Spectral Properties Of Sturm-Liouville Problems With Distributional Potentials

As the origin of the differential operator theory, Sturm-Liouville problem has developed to be a significant field of study in mathematics and physics. In quantum mechanics,in order to describe the interaction in the microscopic particle, potential function in the Schr ?dinger equation could be a distribution(for instance, Dirac δ function). However,such problems are beyond the scope of classical Sturm-Liouville problem. To this end, it seems necessary to study the Sturm-Liouville problem with distributional potential. The main object of this paper is to study the spectral properties of Sturm-Liouville problems with distributional potentials, this paper involves five chapters.In the first chapter, we introduce the background of the problems to be investigated and the main results in this paper are presented.In the second chapter, basic concepts and relative properties are presented.The third chapter is devoted to study the eigenvalue problems of self-adjoint SturmLiouville problems with distributional potentials. Under an appropriate topology, the dependence of n-th eigenvalue on the differential operator will be researched, which includes the continuity of the n-th eigenvalue on the boundary conditions, the continuous dependence and differentiability of the n-th eigenvalue with respect to the coe?cients of the operator. After discussions on the inequalities among eigenvalues of different boundary conditions, the oscillation of the eigenfunctions can be obtained. In this chapter, we will construct a sequence of classical Sturm-Liouville operators with regular potentials such that this sequence approximates the Sturm-Liouville operator with a distributional potential in norm resolvent convergence. Moreover, we obtain a relationship between the eigenvalues of classical Sturm-Liouville operators and the eigenvalues of Sturm-Liouville operator with a distributional potential. Based on this innovative methods, the results given in this chapter extend the properties for the classical Sturm-Liouville problems. The final section of this chapter will apply the investigation of this paper to a class of SturmLiouville problems with transmission conditions.In the forth chapter, we consider the finite spectrum of Sturm-Liouville problems with distributional potentials. Firstly, by relaxing the conditions on the coe?cients,which are required in the third chapter, for the separated boundary conditions, the existence of the eigenvalues and the oscillation of the corresponding eigenfunctions are described. Through the special partition on the interval and the coe?cients of each subin-terval to meet certain special conditions, Sturm-Liouville problems with exactly n eigenvalues can be constructed. With the approximation by the operators investigated in the third chapter, inequalities among eigenvalues of different boundary conditions can be obtained. Moreover, we also study the equivalence between the Sturm-Liouville problem with finite spectrum and the matrix eigenvalue problem.Finally, in the fifth chapter, the spectral properties for the Sturm-Liouville operators with δ-interactions on a discrete set of an infinite interval are investigated. In this chapter,necessary and su?cient conditions and also simple su?cient conditions are given for the spectrum of the operators to be discrete. And we also prove su?cient conditions on the stability of the continuous spectrum. Our discussions are based on the quadratic form associated with the Sturm-Liouville operators with δ-interactions and a series of embedding inequalities we construct. Our results extend the Molchanov’s discrete spectra criterion to the Sturm-Liouville operators with δ-interactions.