Non-Real Eigenvalues Of Indefinite Discrete Sturm-Liouville Boundary Value Problems

Sturm-Liouville boundary value problems originated in the mid-19th century. This continuous mathematical model was established to describe the thermal conductivity of the solid. Sturm-Liouville boundary value problems have profound physical background, and a large part of them come from the heat conduction problems, string vibration problems, Poisson boundary value problems and other partial differential boundary value problems. Spectral theory for Sturm-Liouville boundary value problems has important physical meaning and practical significance, for example, it is well represented the spectrum of the vibration and the spectrum of the Schrodinger equation. In real life, the physical model we created when simulating certain physical phenomena may be discrete, and some continuous problems were easier to solve by discretization, which prompted the study of discrete problems. The study of discrete Sturm-Liouville boundary value problems has a very important role to problems which can be described by discrete and continuous mathematical models.Sturm-Liouville boundary value problem is in self-adjoint form if and only if the potential function is a real function. When the coefficients are real values, Sturm-Liouville boundary value problems had in-depth study in the essays [1-5] and the relevant essays. Similar to a regular self-adjoint boundary value problem, when the Sturm-Liouville equation is not in self-adjoint form, the use of spectral theory of compact operators in Hilbert space shows that Sturm-Liouville boundary value problems have only countable eigenvalues and have no limited accumulation points. When the imaginary part of the potential function is not zero, Sturm-Liouville boundary value problems may have uncountable non-real eigenvalues (see [6, Theorem 1.1]). Progressive characteristics of Sturm-Liouville boundary value problems were also studies (e.g. [7-9]). Sufficient conditions for singular eigenvalues of Sturm-Liouville equation with period, anti-period, Dirichlet and Neumann boundary conditions were given especially in the Essay [10]. Essays [11-14] and the relevant essays also gave some results of non-self-adjoint differential operators. When Sturm-Liouville equation is in self-adjoint form, using comparison theorem, the essay [4] gave the bounds of the eigenvalues of Sturm-Liouville boundary value problems relying on the coefficients of the equation, the coefficients of the comparative equation and the eigenvalues of comparative equation. In the essay [3], the authors used the Rayleigh-Ritz method to give the bounds of eigenvalues of Sturm-Liouville boundary value problems when potential function was greater than zero and weight function was equal to one.For discrete Sturm-Liouville boundary value problems, F. V. Atkinson began the study the spectral problem of self-adjoint difference systems in the essay [15] in 1964. The essay [16] studied oscillation and non-oscillation of second order self-adjoint difference equations. Essays [17] and [18] also examined the oscillation of second order linear difference equations. The essay [19] studied the Green’s functions and conjugation of second order linear difference equations. The essay [20] studied the Prufer transform in discrete form. Spectral problems of discrete Sturm-Liouville boundary value problems were also studied such as in essays [21-22]. The essay [23] studied regular second order vector difference boundary value problems, and got the number of eigenvalues (finite number), orthogonality of eigenvectors, the max-minimum principle and so on. Subsequently, the essay [24] further promoted and deepened the spectral theory of linear self-adjoint difference systems, allowing the spectral theory of linear self-adjoint difference systems tends to improve.We note the following problems. First, most researches on discrete Sturm-Liouville boundary value problems are self-adjoint. In this case, the discrete Sturm-Liouville boundary value problems have no non-real eigenvalues. When weight function changes signs, there may exist non-real eigenvalues. To study the non-real eigenvalues of continuous Sturm-Liouville boundary value problems, "eigencurve" method was used(e.g. [25]). But there are no sufficient conditions for the existence of non-real eigenvalues of discrete Sturm-Liouville boundary value problems. Second, there are many essays on estimating the bounds of eigenvalues of continuous Sturm-Liouville boundary value problems (e.g.[3], [4], [26], [27]), while for discrete case, there are no corresponding conclusions about the bounds of eigenvalues. Third, results of estimating the bounds of eigenvalues of continuous Sturm-Liouville boundary value problems are under the condition of real coefficients (e.g.[26], [27]), but when the coefficients are complex, there are no similar results.This paper studies non-real eigenvalues of indefinite discrete Sturm-Liouville boundary value problems and complex Sturm-Liouville boundary value problems. In this paper, some spectral properties of right-defined discrete Sturm-Liouville boundary value problems are given; smoothness of eigencurves and the intersection of a straight line and eigencurves and the critical points of the eigencurves of two-parameter discrete Sturm-Liouville boundary value problems are given; using "eigencurve" method, sufficient conditions for the existence of non-real eigenvalues of indefinite discrete Sturm-Liouville boundary value problems are given; the bounds of real and imaginary parts of non-real eigenvalues of indefinite discrete Sturm-Liouville boundary value problems are given with respect to the equation coefficients; the lower bounds of the real part of all eigenvalues of continuous Sturm-Liouville boundary value problems with respect to the equation coefficients and the bounds of imaginary part of every eigenvalue with respect to the equation coefficients and the real part of the eigenvalue are given.The present paper is organized as follows. The first chapter gives the advanced knowledge, mainly including the spectral theory of second-order vector difference equations; in the second chapter, some properties of corresponding right-defined problems of indefinite discrete Sturm-Liouville boundary value problems and some properties of eigencurves were given and a sufficient condition for such problems to admit non-real eigenvalues was found using the "eigencurve" method; in the third chapter, relaying on the coefficients of the equation, a prior bounds on the real parts and imaginary parts of possible non-real eigenvalues of indefinite discrete Sturm-Liouville boundary value problems was given; in the fourth chapter, the bounds of complex Sturm-Liouville boundary value problems was obtained including the lower bounds of the real parts of all eigenvalues relaying on the coefficients of the equation and the bound of the imaginary part of every non-real eigenvalue relaying on the coefficients of the equation and the real part of the eigenvalue; the fifth chapter is summary and prospect.