.Comment: 14 pages, 10 figure

Ising model in a light-induced quantized transverse field

We investigate the influence of light-matter interactions on correlated quantum matter by studying the paradigmatic Dicke-Ising model. This type of coupling to a confined, spatially delocalized bosonic light mode, such as provided by an optical resonator, resembles a quantized transverse magnetic field of tunable strength. As a consequence, the symmetry-broken magnetic state breaks down for strong enough light-matter interactions to a paramagnetic state. The nonlocal character of the bosonic mode can change the quantum phase transition in a drastic manner, which we analyze quantitatively for the simplest case of the Dicke-Ising chain geometry. The results show a direct transition between a magnetically ordered phase with zero photon density and a magnetically polarized phase with superradiant behavior of the light. Our predictions are equally valid for the dual quantized Ising chain in a conventional transverse magnetic field

The wave equation on singular space-times

We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.Comment: Latex, 19 pages, 1 figure. Discussion of class of metrics covered by our results and some examples added. Conclusion more detailed. Version to appear in Communications in Mathematical Physic

Isomorphisms of algebras of Colombeau generalized functions

We show that for smooth manifolds X and Y, any isomorphism between the special algebra of Colombeau generalized functions on X, resp. Y is given by composition with a unique Colombeau generalized function from Y to X. We also identify the multiplicative linear functionals from the special algebra of Colombeau generalized functions on X to the ring of Colombeau generalized numbers. Up to multiplication with an idempotent generalized number, they are given by an evaluation map at a compactly supported generalized point on X.Comment: 10 page