Defect index of the square of a formally self-adjoint second-order difference expression
© Ren; licensee Springer. 2014
Received: 24 August 2013
Accepted: 10 January 2014
Published: 30 January 2014
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.
Δ and ∇ are forward and backward difference operators, respectively, i.e. and ; the discrete time interval ℐ is bounded from below; without loss generality, we denote ; and the functions , are all real-valued and for .
According to the classical von Neumann theory (cf. [1, 2]) and its generalization (cf. ), 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 . It is well known that the defect index of ℒ is equal to the number of linearly independent square integrable solutions of equation for each , where ℂ and ℝ denote the sets of the complex and real numbers. Later, some authors studied the defect index of , 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].
for any . The value range of defect index is one or two. The second-order difference expression L is called a limit-point case at if , that is, (1.3) has just one solution which is square summable for any point ; and it is a limit-circle case at if , 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 are all real-valued for , and for all .
for any . For such values of λ, the difference equation (1.5) may have two, three or four linearly independent solutions in 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 ℑλ 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 .
In this paper, we focus on the fourth-order difference expression and discuss the relationship of the limit cases between L and . 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 . Sufficient and necessary conditions for 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 . In the special case , which covers a number of examples which arise in practice, we establish a criterion for both L and 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 considered in Section 3 can be realized.
It can easily be verified that if and only if T can determine a unique linear operator from into X whose graph is just T. For convenience, we will identify a linear operator in X with a subspace in via its graph.
Definition 2.1 
- (1)The adjoint of T is defined by
T is said to be a Hermitian subspace if . T is said to be a self-adjoint subspace if .
Let T be a Hermitian subspace. is said to be a self-adjoint subspace extension (SSE) of T if and is a self-adjoint subspace.
Definition 2.2 
Let T be a subspace in . is called the defect space of T and λ, and is called the defect index of T and λ.
Lemma 2.1 A Hermitian subspace T in with is a self-adjoint subspace.
Proof It suffices to show that .
Again by , it follows that , which implies that , and consequently, . Hence, T is a self-adjoint subspace. □
Lemma 2.2 If T is a closed subspace in . Then is a self-adjoint subspace in .
It is clear that U is a unitary operator and , where I is the identity operator in .
where I is the identity operator in X. Thus, the range of coincides with the whole space X. Therefore, by Lemma 2.1, is a self-adjoint subspace in and since , one sees that also is self-adjoint. □
Corollary 2.1 If T is a self-adjoint subspace in , then so is , where is any integer.
where in if and only if , i.e., , , while is the induced norm.
respectively, and the minimal subspace was defined by , where is the closure of . It has been shown in  that may be multi-valued, and and are only non-densely defined Hermitian operators in .
A sufficient condition for L to be a limit-point case has been given.
Lemma 2.3 
A complete characterization of self-adjoint extension of has been given in terms of boundary conditions. Here we recall one result which will be used.
Lemma 2.4 
and the minimal subspace is defined by . The following is a characterization of self-adjoint extension of in the case when M is a limit-point case at .
Lemma 2.5 
and is the identity matrix.
3 Main results
In this section, we first establish a sufficient and necessary condition for to be limit-circle case.
Theorem 3.1 is a limit-circle case at if and only if L is a limit-circle case at .
Since L is a limit-circle case at , , , . Hence, is a limit-circle case at .
Next we consider necessity. If is a limit-circle case at , we can choose μ and as above and conclude that all solutions of difference equation are in . Therefore, L is a limit-circle case at . The proof is complete. □
The following is a direct conclusion derived from Theorem 3.1.
Corollary 3.1 is a limit-point case or limit-3 case at if and only if L is a limit-point case at .
Theorem 3.1 completely describes the limit classification of in the limit-circle case. So we will assume, for the rest of this paper, that L is a limit-point case at . In addition, we will assume, in the following of this paper, that μ is complex with , and that , , are the two distinct complex roots of with for . Such a choice of μ is always possible.
A sufficient and necessary condition for to be a limit-point case is obtained.
Proof First, we consider the sufficiency. Suppose that (3.3) holds.
Consider then the two difference equations (3.1) for . Since L is in the limit-point case at and each , there will be two linearly independent solutions, and , of (3.1) which satisfy and . In addition, by repeated application of L to (3.1), we see that and are also solutions of (3.2).
Since with , it can easily be shown that these four solutions, , of (3.2) are linearly independent on ℐ and so forms a basis of solutions for (3.2).
such that not all the are zero and .
is non-singular, one can see from (3.4) that for each . Since for all such j, it then follows that and this is a contradiction with the assumption. Hence is a limit-point case at .
Next we consider the necessity. Suppose that is a limit-point case at . Since L is a limit-point case at , by Lemma 2.4, , defined by (2.1), is a self-adjoint subspace extension of .
one can verify that , , and . So by Lemma 2.5, is a self-adjoint subspace derived from .
Thus, and are both self-adjoint subspaces derived from with . Since no self-adjoint subspace can have a strict self-adjoint restriction, one must have . If now we compare and , we see that implies that .
For any satisfying , we can redefine (if necessary) the values of x on , which are finite points, so that the boundary conditions in (3.5) are satisfied. Let the new function be . Then and consequently, . Since Lx and are different at finite points, it follows that . 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 to be a limit-3 case at .
Proof Here it is worth noting that since and it follows that if (3.6) holds, then k is unique and not zero.
It follows that since k is unique and not zero, while . Now if was in the limit-point case, then it would follow from Theorem 3.2 that , which is a contradiction. In addition, it follows from Theorem 3.1 that is not a limit-circle case. So it must be a limit-3 case at .
It is evident that since . Define . Then is of form (3.6).
The whole proof is complete. □
Theorem 3.4 Assume that the equation on ℐ has exactly three linearly independent solutions in . Then is a limit-3 case at and L is a limit-point case at .
Proof Since the equation has exactly three linearly independent solutions in , it follows that by [, Corollary 5.2]. In addition, one has , otherwise, would have four linearly independent solutions in . Hence, and is a limit-3 case at , and consequently L is a limit-point case at by Corollary 3.1. □
In the special case when , we have the following result.
Theorem 3.5 Assume that for . If is bounded on ℐ, then both L and are a limit-point case.
Firstly, L is a limit-point case by Lemma 2.4, since for . Secondly, is bounded on ℐ implies that condition (3.1) in  is satisfied (with respect to and ). Hence, is a limit-point case by [, Theorem 3.3]. □
- (1)Both L and are a limit-point case at if
- (2)Both L and are a limit-circle case at if
- (3)L is a limit-point case and is a limit-3 case at if
and then are two solutions of . It is evident that and . Thus L is not a limit-circle case and so it must be a limit-point at .
If and are defined as before and for , then we can see that for , but is not in . Thus equation has just three linearly independent solutions in . This implies that is not a limit-point case at .
Since and , it follows that . This shows that cannot imply . So by Theorem 3.2, is not a limit-point case. Hence, is a limit-3 case at .
The author worked on the results independently.
This research was supported by the NNSF of China (Grants 11226160, 11301304, 11071143, 11101241).
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