The Development Of The Theory Of Ordinary Differential Operators

The theory of ordinary differential operators is playing an important role in the modern quantum mechanic, and is also an important facility in mathematical-physical equations and other applied technology fields. It is a systematic and comprehensive mathematical branch, which includes theories and methods in ordinary differential equations, functional analysis, operator algebra and function spaces. Some principal problems, including description of self-adjoint domains, analysis of spectrum, deficiency index, and inverse spectrum problemetc, are investigated in the theory of ordinary differential operators.Based on the intensive reading of original sources and research literatures on the subject in consideration, this dissertation investigates systematically and thoroughly into the development history of the theory of ordinary differential operators by using the method of source analysis and the comparative approach of the literature. The main results of this paper are as follows:First, Based on the brief introduction of the well-known Sturm and Liouville's results, the author systematically and thoroughly discuss the origins of the ordinary differential operators theory. The main contents of this part are as follows:(1) the origins of the Sturm-Liouville theory; (2) the work of Sturm and the work of Liouville; (3) the influence of the Sturm-Liouville theory;(4) the later development of the Sturm-Liouville theory.Second, By reviewing carefully the early work about the singular Sturm-Liouville boundary value problems, which took place during the years from 1900 to 1950, in this chapterâ…¢there are detailed discussions of contributions to from:Weyl(1910), Dixon(1912), Stone (1932) and Titchmarsh (1940-1950). The results of Weyl and Titchmarsh are essentially derived within classical, real and complex mathematical analysis. The results of Stone apply to examples of self-adjoint operators in the abstract theory of Hilbert spaces and in the theory of ordinary linear differential equations. The main contents of this part are as follow:1. In 1910, the results of Weyl are the first to consider the singular case of the S-L differential equations it is the first structured consideration of the analytical properties of the equation. The rang of new definitions and results is remarkable and set the stage for the full development of S-L theory in the 20th century, as to be seen in the later theory of differential operators in the work of von Neumann and Stone, and in the application of complex variable techniques by Titchmarsh.2. In 1912, the paper of Dixon seems to be the first paper in which the continuity conditions on the coefficientsp,q,ware replaced by the Lebesgue integrability conditions; these latter conditions are the minimal conditions to be satisfied by p,q,w within the environment given by the Lebesgue integral.3. In 1932, the results of Stone seems to be the first extended account of the properties of S-L differential operators in the Hilbert function spaces, under the Lebesgue minimal conditions on the coefficients of differential equation.4. Titchmarsh considered both the regular and singular case of S-L problems by applying the extensive theory of functions of a single complex variable.Third, By analying the literatures in the past on characterization of self-adjoint domains, the author discusses the important results in the development of the theory of self-adjoint domains. The main content of this part is as follow:1. In the regular case, in 1954, Coddington used the matrix theory and the conjugate boundary conditions, and given the description of self-adjoint domains of even order symmetric expressions. Naimark considered a symmetric quasi-differential expressions with two regular the boundary conditions. When coefficients are sufficiently smooth, Coddington's condition and Naimark's condition were proved equivalant. In 1962, Everitt given a result using linearly independent solutions of differential equation. This result is similar to Coddington's, and above conditions and Everitt's are equivalent. 2. In the singular case, by studying deeply into the Titchmarsh's result Everitt's result, cao zhi Jiang's result and sun jiong's result, it is pointed out that cao zhi Jiang's result is a complete and direct description of self-adjoint domains with the ones exposed by Everitt as special.Fourth, based on the analysis of a number of papers on spectra analysis of ordinary differential operators, in particular, the author generalize the important criteria of the discrete spectrum of the real self-adjoint differential operators, weighted differential operators and J- self-adjoint differential operators, and of the essential spectrum of the classes of differential operators.Fifth, through deep study on the first-hand information, the author systematically and thoroughly discuss the development on the deficiency indices theory for singular symmetric differential operators. In particular two important criterias were drawn as the core of the deficiency indices theory: the limit-point case and the limit-circle case of 2-nd order or higher order differential operators.