Dirac Equation In Whole Space And Half Space

This thesis finds the essential spectrum of Dirac operator in whole space and half space in two and three dimensions and solve the eigenvalue problem of Dirac operator with magnetic fields.This thesis consists of three subjects.In the first part,we introduce some basic knowledge which will be used in this article.The first chapter is an introduction to the background of Dirac equation and our main results.In the second chapter,we introduce the relevant knowledge including essential spectrum,Dirac operator,Fourier transformation,Delta function,Sobolev space.In the second part,we find the essential spectrum of Dirac operator in whole space and half space in two and three dimensions.In section 3.1,we find the essential spectrum of Dirac operator in the whole space in two dimensions.In section 3.2,we use Fourier transformation to get the solutions to Dirac equation in half space and prove the existence of the solutions to Dirac equation with nonzero boundary conditions.In section 3.2.1,we prove the existence of the solutions to Dirac equation with external field.In section 3.2.2,we get the essential spectrum of the Dirac equation with two components.In section 3.2.3,we find the essential spectrum of the Dirac equation with four components by the conclusions in section 3.2.1.In section 3.3 and section 3.4,we also find the essential spectrum of Dirac operator in whole and half space in three dimensions.In the third part,we solve the eigenvalue problem of the Dirac operator with magnetic field with Neumann boundary condition and give the eigenfunctions.