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# Defect index of the square of a formally self-adjoint second-order difference expression

- Guojing Ren
^{1}Email author

**2014**:48

https://doi.org/10.1186/1687-1847-2014-48

© Ren; licensee Springer. 2014

**Received:**24 August 2013**Accepted:**10 January 2014**Published:**30 January 2014

## Abstract

This paper is concerned with the defect index of the square of a formally self-adjoint second-order difference expression with real coefficients, which, in fact, is a class of formally self-adjoint fourth-order difference expressions. Sufficient and necessary conditions for such fourth-order difference expression to be a limit-2 case, a limit-3 case, and a limit-4 case are given, with respect to the limit case of the second-order difference expression. These results parallel the well-known results of Everitt and Chaudhuri for differential expressions.

**MSC:**39A10, 34B20.

## Keywords

- formally self-adjoint difference equation
- defect index
- limit-point case
- limit-circle case

## 1 Introduction

Δ and ∇ are forward and backward difference operators, respectively, *i.e.* $\mathrm{\Delta}x(t)=x(t+1)-x(t)$ and $\mathrm{\nabla}x(t)=x(t)-x(t-1)$; the discrete time interval ℐ is bounded from below; without loss generality, we denote $\mathcal{I}={[0,+\mathrm{\infty})}_{\mathbb{Z}}$; and the functions $p(t)$, $q(t)$ are all real-valued and $p(t)\ne 0$ for $t\ge -1$.

According to the classical von Neumann theory (*cf.* [1, 2]) and its generalization (*cf.* [3]), a symmetric operator or a non-densely defined Hermitian operator has a self-adjoint extension if and only if its positive and negative defect indices are equal and its self-adjoint extension domains have a close relationship with its defect index. So it is very important to determine the defect indices of both differential and difference expressions in the study of self-adjoint extensions.

was first studied by Weyl [4]. It is well known that the defect index of ℒ is equal to the number of linearly independent square integrable solutions of equation $\mathcal{L}y=\lambda y$ for each $\lambda \in \mathbb{C}\setminus \mathbb{R}$, where ℂ and ℝ denote the sets of the complex and real numbers. Later, some authors studied the defect index of ${\mathcal{L}}^{2}$, which, in fact, is a class of fourth-order formally self-adjoint linear differential expressions with real coefficient, and they have obtained a few excellent results [5–8].

*L*began with Atkinson’s work [9] and the properties of its defect index have been sufficiently discussed. It is well known that the defect index, say ${d}_{L}$ is equal to the number of linearly independent solutions which are square summable of the difference equation

for any $\lambda \in \mathbb{C}\setminus \mathbb{R}$. The value range of defect index ${d}_{L}$ is one or two. The second-order difference expression *L* is called a limit-point case at $t=+\mathrm{\infty}$ if ${d}_{L}=1$, that is, (1.3) has just one solution which is square summable for any point $\lambda \in \mathbb{C}\setminus \mathbb{R}$; and it is a limit-circle case at $t=+\mathrm{\infty}$ if ${d}_{L}=2$, that is, (1.3) has two linearly independent solutions which are square summable for any *λ*, real or complex. All difference expressions of the form (1.2) come within the limit-point, limit-circle classification which depends only on the coefficients *p* and *q* and not on the parameter *λ*. Several criteria of the limit-point and limit-circle cases have been established [10–13].

where ${p}_{j}$ are all real-valued for $0\le j\le 2$, and ${p}_{2}(t)\ne 0$ for all $t\in \mathcal{I}$.

*L*, the defect index, say ${d}_{M}$, of

*M*is equal to the number of linear independent solutions which are square summable of the difference equation

for any $\lambda \in \mathbb{C}\setminus \mathbb{R}$. For such values of *λ*, the difference equation (1.5) may have two, three or four linearly independent solutions in ${l}^{2}$ and for any particular *M* the number of such solutions is independent of *λ*. So all such difference expressions as *M* can be classified into three cases: limit-2 (limit-point), limit-3 and limit-4 (limit-circle) cases. In the limit-circle case all solutions are square summable whether ℑ*λ* is zero or not.

In comparison with the second-order difference expressions *L*, fewer criteria for the limit case of the difference expressions *M* have become known. Recently, it has been shown that all values of the defect index from 2 and 4 can be realized and some criteria for the limit-point case were given in [14].

In this paper, we focus on the fourth-order difference expression ${L}^{2}$ and discuss the relationship of the limit cases between *L* and ${L}^{2}$. It is worth noting that, different from the differential expression, the maximal operator generated by the difference expression *L* or *M* may be multi-valued, and the minimal operator may be non-dense [15, 16]. To solve this problem, we will apply the theory of subspaces to discuss the spectral theory of such a difference expression.

The rest of the paper is organized as follows. In Section 2, some preliminary work is given, including some basic concepts and useful results as regards subspaces, and the known result of the second-order difference expression *L* and the fourth-order difference expression *M*. In Section 3, we pay attention to the defect index of the difference expression ${L}^{2}$. Sufficient and necessary conditions for ${L}^{2}$ to be limit-2 (limit-point), limit-3 and limit-4 (limit-circle) cases are given, separately. These results parallel the Chaudhuri and Everitt’s result for differential expressions [5]. In the special case $p(t)\equiv 1$, which covers a number of examples which arise in practice, we establish a criterion for both *L* and ${L}^{2}$ to be a limit-point case. In the final section, *i.e.*, Section 4, some examples are given to show that all the cases of the difference expressions *L* and ${L}^{2}$ considered in Section 3 can be realized.

## 2 Preliminaries

In this section, we first recall some basic concepts and useful results about subspaces. The readers are referred to [3, 16].

*X*be a complex Hilbert space equipped with inner product $\u3008\cdot ,\cdot \u3009$,

*T*be a linear subspace (briefly, subspace) in ${X}^{2}:=X\times X$, and $\lambda \in \mathbb{C}$. Denote

It can easily be verified that $T(0)=\{0\}$ if and only if *T* can determine a unique linear operator from $D(T)$ into *X* whose graph is just *T*. For convenience, we will identify a linear operator in *X* with a subspace in ${X}^{2}$ via its graph.

**Definition 2.1** [3]

*T*be a subspace in ${X}^{2}$.

- (1)The adjoint of
*T*is defined by${T}^{\ast}:=\{(y,g)\in {X}^{2}:\u3008y,f\u3009=\u3008g,x\u3009\text{for all}(x,f)\in T\}.$ - (2)
*T*is said to be a Hermitian subspace if $T\subset {T}^{\ast}$.*T*is said to be a self-adjoint subspace if $T={T}^{\ast}$. - (3)
Let

*T*be a Hermitian subspace. ${T}_{1}$ is said to be a self-adjoint subspace extension (SSE) of*T*if $T\subset {T}_{1}$ and ${T}_{1}$ is a self-adjoint subspace.

**Definition 2.2** [16]

Let *T* be a subspace in ${X}^{2}$. $R{(T-\lambda I)}^{\mathrm{\perp}}$ is called the defect space of *T* and *λ*, and $dim(R{(T-\lambda I)}^{\mathrm{\perp}})$ is called the defect index of *T* and *λ*.

**Lemma 2.1** *A Hermitian subspace* *T* *in* ${X}^{2}$ *with* $R(T)=X$ *is a self*-*adjoint subspace*.

*Proof* It suffices to show that ${T}^{\ast}\subset T$.

Again by $R(T)=X$, it follows that $h=y$, which implies that $(y,g)\in T$, and consequently, ${T}^{\ast}\subset T$. Hence, *T* is a self-adjoint subspace. □

*T*and

*S*be a subspace in ${X}^{2}$. The product of

*T*and

*S*is defined by

**Lemma 2.2** *If* *T* *is a closed subspace in* ${X}^{2}$. *Then* ${T}^{\ast}T$ *is a self*-*adjoint subspace in* ${X}^{2}$.

*Proof*To show the result, we introduce an operator

*U*in ${X}^{2}$, similarly to that given for a graph of an operator (see [[17], §51]), by putting

It is clear that *U* is a unitary operator and ${U}^{2}=-I$, where *I* is the identity operator in ${X}^{2}$.

*T*is a closed subspace in ${X}^{2}$, so is

*UT*. Therefore, the following formula is true:

*U*and ${U}^{2}T=T$, we have

where *I* is the identity operator in *X*. Thus, the range of ${T}^{\ast}T+I$ coincides with the whole space *X*. Therefore, by Lemma 2.1, ${T}^{\ast}T+I$ is a self-adjoint subspace in ${X}^{2}$ and since ${T}^{\ast}T=({T}^{\ast}T+I)-I$, one sees that ${T}^{\ast}T$ also is self-adjoint. □

**Corollary 2.1** *If* *T* *is a self*-*adjoint subspace in* ${X}^{2}$, *then so is* ${T}^{n}$, *where* $n\ge 2$ *is any integer*.

where $y=z$ in ${l}^{2}$ if and only if $\parallel y-z\parallel =0$, *i.e.*, $y(t)=z(t)$, $t\in \mathcal{I}$, while $\parallel \cdot \parallel $ is the induced norm.

*L*is

*L*were defined in some existing literature (

*e.g.*, [16]) by

respectively, and the minimal subspace was defined by ${H}_{L,0}={\overline{H}}_{L,00}$, where ${\overline{H}}_{L,00}$ is the closure of ${H}_{L,00}$. It has been shown in [16] that ${H}_{L}$ may be multi-valued, and ${H}_{L,00}$ and ${H}_{L,0}$ are only non-densely defined Hermitian operators in ${l}^{2}$.

A sufficient condition for *L* to be a limit-point case has been given.

**Lemma 2.3** [10]

*L*

*is a limit*-

*point case at*$t=+\mathrm{\infty}$

*if*

A complete characterization of self-adjoint extension of ${H}_{L,0}$ has been given in terms of boundary conditions. Here we recall one result which will be used.

**Lemma 2.4** [16]

*Assume that*

*L*

*is a limit*-

*point case at*$t=+\mathrm{\infty}$. ${H}_{L,1}$

*is a SSE of*${H}_{L,0}$

*if and only if there exist real numbers*

*a*

*and*

*b*

*with*${a}^{2}+{b}^{2}\ne 0$

*such that*

*M*. The Green formula for

*M*is

*M*are defined as follows:

and the minimal subspace is defined by ${H}_{M,0}={\overline{H}}_{M,00}$. The following is a characterization of self-adjoint extension of ${H}_{M,0}$ in the case when *M* is a limit-point case at $t=+\mathrm{\infty}$.

**Lemma 2.5** [18]

*Assume that*

*M*

*is a limit*-

*point case at*$t=+\mathrm{\infty}$. ${H}_{M,1}$

*is a SSE of*${H}_{M,0}$

*if and only if there exists a matrix*${G}_{2\times 4}$ ,

*satisfying*$rankG=2$, $GJ{G}^{\ast}=0$,

*and*

*where*

*and* ${I}_{2}$ *is the* $2\times 2$ *identity matrix*.

*M*. In fact, from (1.2), one has

*L*to (2.3), one has

## 3 Main results

In this section, we first establish a sufficient and necessary condition for ${L}^{2}$ to be limit-circle case.

**Theorem 3.1** ${L}^{2}$ *is a limit*-*circle case at* $t=+\mathrm{\infty}$ *if and only if* *L* *is a limit*-*circle case at* $t=+\mathrm{\infty}$.

*Proof*We first consider the sufficiency. Suppose that

*L*is a limit-circle case at $t=+\mathrm{\infty}$. Choose a complex number

*μ*such that the roots of ${\lambda}^{2}=\mu $, say ${\lambda}_{1}$ and ${\lambda}_{2}$, are distinct. For each $j=1,2$, let ${x}_{1}^{j}$, ${x}_{2}^{j}$ be a fundamental set of solution of the difference equation

Since *L* is a limit-circle case at $t=+\mathrm{\infty}$, ${x}_{k}^{j}\in {l}^{2}$, $k=1,2$, $j=1,2$. Hence, ${L}^{2}$ is a limit-circle case at $t=+\mathrm{\infty}$.

Next we consider necessity. If ${L}^{2}$ is a limit-circle case at $t=+\mathrm{\infty}$, we can choose *μ* and ${\lambda}_{j}$ as above and conclude that all solutions of difference equation $L[x](t)={\lambda}_{1}x(t)$ are in ${l}^{2}$. Therefore, *L* is a limit-circle case at $t=+\mathrm{\infty}$. The proof is complete. □

The following is a direct conclusion derived from Theorem 3.1.

**Corollary 3.1** ${L}^{2}$ *is a limit*-*point case or limit*-3 *case at* $t=+\mathrm{\infty}$ *if and only if* *L* *is a limit*-*point case at* $t=+\mathrm{\infty}$.

Theorem 3.1 completely describes the limit classification of ${L}^{2}$ in the limit-circle case. So we will assume, for the rest of this paper, that *L* is a limit-point case at $t=+\mathrm{\infty}$. In addition, we will assume, in the following of this paper, that *μ* is complex with $\mathrm{\Im}\mu \ne 0$, and that ${\lambda}_{j}$, $j=1,2$, are the two distinct complex roots of ${\lambda}^{2}=\mu $ with $\mathrm{\Im}{\lambda}_{j}\ne 0$ for $j=1,2$. Such a choice of *μ* is always possible.

A sufficient and necessary condition for ${L}^{2}$ to be a limit-point case is obtained.

**Theorem 3.2**

*Let*

*L*

*be a limit*-

*point case at*$t=+\mathrm{\infty}$. ${L}^{2}$

*is a limit*-

*point case at*$t=+\mathrm{\infty}$

*if and only if*

*Proof* First, we consider the sufficiency. Suppose that (3.3) holds.

Consider then the two difference equations (3.1) for $j=1,2$. Since *L* is in the limit-point case at $t=+\mathrm{\infty}$ and each $\mathrm{\Im}{\lambda}_{j}\ne 0$, there will be two linearly independent solutions, ${\varphi}_{j}$ and ${\psi}_{j}$, of (3.1) which satisfy ${\varphi}_{j}\in {l}^{2}$ and ${\psi}_{j}\notin {l}^{2}$. In addition, by repeated application of *L* to (3.1), we see that ${\varphi}_{j}$ and ${\psi}_{j}$ are also solutions of (3.2).

Since ${\lambda}_{i}\ne {\lambda}_{j}$ with $i\ne j$, it can easily be shown that these four solutions, ${\{{\varphi}_{j},{\psi}_{j}\}}_{j=1}^{2}$, of (3.2) are linearly independent on ℐ and so forms a basis of solutions for (3.2).

*i.e.*that ${L}^{2}$ is not a limit-point case at $t=+\mathrm{\infty}$. Then, from the Titchmarsh-Weyl theory of difference equations [19] or [20, 21], it follows that (3.2) must have exactly three linearly independent solutions in ${l}^{2}$. Since ${\{{\varphi}_{j},{\psi}_{j}\}}_{j=1}^{2}$ is a basis of solutions for (3.2) and since ${\varphi}_{j}\in {l}^{2}$ for $j=1,2,\dots ,n$, it follows that there must be at least one linearly independent solution, say

*ψ*, of (3.2) which is of the form

such that not all the ${c}_{j}$ are zero and $\psi \in {l}^{2}$.

*ψ*is a solution of (3.2), it follows that ${L}^{2}\psi \in {l}^{2}$. So by the assumption (3.3), $L\psi \in {l}^{2}$. So we have from this and the fact that $\psi \in {l}^{2}$,

is non-singular, one can see from (3.4) that ${c}_{j}{\psi}_{j}\in {l}^{2}$ for each $j=1,2$. Since ${\psi}_{j}\notin {l}^{2}$ for all such *j*, it then follows that ${c}_{1}={c}_{2}=0$ and this is a contradiction with the assumption. Hence ${L}^{2}$ is a limit-point case at $t=+\mathrm{\infty}$.

Next we consider the necessity. Suppose that ${L}^{2}$ is a limit-point case at $t=+\mathrm{\infty}$. Since *L* is a limit-point case at $t=+\mathrm{\infty}$, by Lemma 2.4, ${H}_{L,1}$, defined by (2.1), is a self-adjoint subspace extension of ${H}_{L,0}$.

one can verify that $rankG=2$, $GJ{G}^{\ast}=0$, and $Gu(0)=0$. So by Lemma 2.5, ${H}_{{L}^{2},1}$ is a self-adjoint subspace derived from ${L}^{2}$.

Thus, ${H}_{L,1}^{2}$ and ${H}_{{L}^{2},1}$ are both self-adjoint subspaces derived from ${L}^{2}$ with ${H}_{L,1}^{2}\subset {H}_{{L}^{2},1}$. Since no self-adjoint subspace can have a strict self-adjoint restriction, one must have ${H}_{L,1}^{2}={H}_{{L}^{2},1}$. If now we compare $D({H}_{L,1}^{2})$ and $D({H}_{{L}^{2},1})$, we see that $x\in D({H}_{{L}^{2},1})$ implies that $Lx\in {l}^{2}$.

For any $x\in {l}^{2}$ satisfying ${L}^{2}x\in {l}^{2}$, we can redefine (if necessary) the values of *x* on ${[-2,2]}_{\mathbb{Z}}$, which are finite points, so that the boundary conditions in (3.5) are satisfied. Let the new function be $\tilde{x}$. Then $\tilde{x}\in D({H}_{{L}^{2},1})$ and consequently, $L\tilde{x}\in {l}^{2}$. Since *Lx* and $L\tilde{x}$ are different at finite points, it follows that $Lx\in {l}^{2}$. Hence, condition (3.3) holds.

The whole proof is complete. □

By the proof of Theorem 3.2, one can find some relationship of solutions of equations (3.1) and (3.2). Next, by exploiting this relationship, we prove a result which gives a necessary and sufficient condition for ${L}^{2}$ to be a limit-3 case at $t=+\mathrm{\infty}$.

**Theorem 3.3**

*Assume that*

*L*

*is a limit*-

*point case at*$t=+\mathrm{\infty}$.

*Let*${\psi}_{j}\notin {l}^{2}$

*be a solution of the equation*$Lx={\lambda}_{j}x$

*for*$j=1,2$.

*Then*${L}^{2}$

*is a limit*-3

*case at*$t=+\mathrm{\infty}$

*if and only if there exists a unique constant*$k\ne 0$,

*such that*

*Proof* Here it is worth noting that since ${\psi}_{1}\notin {l}^{2}$ and ${\psi}_{2}\notin {l}^{2}$ it follows that if (3.6) holds, then *k* is unique and not zero.

It follows that $L\psi \notin {l}^{2}$ since *k* is unique and not zero, while ${L}^{2}\psi \in {l}^{2}$. Now if ${L}^{2}$ was in the limit-point case, then it would follow from Theorem 3.2 that $L\psi \in {l}^{2}$, which is a contradiction. In addition, it follows from Theorem 3.1 that ${L}^{2}$ is not a limit-circle case. So it must be a limit-3 case at $t=+\mathrm{\infty}$.

*ψ*, of (3.2), which is a linear combination of ${\psi}_{1}$ and ${\psi}_{2}$,

*i.e.*,

It is evident that ${c}_{1}\ne 0$ since ${\psi}_{2}\notin {l}^{2}$. Define $\tilde{\psi}={c}_{1}^{-1}\psi $. Then $\tilde{\psi}$ is of form (3.6).

The whole proof is complete. □

**Theorem 3.4** *Assume that the equation* ${L}^{2}x=0$ *on* ℐ *has exactly three linearly independent solutions in* ${l}^{2}$. *Then* ${L}^{2}$ *is a limit*-3 *case at* $t=+\mathrm{\infty}$ *and* *L* *is a limit*-*point case at* $t=+\mathrm{\infty}$.

*Proof* Since the equation ${L}^{2}[x](t)=0$ has exactly three linearly independent solutions in ${l}^{2}$, it follows that ${d}_{{L}^{2}}\ge 3$ by [[15], Corollary 5.2]. In addition, one has ${d}_{{L}^{2}}\ne 4$, otherwise, ${L}^{2}[x](t)=0$ would have four linearly independent solutions in ${l}^{2}$. Hence, ${d}_{{L}^{2}}=3$ and ${L}^{2}$ is a limit-3 case at $t=+\mathrm{\infty}$, and consequently *L* is a limit-point case at $t=+\mathrm{\infty}$ by Corollary 3.1. □

In the special case when $p(t)\equiv 1$, we have the following result.

**Theorem 3.5** *Assume that* $p(t)\equiv 1$ *for* $t\ge -1$. *If* $|q(t)+q(t-1)|$ *is bounded on* ℐ, *then both* *L* *and* ${L}^{2}$ *are a limit*-*point case*.

*Proof*By (2.4) and (2.5), ${L}^{2}$ takes the form

Firstly, *L* is a limit-point case by Lemma 2.4, since $p(t)\equiv 1$ for $t\ge -1$. Secondly, $|q(t)+q(t-1)|$ is bounded on ℐ implies that condition (3.1) in [14] is satisfied (with respect to $n=2$ and $\sigma (t)=1$). Hence, ${L}^{2}$ is a limit-point case by [[14], Theorem 3.3]. □

## 4 Examples

*L*and ${L}^{2}$ can be realized.

- (1)Both
*L*and ${L}^{2}$ are a limit-point case at $t=+\mathrm{\infty}$ if$p(t)=1,\phantom{\rule{2em}{0ex}}q(t)=0,\phantom{\rule{1em}{0ex}}t\ge -1$

*L*is a limit-point case by Lemma 2.3 and ${L}^{2}$ is a limit-point case by Theorem 3.5.

- (2)Both
*L*and ${L}^{2}$ are a limit-circle case at $t=+\mathrm{\infty}$ if$p(t)=q(t)={4}^{t-1},\phantom{\rule{1em}{0ex}}t\ge 0.$

*L*is a limit-circle case by [[10], Example 3.2], and consequently ${L}^{2}$ is a limit-circle case by Theorem 3.1.

- (3)
*L*is a limit-point case and ${L}^{2}$ is a limit-3 case at $t=+\mathrm{\infty}$ if$p(t)={2}^{t},\phantom{\rule{2em}{0ex}}q(t)={2}^{t},\phantom{\rule{1em}{0ex}}t\ge -1.$

*L*is a limit-point case. The solutions of the difference equation $Lx=0$ are of the form ${a}^{t}$ with

*a*satisfying the equation $2{a}^{2}-4a+1=0$. The two roots of this equation are

and then ${x}_{i}(t)={a}_{i}^{t}$ are two solutions of $L[x]=0$. It is evident that ${x}_{1}\notin {l}^{2}$ and ${x}_{2}\in {l}^{2}$. Thus *L* is not a limit-circle case and so it must be a limit-point at $t=+\mathrm{\infty}$.

*a*satisfying

If ${x}_{1}$ and ${x}_{2}$ are defined as before and ${x}_{j}(t)={a}_{j}^{t}$ for $j=3,4$, then we can see that ${x}_{j}\in {l}^{2}$ for $j=2,3,4$, but ${x}_{1}$ is not in ${l}^{2}$. Thus equation ${L}^{2}[x]=0$ has just three linearly independent solutions in ${l}^{2}$. This implies that ${L}^{2}$ is not a limit-point case at $t=+\mathrm{\infty}$.

Since ${a}_{3}^{2}-4{a}_{3}+1\ne 0$ and $|2{a}_{3}|>1$, it follows that $L({x}_{3})\notin {l}^{2}$. This shows that ${L}^{2}[{x}_{3}]\in {l}^{2}$ cannot imply $L[{x}_{3}]\notin {l}^{2}$. So by Theorem 3.2, ${L}^{2}$ is not a limit-point case. Hence, ${L}^{2}$ is a limit-3 case at $t=+\mathrm{\infty}$.

## Author’s contributions

The author worked on the results independently.

## Declarations

### Acknowledgements

This research was supported by the NNSF of China (Grants 11226160, 11301304, 11071143, 11101241).

## Authors’ Affiliations

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