Spectrum of a Class of Difference Operators with Indefinite Weights

In this study, we use analytical methods and Sylvester inertia theorem to research a class of second order difference operators with indefinite weights and coupled boundary conditions. The eigenvalue problem with sign-changing weight has lasted a long time. The number of eigenvalues and the number of sign changes of the corresponding eigenfunctions of discrete equations under different boundary conditions are mainly studied. For the discrete Sturm-Liouville problems, similar conclusions about the properties of eigenvalues and the number of sign changes of the corresponding eigenfunctions are obtained under different boundary conditions, such as periodic boundary conditions, antiperiodic boundary conditions and separated boundary conditions etc. The purpose of this paper is to extend the similar conclusion to the coupled boundary conditions, which is of great significance to the perfection of the theory of the discrete Sturm-Liouville problems. We came to the following conclusions: first, the eigenvalues of the problem are real and single, the number of the positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. Second, under some conditions, we obtain the sign change of the eigenfunction corresponding to the j-th positive/negative eigenvalue.


Introduction
Spectral theory for Sturm-Liouville boundary value problems has important physical meaning and practical significance. The study of discrete Sturm-Liouville boundary value problems has a very important role in problems when it can be λ is the spectrum parameter, In the introduction part, we also mention some other boundary conditions listed below, The study of the eigenvalue problem with sign-changing weight has lasted a long time. In 1914, Bôcher [1] studied such problems in the continuous case.
Subsequently, many scholars have studied the differential operator problem with indefinite weight function and obtained a series of important results (for exam- However, there are few results on the spectra of discrete second-order linear eigenvalue problems when the weight function ( ) m t changes its sign on T.
In 2007, Ji and Yang [4] [5] studied the structure of the eigenvalues of problem (1) (3) when the weight function ( ) m t changes its sign, and they obtained that the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function.
In 2008, Ji and Yang [6] discussed the eigenvalues of (1) (4) and (1) (5); by using the matrix theory, they got a very interesting result: the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function.
In 2013, Ma, Gao and Lu [7] discussed the spectra of the discrete second-order Neumann Eigenvalue problem (1) (6), By the analytical methods, he not only gives the properties of the eigenvalues, but also gives the number of sign changes of the eigenfunction corresponding to the j-th positive/negative eigenvalue.
In 2015, C. Gao, R. Ma [8] studied the problem (1) (4), they find that the problems have T real eigenvalues (including the multiplicity). Furthermore, the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function.
In 2018, R. Ma, C. Gao, Y. Lu [9] studied the problem (1) (7), by the Sylvester inertia theorem, they obtained the following conclusion: a) (1) (7) has real and simple eigenvalues, which can be ordered as follows It is the purpose of this paper to establish the discrete analogue of the above conclusion.
In this paper, we will study the problems (1) (2)

Main Theorem
Theorem 1. The question (1) (2) has T real and simple eigenvalues. And 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. And the eigenvalues can be sorted as follows: Theorem 2. In this paper, let 2) When 12 0 k < If i is even, the number of sign changes of ( ) If i is even, the number of sign changes of ( ) If i is even, the number of sign changes of ( )

Lemma and Proof of the Theorem
, then problem (1) (2) can be transformed into a matrix equation We can know J and j J ( In fact, for every ( ) As we know, finding the eigenvalues of (1) (2) is equivalent to finding the zeros of ( ) , let j + be the number of the elements in , and j − be the number of the elements in The proof is completed. Lemma 5. For , the proof is similar to Lemma 5 of [9].
From the above five lemmas, we can get Theorem 1.
Proof. This proof is similar to the lemma 7 of [9].
Proof. We only prove case 1. The proof of case 2 is similar.
From (12), we know that the sign of ( ) , , T i u k λ + is equal to the sign of ( ) 1) When 12 0 k >