Extremun Problems For Self-adjoint Sturm-Liouville Difference Equations

The boundary problem of second-order Sturm-Liouville differential equa-tions originated from the study of solid beat conduction model in the 19th century,they constitutes a,class of very important,second-orcder differential operators,are widely used in classical physics and modern quantum mechan-ics.The research in this field has a long history and abundant conclusions so far.Second-order Sturm-Liouville difference equations are the discretization of Sturm-Liouville differential equations,the spectral theor-of the second-order difference operators derived from them are also widely used in engineering tech-nology,life sciences and other fields.The inverse spectral theory of differential or differential operators is an important subject in the theory of differential or differential operators,it mainly studies what spectral information can be used to reconstruct differential or differential operators.When given a family of spectral information,all potential functions that make the system satisfy these spectral information are called inverse speetral sets,which play an im-portant role in the reconstruction of operators.This paper mainly studies the inverse spectral of self adjoint Sturm-Liouville differential equations,then obtained some information about the infimum of the element’s norm in the inverse spectral set of a Sturm-Liouville difference equation by using Green’s function and Mercer’s theorem of difference operators,when the first.eigenval-ue of the boundary problem is known,including the expression of the infimum,the accessibility and when to get the infumum.The full paper can be divided into five chapters.Chapter 1 is an introduction,which describes the background and research status of this paper,and points out the main research work and innovative points of this paper.Chapter 2 introduces the spectral theory of boundary problems for second-order Sturm-Liouvillc difference equations,including self-adjoint,conditions:the number and distribution of eigenvalues,continuity cdependence of eigenval-ues,monotonicity of eigenvalue branches and so on.Chapt,er 3 introduces the Green’s formula and Mercer’s theorem for bouncd-ary problems of second-order Sturm-Liouvillc cdifference equation.Based on Mereer’s theorem,the relationships between Green’s function and eigenvalues,potentical function and the Green’s function are obtained.Chapter 4 gets a main conc:lusion of this paper.With Dirichlet boundary conditions,the expression of the infiiunm of the norm of the clcments in the inverse spectral set with respect to the known first eigenvalue is obtained.Chapter 5 obtains a.method for finding the infimum of the norm of the inverse spectral set of known first,eigenvalues under general self-adjoint bound-ary conditions,then obtains explicit,expression of the first eigenvalue for the lower bound of the element norm in the inverse spectral set with some normal boundary conditions.