On the Neumann eigenvalues for second-order Sturm–Liouville difference equations

The paper is concerned with the Neumann eigenvalues for second-order Sturm–Liouville difference equations. By analyzing the new discriminant function, we show the interlacing properties between the periodic, antiperiodic, and Neumann eigenvalues. Moreover, when the potential sequence is symmetric and symmetric monotonic, we show the order relation between the first Dirichlet eigenvalue and the second Neumann eigenvalue, and prove that the minimum of the first Neumann eigenvalue gap is attained at the constant potential sequence.

In this paper, we study the second-order difference equations (1). So far, there have been results on the second-order difference equations (1) which are analogue to the continuous Sturm-Liouville equation (2). Using the information on more than one set of eigenvalues, the potential sequence can be determined uniquely, for example, two sets of eigenvalues [12,13], one set of eigenvalues plus a symmetric potential sequence [10], and one set of eigenvalues plus partial information of the potential sequence [25].
In 1990, Ashbaugh and Benguria [2] studied the comparison of the eigenvalues of two discrete Sturm-Liouville equations whose potential sequences satisfy certain relation. We say the sequence {x k } N-1 k=0 is symmetric if x k = x N-1-k for k = 0, 1, 2, . . . , N -1, and the se- . In particular, the sequence {x k } N-1 k=0 is said to be symmetric increasing if it is symmetric and quasisymmetric increasing. Ashbaugh and Benguria showed that if {q k } N-1 k=0 is symmetric increasing in (1), then the eigenvalue {μ k } N-1 k=1 satisfies Equality holds if and only if q k = q 0 for k = 0, 1, 2, . . . , N -1. Note that if q k = 0 for k = 0, 1, 2, . . . , N -1, then Furthermore, the system of (1) with other self-adjoint boundary conditions has also been investigated [3,18]. Jirari in 1995 showed that problem (1) with the boundary conditions where α, β ∈ R has N real and simple eigenvalues. And recently, Ji and Yang [17] studied the eigenvalue comparison of (1) and (4). In particular, they also showed that if q i = 0 for i = 0, 1, 2, . . . , N -1, then the first eigenvalue is simple and associated with the vector whose entries are of all ones.
In 2005, Wang and Shi [26] (see also [24]) were concerned with the eigenvalues for (1) coupled with the periodic boundary conditions the antiperiodic boundary conditions and the Dirichlet boundary conditions Define the discriminant of (1) by where ϕ n and ψ n are solutions of (1) satisfying the initial conditions By the similar argument as the differential equations (see [3,22]), Wang and Shi showed that the periodic problem (1), (5) and the antiperiodic problem (1), (6) have exactly N real eigenvalues, while the Dirichlet problem (1), (7) has exactly N -1 real eigenvalues. Furthermore, they denoted by {λ k } N-1 k=0 , {λ k } N k=1 , and {μ k } N-1 k=1 the periodic, antiperiodic, and Dirichlet eigenvalues, respectively, and arranged them in the nondecreasing order λ 0 ≤ λ 1 ≤ · · · ≤ λ N-1 ,λ 1 ≤λ 2 ≤ · · · ≤λ N , and μ 1 < μ 1 < · · · < μ N-1 . They showed that a set of these three eigenvalues satisfy the following interlacing properties: if N is odd, and if N is even, In this paper, we consider the second-order difference equations (1) coupled with the Neumann boundary conditions Denote by {ν k } N-1 k=0 the Neumann eigenvalues of the second-order difference equations (1). It is known that if q k = 0 for k = 0, 1, 2, . . . , N -1, then Combined with the result of [26], we will show the interlacing properties of the eigenvalues, which is a discrete analogue result for the continuous Sturm-Liouville problem (see [9,22]). We shall remark that, in the continuous case, we analyze Hill's discriminant H(λ) to obtain the interlacing property (3). But in the discrete case, we need to define another discriminant where ϕ n and ψ n are solutions of (1) satisfying the initial conditions to show the interlacing property for the Neumann eigenvalues in Theorem 1. By analyzing this new discriminant f (λ), we can prove Theorem 1. (1). The eigenvalues satisfy the following interlacing inequality: if N is odd,

Theorem 1 Consider the second-order difference equations
and if N is even, After obtaining Theorem 1, we will consider the order relation of the first Dirichlet eigenvalue μ 1 and the second Neumann eigenvalue ν 1 , and the first Neumann eigenvalue gap ν 1ν 0 . Theorems 2 and 3 can be regarded as discrete analogue results of [6] and [7] respectively for the continuous Sturm-Liouville problem (2). Theorem 2 Consider the second-order difference equation (1). If q k is symmetric and symmetric decreasing and satisfies max k∈[0,N-1] q k ≤ ν 1 , then μ 1 ≤ ν 1 and the equality holds if and only if q k = q 0 for all k ∈ [0, N -1].
The paper is organized as follows. Section 2 gives lemmas about the Wronskian and a variation of constant formula which is used in Sect. 3. In Sect. 3, we study the interlacing properties for the periodic, antiperiodic, and Neumann eigenvalues, and use an argument similar to that in [9,22] to prove Theorem 1. Finally, the proof of Theorem 2 is given in Sect. 4, while the proof of Theorem 3 is given in Sect. 5.

Preliminaries
In this section, we derive some discrete analogous lemmas of the continuous case. One can refer to [4]. Lemmas 1 and 2 have been shown in [18] (see also [26]) by using a similar argument as the continuous case, so we omit the proofs here.
Let λ = μ in Lemma 1, we have the following Wronskian-type identity.

Lemma 2 ([18, Theorem 2.2.8]) Let y and z be solutions of (1). Then the Wronskian
Now, let ϕ n and ψ n be two solutions of (1) satisfying the initial conditions Note that ϕ -1 = ϕ 0ϕ -1 = 0 and ψ -1 = ψ 0ψ -1 = 1. In particular, we find that, by Theorem 2, and it is known that ϕ n , ψ n are two linear independent solutions of (1). The following theorem is similar to [26, Theorem 2.3], but the initial conditions are different.

Theorem 4
For any {f n } N-1 n=0 ⊆ R and for any c 0 , c 1 ∈ R, the initial value problem has a unique solution z, which can be expressed as Proof The technique of the proof is based on the variation of parameters on the differential equation. Let be a solution of (11). Then z n = A n ϕ n + ϕ n+1 A n + B n ψ n + ψ n+1 B n , n ∈ [-1, N -1]. Setting we have z n = A n ϕ n + B n ψ n , n ∈ [-1, N -1].
Since ϕ n and ψ n are two solutions of (1), we find and hence Combined with (11), we find Now, solving system (13) and (14) for ( A n , B n ), we find By (10), we find ϕ n+1 ψ nψ n+1 ϕ n = 1. Hence and then, by defining -1 j=0 • = 0, we have This implies that By (9) and (12), we find Finally, for n = N , we evaluate This proof is complete.

Interlacing properties of eigenvalues
Recall ϕ n and ψ n defined in Sect. 2. It is clear that λ is an eigenvalue of (1) and (5) if and only if c 1 ϕ n (λ) + c 2 ψ n (λ) satisfies the periodic boundary conditions, i.e., By (9), we find The above system has a nontrivial solution (c 1 , c 2 ) if and only if which implies that, combined with (10), Hence, we find that f (λ) = 2 if and only if λ is a periodic eigenvalue. Similarly, it can be showed that f (λ) = -2 if and only if λ is an antiperiodic eigenvalue. In particular, we have the following lemma. (1) and (5), while f (λ) = 0 whenever f (λ) = -2 if and only if λ is a multiple eigenvalue of (1) and (6).
By the above discussion and combining the result of [26, Theorem 3.1], we find that if N is odd, and if N is even,

The order relation of the first Dirichlet eigenvalue and the second Neumann eigenvalue
In this section, we investigate the order relation between the first Dirichlet eigenvalue and the second Neumann eigenvalue. Denote by (ν k , w k ) N-1 k=0 the Neumann eigenpairs of (1) and (8) with w k 2 = 1. In particular, w k can be chosen to have exactly k sign changes. We have the following lemma.

The lower bound of the first Neumann eigenvalue gap
In this section, we give an optimal lower bound of the first Neumann eigenvalue gap. (1) and (8), and let q n (t) be a one-parameter family of potential sequences such that q n (t) exists. Then ν k (t) = N-1 j=0 q j (t)w 2 k,j (t).