Abstract
This article is concerned with the defect indices of singular symmetric linear difference equations of order 2n with complex coefficients and one singular endpoint. We first show that the positive and negative defect indices d+ and d- of a class of singular symmetric linear difference equations of order 2n with complex coefficients satisfy the inequalities n ≤ d+ = d- ≤ 2n and all values of this range are realized. This extends the result for difference equations with real coefficients. In addition, some sufficient conditions for the limit point and the strong limit point cases are given.
AMS Classification: 39A70; 34B20.
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1 Introduction
In this article, we are interested in the positive and negative defect indices of the following singular symmetric linear difference equation with complex coefficients:
where Δ and ∇ are forward and backward difference operators, respectively, i.e., Δy(t) = y(t + 1) - y(t) and ∇y(t) = y(t) - y(t - 1); ; w(t) > 0, p j (t), and q k (t) are real-valued for 0 ≤ j ≤ n, 1 ≤ k ≤ n; for ; and λ is a complex spectral parameter.
By letting u(t) = (u1(t), u2(t), ..., u2n(t))Twith
for 1 ≤ j ≤ n, Equation (1.1) can be converted into the following singular linear discrete Hamiltonian system
where
and
In-1is the (n - 1) × (n - 1) unit matrix, and the right partial shift operator
According to the classical von Neumann theory (cf. [1, 2]) and its generalization [3], a symmetric operator or a Hermitian subspace 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 indices. So it is very important to determine the defect indices of both differential equations and difference equations in the study of self-adjoint extensions.
Consider singular symmetric linear differential equation with complex coefficients:
where p j and q k are all real functions, and w(t) > 0 defined on (a, b). The defect indices of Equation (1.6) has been studied for a long time. It is well known that the positive and negative defect indices d± of (1.6) are equal to the number of linearly independent square integrable solutions of (1.6) with Imλ > 0 and Imλ < 0, respectively. In the special case that the coefficients of (1.6) is real; that is all q k (t) ≡ 0, it is evident that d+ = d- := d. Glazman [4] showed that the defect index d of Equation (1.6) with real coefficients defined on (0, +∞), where x = 0 is a regular endpoint, satisfies the inequalities n ≤ d ≤ 2n and all values of d in this range are realized. Many other results on the defect index d of Equation (1.6) with real coefficients were summarized in [5]. Mcleod gave an example of a fourth-order symmetric ordinary differential equation, whose positive and negative defect indices are different [6]. Kogan and Rofe-Beketov discussed the positive and negative defect indices of Equation (1.6), and showed that the positive and negative defect indices may differ an integer [7, 8].
For the discrete case, Atkinson first studied the number of linearly independent square summable solutions of second-order symmetric linear difference equations with real coefficients [9]. Subsequently, his study was further developed (cf. [10–13]). Sun studied the number of linearly independent square summable solutions of second-order symmetric difference equations with complex coefficients [14]. It has been shown that the positive and negative defect indices of second-order symmetric difference equations with complex coefficients are still equal; that is (d+, d-) = (1, 1) or (2, 2). The positive and negative defect indices of singular symmetric linear difference equation with real coefficients:
has been discussed in [15]. Since the coefficients of (1.7) are all real, it can be easily verified that d+ = d- := d. It has been shown in [15] that n ≤ d ≤ 2n, and all values in this range can be realized. There are seldom results on the positive and negative defect indices of symmetric difference equation with complex coefficients.
In the present article, we study the positive and negative defect indices of Equation (1.1). The rest of the article is organized as follows. In Section 2, two equivalent forms of Equation (1.1) are formulated, and some useful lemmas are stated. In Section 3, we pay attention to the defect indices of Equation (1.1) and its equivalent forms. Theorem 3.1 and Example 3.1 show that the positive and negative defect indices of a class of symmetric difference equations of order 2n with complex coefficients satisfies the inequalities n ≤ d+ = d- ≤ 2n and all values in this range are realized. This extends the corresponding result for symmetric difference equations with real coefficients in [15]. We point out the method used here is different from that in [15]. In addition, several criteria of the limit point and strong limit point cases are established. Most of the results in the present article extend the corresponding results for real coefficient equations in [15].
2 Preliminaries
This section is divided into three sections. In Section 2.1, a classification of limit cases of Equation (1.1) is introduced. In Section 2.2, two equivalent forms of Equation (1.1) are introduced. In Section 2.3, some sufficient and necessary conditions of limit point case and strong limit point case of Equation (1.1) are given. Some of the results in this section can be regarded as extensions of those in [15].
2.1 Classification of limit cases
By ℂ denotes the set of the complex numbers, and by and u* denote the conjugate of z and the complex conjugate transpose of u, respectively.
We now introduce the following space:
with inner product
where the weight function w(t) > 0 on . For , x is said to be equal to y if ∥x - y∥ w = 0, where ∥ · ∥ w := (〈·, ·〉 w )1/2. In this sense, is a Hilbert space with the inner product 〈·, ·〉 w . In the special case of is briefly denoted by .
Similarly to the scalar case, denote
with the inner product
where the weight function W(t) is a 2n × 2n non-negative Hermitian matrix, R(·) is defined by (1.5). Then is a Hilbert space with the inner product 〈·, ·〉 w in the sense that u = v if ∥u - v∥ W = 0, where ∥ · ∥ W := (〈·, ·〉 W )1/2.
As stated in the previous section, (1.1) can be converted into (1.3) by (1.2). It is evident that for any λ ∈ ℂ, if y(t) is a solution of (1.1), then u(t) defined by (1.2) is a solution of (1.3) and conversely, if u(t) is a solution of (1.3), then y(t) = u1(t + 1) is a solution of (1.1). Moreover, it follows from (1.2) and (1.4) that
By d w (λ) and D W (λ) denote the number of the linearly independent solutions of Equation (1.1) in and system (1.3) in , respectively. Then the following result is obtained:
Lemma 2.1. D W (λ) = d w (λ) for any λ ∈ ℂ.
Denote d+ = d w (i) and d- = d w (-i). By Theorem 5.1 of [16], d+ and d- are equal to the positive and negative defect indices of the minimal operator generated by (1.1), respectively. The following result is directly derived from Corollary 4.1 and Theorem 5.4 in [16] and Lemma 2.1:
Lemma 2.2. n ≤ d+, d- ≤ 2n.
Definition 2.1. Equation (1.1) is called in the (d+,d-) case at t = +∞. In the case that d± = n, Equation (1.1) is called in the limit point case at t = +∞; in the case that d± = 2n, Equation (1.1) is called in the limit circle case at t = +∞.
Lemma 2.3 [14]. For second-order symmetric difference equation:
where c(t) is a complex-valued function and b(t) is a real-valued function, and w(t) > 0 on , the positive and negative defect indices (d+, d-) are equal to either (1, 1) or (2, 2).
2.2 Two equivalent forms of Equation (1.1)
In this section, we formulate two equivalent forms of (1.1).
For convenience, we first introduce the following useful formulae:
where is the binomial coefficient.
First, by using (2.2) and (2.3), Equation (1.1) can be rewritten as
where
or
Equation (2.4) is a equivalent form of Equation (1.1). In addition, it is evident that Equation (2.1) is a special case of Equation (2.4) with n = 1.
Second, setting
and multiplying w-1/2(t) on the both side of (2.4) we get
where
Further, (2.8) can be rewritten as
where and are determined by (2.6) with P j (t), p s (t), Q j (t), and q s (t) replaced by , and , respectively. It is evident that , 0 ≤ j ≤ n, and , 1 ≤ k ≤ n, are all real-valued, and for all t ≥ n.
Equation (2.10) is an another equivalent form of Equation (1.1). It is evident that for any λ ∈ ℂ, y(t) is a solution of Equation (1.1) if and only if x(t) = w1/2(t)y(t) is a solution of Equation (2.10). Moreover, it follows that
if and only if
By d(λ) denotes the numbers of the linearly independent solutions of Equation (2.10) in . Then we have that
This means that Equation (1.1) has the same positive and negative defect indices as those of Equation (2.10).
2.3 Sufficient and necessary conditions of limit point case and strong limit point case
The natural difference operator corresponding to Equation (1.1) is defined by
and the bilinear form [·, ·] associated with (1.1) is defined by
Then, it follows from [16, Lemma 2.2] that for any x(t) and y(t) defined in we have
Denote
which is the domain of the maximal operator corresponding to operator . It yields from (2.11) that limt→+∞[x, y](t) = c for any , where c is a finite constant. The following lemma is a direct consequence of [16, Theorem 6.15].
Lemma 2.4. Equation (1.1) is in l.p.c. at t = +∞ if and only if for all ,
Following the concept of the strong limit point case for singular discrete Hamiltonian system given in [17], we give the following concept:
Definition 2.2. Equation (1.1) is said to be in the strong limit point case (s.l.p.c.) at t = +∞ if for all ,
For convenience, denote
The following result is a direct consequence of Theorem 2.1 in [17].
Lemma 2.5. Equation (1.1) is in s.l.p.c. at t = +∞ if and only if limt→+∞S(y)(t) = 0 for all .
Lemma 2.6. Assume that there exist a constant c and an integer t0 ≥ 0 such that p j (t) (1 ≤ j ≤ n) and p0(t) - cw(t) are either non-negative for all t ≥ t0 or non-positive for all t ≥ t0. Then limt→+∞S(y)(t) exists, finite or infinite, for all .
Proof. Using (2.2) and (2.3), we have that
Summing up above relation from t0 to m, we get
For any , it follows from Cauchy's inequality that the second and third items on the right side of (2.13) converge. Since p j (t) (1 ≤ j ≤ n) and p0(t)-cw(t) are either non-negative for all t ≥ t0 or non-positive for all t ≥ t0, the last two items on the right side of (2.13) are monotonic. Therefore, limt→+∞S(y)(t) exists, finite or infinite. The proof is complete.
Remark 2.1. Note that the assertion of Lemma 2.6 in the present article are the same that of [15, Lemma 2.7]. Moreover, we point out that none conditions of Lemma 2.6 impose a restriction on q j (t), 1 ≤ j ≤ n.
3 Main results
In this section, several results on the positive and negative defect indices of Equation (1.1) and its equivalent forms; that is, (2.4) and (2.10), are given. It is first shown that for a class of symmetric difference equations in form of (2.4), the positive and negative defect indices are equal; that is n ≤ d+ = d- ≤ 2n, and all values in this range are realized. Next, several sufficient conditions of the limit point case for Equations (2.10) and (1.1) are given. Finally, two criteria of the strong limit point case for Equations (2.10) and (1.1) are established.
3.1 The range of the positive and negative defect indices
In this section, we consider a special case of Equation (2.4):
where P(t) is a complex-valued function, and w(t) > 0 in .
Theorem 3.1. The positive and negative defect indices d± of Equation (3.1) satisfy that n ≤ d+ = d- ≤ 2n, and all the values in this range are realized.
Proof. We only prove the assertion holds for n = 2, and the other cases can be shown similarly.
In the case that n = 2, Equation (3.1) can be written as
It can be divided into the following two second order symmetric difference equations:
where
For any given λ ∈ ℂ, let x(t) be a solution of Equation (3.3) and z(t) be a solution of Equation (3.4). Denote
It can be easily verified that y' and y" are two linearly independent solutions of Equation (3.2). Further, since
It follows that if and only if , and if and only if . By d±, and denote the positive and negative defect indices of Equations (3.2), (3.3), and (3.4), respectively. Then it follows that
On the other hand, one has by Lemma 2.3 that
and all the values in this range can be realized. Inserting (3.6) into (3.5), one has that
and all the values in this range can be realized.
For the general Equation (3.1), we can write it into n second-order symmetric difference equations with complex-valued coefficients. The positive and negative defect indices are equal to the sum of those of the n second-order symmetric difference equations, respectively. Then by Lemma 2.3 one can prove the assertion in Theorem 3.1. The proof is complete.
Example 3.1. Let w1(t) ≡ 1 for . It has been shown in [14] that for Equation (3.3), if Q1(t) ≡ 1 for , then Equation (3.3) is in the limit point case at t = +∞; if Q1(t) = -4t + 4ti for , then Equation (3.3) is in the limit circle case at t = +∞. Thus, if
then the positive and negative defect indices of Equation (3.2) are (2, 2); if
then the positive and negative defect indices of Equation (3.2) are (3, 3); if
then the positive and negative defect indices of Equation (3.2) are (4, 4). This example shows that the positive and negative defect indices of Equation (3.1) satisfies the inequality n ≤ d+ = d- ≤ 2n and all values in this range are realized.
3.2 Criteria of the limit point and the strong limit point cases
In this section, we give some sufficient conditions for the limit point case and the strong limit point case.
First, we consider the criteria of the limit point case. Similarly to the notations in Section 2, we introduce the following notations for Equation (2.10):
Theorem 3.2. Assume that there exist a non-negative function σ(t) defined on , a constant M > 0 and an integer t0 ≥ 0 such that
for all 1 ≤ j ≤ n and 0 ≤ l ≤ j - 1. Then Equation (2.10) is in l.p.c. at t = +∞.
Proof. First, we mention that the main idea of the proof is the same as that of [15, Theorem 3.3]. By Lemma 2.4, it suffices to show limt→+∞[x, y]1(t) = 0 for all . By the discussion before Lemma 2.4, limt→+∞[x, y]1(t) exists, finite, for all . Suppose that there exist such that limt→+∞[x, y]1(t) = c ≠ 0. Then there exists T0 ≥ t0 such that |[x, y]1(t)| ≥ |c|/2 > 0 for all t ≥ T0. It follows from condition (3.7) that
On the other hand, we have from (2.3) that
which, together with condition (3.8) and Cauchy's inequality, implies that
Since and belong to for any k ≥ 0. Hence,
This is a contradiction with (3.9). Therefore, limt→+∞[x, y]1(t) = 0 holds for all . By Lemma 2.4, Equation (2.10) is in l.p.c. at t = +∞.
Reversing the transformation from (1.1) to (2.10), we get the following result:
Theorem 3.3. Assume that there exist a non-negative function σ(t) defined on , a constant M > 0 and an integer t0 ≥ n such that σ(t) satisfies condition (3.7), and
for all 1 ≤ s ≤ n, 1 ≤ j ≤ s, 0 ≤ k ≤ s - j, 0 ≤ v ≤ j - 1, 0 ≤ m ≤ j - 1 - v, and 0 ≤ b ≤ j - 1. Then Equation (1.1) is in l.p.c. at t = +∞.
Proof. Since Equations (1.1) and (2.10) have the same limit case at t = +∞ under the transformation (2.7), it suffices to show that conditions (3.10) and (3.11) can imply that (3.8) holds. By the proof of Theorem 3.4 in [15], condition (3.10) implies that the first condition in (3.8) holds. So, it remains to show that condition (3.11) can imply that the second condition in (3.8) holds.
From (2.5), (2.6), and (2.9) we have
When s = j = n, it follows from (3.11) that
which together with (3.12) yields that
When s = j = n - 1, it follows from (3.11) that
When s = n and j = n - 1, (3.11) implies that
Inserting (3.14)-(3.16) into (3.13) with j = n - 1, we have
With a similar argument one can conclude that
for all 1 ≤ s ≤ n, 0 ≤ b ≤ s - 1 by using (3.13). Therefore, the conditions for q k in (3.8) hold. By Theorem 3.3, Equation (2.10) is in l.p.c. at t = +∞, which is equivalent to that Equation (1.1) is in l.p.c. at t = +∞. The proof is complete.
Note that if there exist a constant K > 0 and an integer t0 ≥ 0 such that |f(t)| ≤ Kt for t ≥ t0, then there exists a constant N > 0 such that |Δmf(t + v)| ≤ Nt for t ≥ t0 + 1 and 0 ≤ m, v ≤ n. Thus the following result is a direct consequence of Theorem 3.4 by taking σ(t) = t-1:
Corollary 3.1. If there exist a constant M > 0 and an integer t0 ≥ n such that
for all 1 ≤ s ≤ n, 1 ≤ j ≤ s and 0 ≤ k ≤ s - j, then Equation (1.1) is in l.p.c. at t = +∞.
Remark 3.1. In the case that n = 1, the conditions of Theorem 3.3 are equivalent to that of [14, Theorem 3.1]; that is
In fact, the conditions of Theorem 3.3 in the case that n = 1 are
If (3.19) holds, then (3.20) holds with
and M = 1. On the other hand, suppose that (3.20) holds. Then it follows from the second and third conditions in (3.20) that
which, together with the first condition in (3.20), implies that (3.19) holds.
At the end of this section, two criteria of the strong limit point case for Equations (2.10) and (1.1) are established, respectively.
Theorem 3.4. If , and , satisfy all the conditions in Theorem 3.2 and Lemma 2.6, then Equation (2.10) is in s.l.p.c. at t = +∞.
Proof. Since q j (t), 0 ≤ j ≤ n, satisfy the conditions in Lemma 2.6, limt→+∞S1(y) (t) exists, finite or infinite, for all . With a similar argument to that used in the proof of Theorem 3.2, it follows that limt→+∞S1(y)(t) = 0 for all . By Lemma 2.5, (2.10) is in s.l.p.c. at t = +∞. The proof is complete.
Theorem 3.5. If p j (t), 0 ≤ j ≤ n, q k (t), 1 ≤ k ≤ n, and w(t) satisfy all the conditions in Theorem 3.3 and Lemma 2.6, then Equation (1.1) is in s.l.p.c. at t = +∞.
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Acknowledgements
This research was supported by the NNSF of China (Grant 11071143, 11101241), and the NNSF of Shandong Province (Grant J10LA62).
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Ren, G. Defect indices of singular symmetric linear difference equations with complex coefficients. Adv Differ Equ 2012, 27 (2012). https://doi.org/10.1186/1687-1847-2012-27
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DOI: https://doi.org/10.1186/1687-1847-2012-27