Self-Adjoint Boundary-Value Problems For Products Of Differential Operators And Spectral Properties Of A Class Of Differential Operators

The self-adjoint boundary-value problems and spectral theory of differential operators are important and fundamental problems in the operator theory. The self-adj oint boundary-value problems for products of differential operators and spectral properties of a class of differential operators are investigated in this dissertation.Chapter 1 deals with the background and advance of the study on the differential operators, and the main topic, some important results obtained and employed methods in this paper. Chapter 2 is some preliminaries, symbols and some lemmas.In Chapter 3, we discuss the self-adjoint boundary-value problems for products of m differential operators generated by the same symmetric differential expression of order n defined on [a, b) (-∞ < a < b ≤ ∞), endowed with some boundary conditions. Combining the general construction theory of self-adjoint extensions of ordinary differential operators, we give, respectively, analytic characterization for self-adjoint boundary conditions of products of two 4th-order differential operators, two nth-order differential operators, and m nth-order differential operators at the regular (b < ∞) and the singular (b = ∞) cases, and obtain the necessary and sufficient conditions for self-adjointness of products of differential operators. Furthermore, some useful results concerning self-adjointness of the product operator are obtained. Meanwhile, in Chapter 4, for a regular symmetric vector-valued differential expression l(y) of order n on (a,b) (-∞ < a < b ≤ ∞), under the assumption that the power expression l~2(y) is partially separated in weighted function space L_r~2(a,b), the boundary conditions determining Friedrichs extension of the minimal operator generated by l~2(y) are identified explicitly.In Chapter 5, using the approach of operators, we discuss the discreteness of the spectrum of J—self-adjoint differential operators that are generated by J—symmetric differential expression with complex-valued coefficients. Some criteria for the discrete spectrum of J—self-adjoint differential operators are obtained, which provide some criteria for the discreteness of spectrum of the differential operators. Meanwhile, we also investigate the essential spectrum of J—self-adjoint differential operators that are extended by J—symmetric differential expression with complex-valued periodic function coefficients where the underlying Hilbert space is weighted. The existence region for the essential spectrum of J—self-adjoint differential operators defined above is located.