The Dirac equation is an important model in relativistic quantum mechanics. In the semi-classical regime ε≪1, even a spatially spectrally accurate time splitting method [6] requires the mesh size to be O(ε), which makes the direct simulation extremely expensive. In this paper, we present the Gaussian beam method for the Dirac equation. With the help of an eigenvalue decomposition, the Gaussian ...

We generalize the well-known lower estimates for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold proved by Th. Friedrich (1980) and O. Hijazi (1986, 1992). The special solutions of the Einstein-Dirac equation constructed recently by Friedrich/Kim are examples for the limiting case of these inequalities. The discussion of the limiting case of these estimates yiel...

We study the Einstein-Dirac equation as well as the weak Killing equation on Riemannian spin manifolds with codimension one foliation. We prove that, for any manifold M admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics η on M × R such that (M × R, η) admit Einstein spinors (resp. weak Killing spinors). To prove the result we split the Einstein-Dirac equ...

most physical phenomena that occur in nature. Its basic equation, the Schrödinger equation, accurately describes the electronic properties of solids—of great importance in modern technology.To describe the behavior of particles that travel near the speed of light, such as neutrinos, the theory must be modified to take Einstein’s theory of relativity into account. This was accomplished by physic...

This paper is devoted to the derivation of absorbing boundary conditions for the KleinGordon and Dirac equations modeling quantum and relativistic particles subject to classical electromagnetic fields. Microlocal analysis is the main ingredient in the derivation of these boundary conditions, which are obtained in the form of pseudodifferential equations. Basic numerical schemes are derived and ...