Sturm-Liouville Operators With Transmission Conditions And The Spectrum Of The Schrodinger Operators With Point Interactions

The differential operators are essentially a class of closed, unbounded linear operators. A lots of mathematical, physical and other problems can be transformed into the differential operators. Some of these opera-tors are discontinuous, i.e., the operators with transmission conditions, so it is important to investigate the differential operators with transmission conditions. In this paper, we study the Sturm-Liouville operators with transmission conditions, and the spectral analysis of the Schrodinger op-erators with point interactions using the theory of the singular self-adjoint differential operators with transmission conditions.We study the spectrum of the Schrodinger operators with point in-teractions with the use of the theory of the singular self-adjoint differential operators with transmission conditions. The Schrodinger operators with point interactions are the Schrodinger operators in which the potential function is a generalized function. We discuss the Schrodinger operators with point interactions in which the potential functions are Dirac func-tion δ and δ’, the derivative of Dirac function δ. Taking the action point0of the potential functions as a discontinuous point, we characterize the self-adjoint domain of the singular differential operators with transmission conditions. The correspondence between the Schrodinger operators with point interactions and the singular self-adjoint differential operators with transmission conditions is given. Then we discuss the spectrum of the singular self-adjoint differential operators with transmission conditions by using the relations of the spectral function and the Weyl function of the sin-gular self-adjoint differential operators. The spectrum of the corresponding Schrodinger operators with point interactions is obtained.Furthermore, we study some Sturm-Liouville(S-L) operators with trans-mission conditions. Here, first Sturm-Liouville(S-L) operators with sepa-rate boundary conditions and transmission conditions are investigated in the abutted two intervals. We define an inner product associated with the transmission conditions and construct a new Hilbert space. In this Hilbert space, the S-L operators with separate boundary conditions and transmis-sion conditions are discussed. The necessary and sufficient condition for parameter A being an eigenvalue of the S-L operators is obtained, and the multiplicity of the eigenvalues is also discussed. With the constructing of the Green’s function of the S-L operators with separate boundary condi-tions and transmission conditions, we prove the modified Parseval equality, the Parseval equality of the S-L operators with transmission conditions.Secondly, the S-L operators in the disjoint two intervals are consid-ered. Viewing the boundary conditions between the right endpoint of the left interval and the left endpoint of the right interval as the transmission conditions, we study the corresponding S-L operators with transmission conditions. Constructing the solutions of the S-L equation, which satisfy the transmission conditions and some of the boundary conditions, we ob-tain the asymptotic formulas of the solutions using the method given by Titchmarsh. We study the asymptotic behavior of the eigenvalues and eigenfunctions of the S-L operators with transmission conditions by the asymptotic formulas of the solutions. Using the Eigenfunctions expansion of the Green’s function of the considered S-L operators, we establish the Parseval equality of the S-L operators with transmission conditions.Finally, we study the S-L operators with coupled boundary conditions and transmission conditions in the abutted two intervals. The maximal and minimal operators associated with the coupled boundary conditions and transmission conditions are defined. We prove the considered S-L opera-tors is self-adjoint if the matrix coefficient in the coupled boundary con-ditions satisfy some certain conditions. Constructing the solutions of the S-L equation, we obtain an entire function whose zeros are the eigenvalues of the S-L operators with the coupled boundary conditions and transmis-sion conditions, and the multiplicity of the eigenvalues. The absolutely and uniformly convergence of the Eigenfunctions expansion of the Green’s function of the considered S-L operators is proved. The Parseval equality of the considered problem is given.This paper contians six chapters. In the first chapter, we introduce the background of the problems to be investigated and the main results in this paper are presented. The second chapter introduces some funda-mental concepts and lemmas used in this paper. In the third chapter, we investigate the Schrodinger operators with point interactions by using the theory of the singular self-adjoint differential operators with transmission conditions, and give the spectrum of the Schrodinger operators with point interactions. In the forth chapter, we study the S-L operators with the separate boundary conditions and transmission conditions in the abutted two intervals [-1,0) and (0,1]. In the fifth chapter, we consider the S-L operators with transmission conditions in the disjoint two intervals (a1, b1) and (a2, b2). In the sixth chapter, the S-L operators with coupled bound-ary conditions and transmission conditions are discussed in the abutted two intervals [-1,0) and (0,1].