Spectral Problems Of Left-Definite Discrete Sturm-Liouville Operators

In this paper, spectral theory of left-definite discrete Sturm-Liouville operators willbe considered.The paper is mainly divided into four chapters.The first section is the introduction of the whole paper. We talk about the back-ground of this problem, and make plans for the research of the problems.Next section of the paper mainly introduces diference operators, and their self-adjointness and investigates a class of self-adjoint Sturm-Liouville diference operatorswith either a non-Hermitian leading coefcient function, or a non-Hermitian potentialfunction, or a non-definite weight function, or a non-self-adjoint boundary condition.The minimum conditions for such diference operators to be self-adjoint with respectto a natural quadratic form is obtained. It is shown that a discrete Sturm-Liouvilleproblem admits a diference operator realization if and only if it does not have allcomplex numbers as eigenvalues.The third section mainly investigates the spectral theory of left-definite discreteSturm-Liouville operators. Some fundamental spectral results, which include the resultthat its eigenvalues are real, are obtained by means of the definition of new inner productand the structure of self-adjoint subspace.At last, we summarize the results of the whole paper.