The Friedrichs Extensions Of Singular Differential Operators And The Products Of Operators

The research of linear Hamiltonian systems originates from various of parts of applied mathematics and physics. It is one of the most active fields that is studied in nonlinear functional analysis. The systems of differential equations are important parts in differential equations. The structures they present have the profound significance of physical backgrounds and mathematical models. So, the study of differential equation and furthermore the systems of differential equations has profoundly intrinsic value. Spectral theory of boundary value problems for (systems of) differential equations are the combination of the two. They are both new and vital branches. They play important parts in applied mathematics and engineering, especially in the biochemistry aspects. So, it becomes important to study the solutions of them. Moreover, the study of the nature of the solutions is important as well. The thesis is divided into four sections according to the contents.Chapter1Preference.Chapter2We consider a formally self-adjoint differential express where pij are real valued functions and locally integrable in (a,∞). If L is bounded be-low and the deficiency indices of L are equal, in this case we characterize its Friedrichs extension in terms of solutions which are both square integrable and principal at infin-ity.Chapter3In this section, the product of two Hamiltonian operators of order2n is transformed to a Hamiltonian operator of order4n. Using the principal solution, the Friedrichs extension of products of two Hamiltonian operators which is defined on [a, b) is considered.Chapter4Based on the former two chapters, in this section we consider the product of four Hamiltonian operators of order2n is transformed to a Hamiltonian operator of order8n. Using the principal solution, the Friedrichs extension of products of four regular Hamiltonian operators which is defined on (a,∞) is considered.