Analytic Discription Of Self-Adjointness Of General C-Symmetric Differential Operators

In this dissertation,the problem of self-adjoint extension of C-symmetric differential operators is studied according to the characterization of the boundary conditions in the self-adjoint domains,starting from discussing the new unified canonical form of self-adjoint domains of second and fourth order differential operators.Linear operator theory is an important part of functional analysis.It is a branch of mathematics that deeply reflects the essence of many mathematical problems.It has a very wide application background and research significance.Among them,linear differen-tial operators,as the most basic operation in modern mathematics,plays an unexpected role in linear operator theory and other branches of mathematics.In general,the linear differential operator,is a linear differential equation and homogeneous linear boundary conditions given to it.Since the spectrum of self-adjoint operators is real,and therefore it has a special importance in applications.Differential operators are densely defined opera-tors generated by the differential expression.They are a class of unbounded closable linear operators,while self-adjoint differential operators are a class of unbounded closed opera-tors.According to closed graph theorem in functional analysis,the domain of self-adjoint differential operators cannot be the whole space,thus the choice of the self-adjoint domains is always a extremely difficult problem in the theory of self-adjoint differential operator.The problem of characterization of the self-adjoint differential operators depends not only on the differential equations generated,but also on the description of the spatial range in which it acts.As usual,symmetric operators are the basis for further study of other types of operators.The self-adjoint problem of differential operators is ultimately reflected in the restriction of the domain.The spectrum decomposition of differential operators with different definition domains is quite different,especially for discrete spectrum.Therefore,the canonical form of self-adjoint boundary conditions for symmetric differential operators is the theoretical basis for studying the influence of boundary conditions of differential operators on spectral distribution.Canonical form of boundary conditions has a basic and unique position in studying of the effects of boundary conditions on the spectral distribution of differential operators.In recent years,some mathematicians have given two different canonical forms of coupled self-adjoint boundary conditions and separated self-adjoint boundary conditions for second-order differential operators.The classification of canonical forms of self-adjoint boundary conditions for the fourth-order differential operators and their specific forms also have been studied.It is noted that the two canonical forms of coupled and separated for the second-order differential operators have completely different forms each other,and therefore their application(including the study of the dependence of eigenvalues on boundary conditions)will be limited to a certain extent.In this paper,we obtain a new unified canonical form of the second order self-adjoint boundary conditions,which can be transformed into a coupled form or a separated form by choosing the coefficients of the canonical form.On this basis,by studying new canonical forms of self-adjoint boundary conditions for the fourth order differential operators,the fourth order case is completely consistent with the second order case in form,and it contains each of their own type of canonical forms.This provides a nice basis for studying the canonical form of self-adjoint boundary conditions of general even order symmetric differential operators.The characterization of the self-adjoint domains,i.e.,the restriction of boundary condition,is a great meaningful and fundamental problem in the theory of differential operators,which has been widely explored by many Chinese and foreign scholars.In the process of studying the canonical forms of self-adjoint boundary conditions,we note that professors M.A.Naimark and A.Zettl introduced the different symmetric differential expressions respectively,so based on these works,we consider and introduce the concept of C-symmetry to unify these two different differential operators.Futhermore,we study the general even or odd order C-symmetric differential expressions,where C is a skew-diagonal constant matrix satisfying C-1 =-C = C*,this expands the mathematical implication of symmetric form and gives a more complete new symmetric form for differential operators.With the demand of application,the study of self-adjoint differential operators in direct sum spaces has been greatly expanded.Since the self-adjoint extensions problem for the two-interval second-order Sturm-Liouville problem is studied,these two-interval theories are extended to the characterization of self-adjoint domains for higher-order dif-ferential operators,and to the higher-order differential equations on any finite or infinite number of intervals.Since the spectrum of self-adjoint operators is real,the application of real parameter square-integrable solution to characterize the self-adjoint problem will generate information related to the spectrum of differential operators.The two-interval case is studied in this paper.That is,in the direct sum framework of Hilbert space,apply the real parameter square-integrable solution of the differential equation,and give an ex-plicit characterization of all self-adjoint extensions of C-symmetric differential operators on two intervals which all the four endpoints are singular in the direct sum of Hilbert spaces.Through the above research,noting the fundamental characteristics of the matrix describing the boundary conditions of differential operators,we sum up a class of matrix groups named C-symplectic groups acting on boundary conditions of self-adjoint differen-tial operators,and study the properties of these C-symplectic groups and the distribution of their eigenvalues.Further,from the perspective of C-symplectic groups,we study the characterization of all self-adjoint extensions for general even-order C-symmetric differen-tial operators and the canonical boundary forms for these boundary matrices.The study of the properties of C-symplectic groups provides a new way for us to study and understand self-adjoint extensions.