Eigenvalues Of Indefinite Difference Sturm-Liouville Problems

In the mid-19th century,the Sturm-Liouville problem was developed as a mathemat-ical model for describing the heat conduction of solids.The problem has an important place in many fields,such as classical physics,quantum mechanics,geophysics,mete-orological physics,engineering technology and so on.With the continuous solution of Sturm-Liouville problem,mathematicians begin to pay attention to difference problem.In the early days,difference equations were used to describe discrete phenomena in nature,such as random sequence,combinatorial analysis,genetics,biology and so on.Therefore,there are many conveniences in application.Moreover,the mathematical models of some physical phenomena that we have established may be differential,and some continuous problems are easier to solve by discretization,this makes the research of the difference Sturm-Liouville problem very important.In 1964,Atkinson studied the spectral problem of self-adjoint difference systems[2].In 1978,Hartman studied the spectrum of the discrete Sturm-Liouville problem[12],and in the same year,Hinton and Lewis studied the second order regular vector difference boundary value problem[15].In 1987,Hooker studied the Guerin function of difference equations and its conjugacy[16].In 1989,Chen and Erbe studied the oscillation and non-oscillation forms of second order difference systems with self adjoint boundary conditions[6].In 1991,Peterson and Ridenhour studied Prufer transformations in discrete form[32].In 2010,R.Ma,C.Gao,and Y.Lu studied the spectral structure of second order difference operators with sign changing weights,and Applied Sylvester’s law of inertia to prove that the spectra contain real eigenvalues and simple eigenvalues,and that the number of positive eigenvalues is equal to the number of positive elements in the weight function,the number of negative eigenvalues is equal to the number of negative elements in the weight function[25].So far,the theory of differential system spectrum is becoming more and more perfect.In this paper,the eigenvalue problem of the indefinite difference Sturm-Liouville equation under the boundary condition of general separation type is studied.Firstly,we give some properties of the right-definite difference Sturm-Liouville problem.Secondly,a priori estimate of upper bound of non-real eigenvalues is obtained by using the above results and Schwarz inequality.An example is given to illustrate the convenience of finding the upper bound of non-real eigenvalues.Then,a priori estimate of the lower bound of real eigenvalues is given.Finally,the existence of non-real eigenvalues is proved by using PT symmetry method.The major work arrangements are as follows:Chapter One,introduction.This paper mainly introduces the background,actual source and present situation of the research.Chapter two,prepare knowledge.The theoretical knowledge of the right-definite difference Sturm-Liouville problem and the relevant conclusions needed in this paper are given.In Chapter three,by using Schwarz inequality and Krein space,we estimate the bounds of eigenvalues of the indeterminate difference Sturm-Liouville problem under the general separated boundary conditions,including the upper bound of the non-real eigen-values and the lower bound of the real eigenvalues.In Chapter four,we study the existence of non-real eigenvalues for indeterminate difference Sturm-Liouville problems with general separated boundary conditions.First,some properties of the indefinite difference Sturm-Liouville problem are given.Then,it is proved that the number of non-real eigenvalues of the indefinite difference Sturm-Liouville problem is related to the number of negative eigenvalues of the right definite difference Sturm-Liouville problem,the existence of non-real eigenvalues for the indefinite difference Sturm-Liouville problem is proved by using the PT symmetric method.