Friedrichs Expansion Of Sturm - Liouvill Operator For Vector Differential Equation

As the continuous deepening and development of the scientific research, people have payed extensive attention to more kinds of problems for differential equation, and the opera-tors theory for differential equation has been one of the important research directions along the present mathematics. The operators theory for differential equation takes the quantum mechanics of physics as the background, synthesize the ordinary differential equation, func-tion of real variable, partial differential equation, functional analysis, abstract algebra and other theoretical branches and methods to develop a systematic mathematic research grad-ually. It is used to solve many problems of mathematical physics equation and technology, and has been an important instrument of math. After looking up a large number of relevant books and original literature, applying the method of analysis and comparison, we study the vector differential equation in the following aspects.According to contents, the thesis is divided into five sections.Chapter I Preference, the objective is to research the development and the current situation of the problems.Chapter 2 In this section, we research some basic properties of the vector differential equation By the symmetry of P(t) and Q(t), we study a series of characters of the differential expres-sion τ.Chapter 3 In this section, we study of Sturm-Liouville operators for the vector differen-tial equation as a key. Firstly, we define the inner product in the Hilbert space L2((l, m); dt), then we define the maximal operator Tmax, the minimal operator Tmin and the operator with compact support To of the Hilbert space.Chapter 4 In this section, in order to research the principal solutions of the vector differential equation, we change the vector differential equation into the equivalent differential systems of the form then it will be more convenient to study with the matrix differential systems At last, we can get the corresponding principal solutions due to the Hamiltonian system.Chapter 5 In this section, we use the principal solutions of the matrix differential equation to get the boundedness from below and a new Friedrichs extension of the vector differential equation.