Caustics, counting maps and semi-classical asymptotics

Semi-classical Asymptotics for the Counting Functions and Riesz Means of Pauli and Dirac Operators with Large Magnetic Fields

We study the asymptotic behavior, as Planck’s constant ~ → 0, of the number of discrete eigenvalues and the Riesz means of Pauli and Dirac operators with a magnetic field μB(x) and an electric field. The magnetic field strength μ is allowed to tend to infinity as ~ → 0. Two main types of results are established: in the first μ~ ≤ constant as ~ → 0, with magnetic fields of arbitrary direction; t...

Semi-classical Counting Function with Application to Quantum Current

By adapting methods developed in SV] to the semi-classical situation we obtain a new two-term asymptotic formula for the counting function of eigenvalues of h pseudodiierential operators in the limit as h ! 0. Recent results on clustering of eigenvalues obtained in PP2] follow in Corollary and, as an application, we consider the semi-classical behaviour of the quantum current studied recently i...

Rendering of Caustics with Photon Maps

A number of techniques exist for approximating the rendering equation in order to create realistic detailed images. Photon mapping is a method that allows for the accurate rendering of a wide range of reflective and refractive phenomena. We explore the basis of the photon mapping technique with the goal of constructing a system for rendering such effects, particularly in regard to caustics. Our...

Asymptotics of Classical Spin Networks

A spin network consists of a cubic ribbon graph (i.e., an abstract trivalent graph with a cyclic ordering of the vertices around each edge) and an admissible coloring of its edges by natural numbers. The evaluation of a spin network, with our normalization, is an integer number. The main results of our paper are: (a) an existence theorem for the asymptotics of evaluations of arbitrary spin netw...

Semi-θ-Open Sets and New Classes of Maps