Some Kinds Of Extensions Of The Symmetric Differential Operators

In this paper, we study the extensions of the symmetric differential operators.The differential operators are essentially the unbounded closable op-erators, and the domains of the unbounded closable operators must’t be the whole space. Hence the choices of the domains of the differential op-erators are always the very important and difficult. Given a differential expression, the specific demands for the differential operators eventually reflect on the restrictions on the domains. Differential operators with dif-ferent domains will have different spectral distributions especially the dis-crete spectrum. Symmetric, self-adjoint, dissipative and bound-preserving are some of the most important ones among the restrictions. Symmetric operators usually play an important role in investigating the other oper-ators since the minimal operator is symmetric, and other operators can be obtained by extending the domain of the minimal operator or restrict-ing the domain of the maximal operator. In this paper, we consider the very important problem about how to choose the domains of the differ-ential operators and place great emphasis on characterizing the domains of the self-adjoint extensions, dissipative extensions, Friedrichs extensions by using the latest methods from symplectic geometry and Limit-Circle solutions. We also investigate the eigenvalue problem and completeness of eigenfunctions of the Sturm-Liouville operators with transmission condi-tions and eigenparameter-dependent boundary conditions.In1999W. N. Everitt and L. Markus, using methods from symplec-tic algebra and geometry, characterized the self-adjoint domains in terms of Lagrangian subspaces of symplectic spaces (we refer to this as the EM characterization). In recent years, Wang Aiping, Hao Xiaoling, Sun Jiong and A. Zettl, constructed Limit-Circle (LC) solutions and characterized the self-adjoint domains in terms of LC solutions (we refer to this as the LC charaterization), furthermore, investigated the discrete spectrum of the singular differential operators, using the properties of the LC solutions and approaching of regular operator in the sense of strong resolvent conver-gence. Then we consider the following problems in this paper:what is the relationship between the LC characterization and the EM characterization? Being a very effective method of characterizing the self-adjoint extensions, could the symplectic geometry be used to characterize the dissipative ex-tensions? Could the LC solutions be used to characterize the dissipative extensions? How about the LC characterization of the Friedrichs exten-sion?In this paper we investigate the above problems. Firstly, we consider the LC characterization of the self-adjoint extensions in terms of symplectic geometry. We use real-parameter LC solutions to describe the complete Lagrangian subspaces of the complex symplectic spaces. Combining ME characterization and LC characterization, we give the description of the self-adjoint domains in terms of symplectic geometry. Also using real-parameter LC solution, we show the necessary and sufficient conditions about the three kinds of classifications of complete Lagrangian subspaces: strictly separated, totally coupled, mixed. Then we try to investigate the spectrum of the differential operators in terms of symplectic geometry, and obtain a necessary condition for a real A to be an eigenvalue of a self-adjoint realization.Secondly, we investigate the dissipative extensions of the symmetric operator in terms of symplectic geometry and real-parameter LC solutions. we define Dissipative (Accretive) subspaces, strictly Dissipative (strictly Accretive) subspaces and maximal Dissipative (maximal Accretive) sub-spaces of complex symplectic spaces and develop their algebraic properties in case the complex symplectic spaces have finite dimension. Then we give the result, which is similar to GKN-EM theory:There is a one to one correspondence between the set of all dissipative (strictly dissipative) ex-tensions of the minimal operator Mmin and the set of all Dissipative (strictly Dissipative) subspaces in the complex symplectic space S=Dmax/Dmin· Then under a basis of the complex symplectic space, which is constructed by real-parameter LC solutions, we give the specific characterizations of a maximal Dissipative subspace and a maximal Accretive subspace and prove that they form a symplectic orthogonal direct sum decomposition of complex symplectic space S=Dmax/Dmin.Thirdly, we characterize the Friedrichs extension of the Sturm-Liouville operator in terms of real-parameter LC solutions. We directly start with the definition of Friedrichs extension, by constructing the proper real-parameter LC solutions, characterize the Friedrichs extensions of both reg-ular and singular cases uniformly. And we establish the characterization as an extension of the minimal domain. Our proof is closer in spirit to the Friedrichs construction. At last, we study the eigenvalue problem and com-pleteness of eigenfunctions of the Sturm-Liouville operators with transmis-sion conditions and eigenparameter-dependent boundary conditions, using the classical method of spectral analysis and operator theory.This thesis consists of six chapters. In chapter I, we introduce the background about the problems, which we study, the main results and innovations of this thesis. Chapter II introduce the associated fundamen-tal definitions and important lemmas. In chapter III, we investigate the LC characterization of the self-adjoint extensions in terms of symplectic geometry. Chapter IV characterizes the dissipative extensions of symmet-ric operators in terms of symplectic geometry and Limit-Circle solutions. In chapter V we give the LC characterization of the Friedrichs extension of the Sturm-Liouville operator. In chapter VI, we study the eigenvalue problem and completeness of eigenfunctions of the Sturm-Liouville opera-tors with transmission conditions and eigenparameter-dependent boundary conditions.