Dirac Equation On Bounded Domain And Whole Line

In this thesis,we study Dirac equation on bounded domains and the whole line.In chapter 1,section 1.1 introduces the physical background of the Dirac equation and the difficulties of studying the problems on bounded domains.Some theorems and definitions which will be used in this thesis are introduced in section 1.2.In chapter 2,we study several forms of homogeneous linear Dirac equations with some kinds of boundary value conditions.In section 2.1,we prove that the homogeneous boundary value problem has only the zero solution.In section 2.2,we study eigenvalue problem and obtain the eigenvalues and the corresponding eigenfunctions.In section 2.3,we study a special kind of boundary value problem.We obtain the necessary and sufficient condition of the existence of the solution and solved the problem.In section 2.4,we obtain the solution of the equation with the periodic boundary condition.In section 2.5,the conclusions we obtained in 2.1,2.2 and 2.3 are generalized to the equation with four components.In section 2.6,we study other forms of homogeneous linear Dirac equation.In chapter 3,we study nonhomogeneous linear Dirac equations.In section 3.1,we find the solution on the whole line by using Fourier transformation.In section 3.2,we use the method of constant variation to obtain the existence and the uniqueness of the solution of the boundary value problem on the bounded domain.In chapter 4,we use Fourier transformation to solve the time dependent Dirac equation and prove the existence of the solution.In chapter 5,we begin with studying the initial value problem of the nonlinear Dirac equation.We obtain the existence and the uniqueness of the solution on closed intervals.Then we study the nonlinear Dirac equation on bounded domains.We obtain the existence of the solution of the boundary value problem with appropriate boundary conditions.