Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 3210983, 12 pages
https://doi.org/10.1155/2019/3210983

Research Article

Characterization of Self-Adjoint Domains for Two-Interval Even Order Singular -Symmetric Differential Operators in Direct Sum Spaces

Qinglan Bao, Xiaoling Hao, and Jiong Sun

Math. Dept., Inner Mongolia University, Hohhot 010021, China

Correspondence should be addressed to Qinglan Bao; baoqinglan19@163.com

Received 30 October 2018; Revised 23 January 2019; Accepted 12 February 2019; Published 17 March 2019

Academic Editor: Nickolai Kosmatov

Copyright © 2019 Qinglan Bao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with the characterization of all self-adjoint domains associated with two-interval even order singular -symmetric differential operators in terms of boundary conditions. The previously known characterizations of Lagrange symmetric differential operators are a special case of this one.

1. Introduction

Self-adjoint differential operators [1–3] in Hilbert space are of interest in mathematics and physics; in Quantum Mechanics they represent observables [4–7]. These operators are generally defined by symmetric expressions and boundary conditions. Two-interval theory of differential equations was developed by W. N. Everitt and A. Zettl [8] in 1986. In 1988, A. M. Krall and A. Zettl [9, 10] generalized the method given by Coddington [11] and obtained the characterizations of self-adjoint domains for Sturm-Liouville differential operators with interior singular points. Afterwards, in [12] the two-interval theory was extended to higher order equations and any finite or infinite number of intervals. In [13] Wang et al. give an explicit characterization of all self-adjoint domains for Lagrange symmetric differential operators in terms of certain solutions for real   for the one-interval case when one endpoint is regular and the other is singular. In analogy with the celebrated Weyl limit-point, limit-circle theory in the second order case, i.e., Sturm-Liouville problems [14], they construct limit-point and limit-circle solutions and characterize the self-adjoint domains in terms of the limit-circle solutions. In [15], Hao et al. give a characterization for Lagrange symmetric differential operators by dividing one interval into two intervals and for some point when both endpoints and are singular. In [16], Suo et al. extend the characterization in [13] to two-interval case for one endpoint of each interval , and is regular, and illustrate the interactions between the regular endpoints and singular endpoints with some examples.

As noted in survey article [17], we observe that a special type of matrix, , plays key role in the characterization of a self-adjoint differential operators, both boundary conditions and symmetric differential operators. What is more interesting is that the symbol difference of this special type matrix is equivalent to skew-diagonal matrix , , which also generates self-adjoint operators. Actually these matrices can be generalized as a fixed nonsingular matrix and preserve their properties. So we can enlarge the known set of these operarors by extending the known symmetric expressions to C-symmetric expressions and charaterize the boundary conditions which determine self-adjoint extensions of these C-symmetric expressions on a single interval case. Remarkably, the same matrices C which generate the expressions also generate their self-adjoint extensions. This paper is based on all the above known works, and the complete characterization of self-adjoint domains of the two-interval case for even order -symmetric differential operators is given when four endpoints are singular or regular. Moreover, it has shown that the previous results in [16, 17] are special cases of ours. Following this introduction, some basic notations and facts are given in Section 2, in Sections 3 and 4 we give our main theorems for characterization of all self-adjoint domains and their proofs, and at last in Section 5 we give some examples to illustrate our main results.

2. Notation and Basic Facts

In this section we summarize some basic facts about general -symmetric quasidifferential expressions of even order (, and real or complex coefficients on one-interval and two-interval cases for the convenience of the reader.

Firstly, let be an interval with and denote the set of complex matrices with entries from a given set .

Set as a skew-diagonal constant matrix satisfying

and let

Let . We define

and

Inductively, for , we define

where , and denotes the set of complex-valued functions which are absolutely continuous on all compact subintervals of . Finally we set

The expression is called the quasidifferential expression associated with . For we also use the notations and .

Definition 1. Let and let be defined as above. Assume that

where

satisfying

i.e.,

with

Then is called a -symmetric differential expression.

Let be positive a.e. on We consider the Hilbert space

with its usual inner product

For the -symmetry , the Green’s formula has the form

where , and the limits always exist and are finite. Here the skew-symmetric sesquilinear form maps .

Every self-adjoint extension of the minimal operator is between the minimal operator and maximal operator ; i.e., we have

Thus these self-adjoint operators are distinguished from one another only by their domains.

Lemma 2 (Lagrange identity). Assume satisfies (8) and let be the corresponding -symmetric differential expression. Then for any we have

and

where

and

is defined by (11).
In fact,

and by (11) we have and ,

Proof. Set , and . Then we infer that

So for ,

is invertible a.e. on .
Since for , , , then

This concludes that .
Since satisfies (8), i.e., .
Now, let , ; then, from (4) and (6) we have

where

So from , we have

After integrating both sides of the above equation, we get

Hence from (15) we have

and

Together with some caculations we have

and has the form (11) and
Then we also have

This completes the proof.

Following this we consider direct sum Hilbert space

where .

The inner product in space is defined by

and is the usual inner product in :

Define two differential expressions with complex-valued coefficients by

Let ; i.e., .

Definition 3 (see [1, 8, 16]). The two-interval maximal and minimal domains and operators are simply the direct sums of the corresponding one-interval domains and operators, i.e.,

and

We also have the following lemma.

Lemma 4 (see [8, 16]). In the direct sum spaces, we have

The minimal operator is a closed, symmetric, densely defined operator in the Hilbert space H with deficiency index given by .

It is interesting to note that Lemma 2 extends to the two-interval case:

where

and has the form (11).

Lemma 5. Let . The number of linearly independent solutions of

lying in is independent of , provided If one endpoint of is regular and the other is singular, then the inequalities

hold. For , the number of linearly independent solutions of lying in is less than or equal to and of lying in is less than or equal to .
Let . If is the deficiency index on , is the deficiency index on and is the deficiency index on , then

Proof. See [15, 16].

W. N. Everitt and A. Zettl extend the well-known single interval GKN characterization of all self-adjoint extensions to the two-interval case for Lagrange symmetric differential expressions in [12], and it is obvious that this extended GKN theorem also can be established for two-interval -symmetric differential expression. It is expressed as follows.

Lemma 6 (GKN). Let be the two-interval minimal operator in and let be the deficiency index of Then a linear submanifold of is the domain of a self-adjoint extension of if and only if there exist vectors in satisfying the following conditions:

(i) are linearly independent modulo ;

(ii) ;

(iii) , .

3. Characterization of All Self-Adjoint Domains for Singular Two-Interval Problems

In this section we assume that are generated by satisfying (8), , , the endpoints and are singular. We give the decomposition of the maximal domain and the characterization of all self-adjoint extensions of the two-interval minimal operator.

First we have the following theorem.

Theorem 7. Let be a -symmetric differential expression on and let . Consider the equations

Assume that for some (45) has linearly independent solutions on which lie in and that for some (45) has linearly independent solutions on which lie in . Then, we have the following:

(1)The solutions can be extended to such that the extended functions, also denoted by , satisfy and is identically zero in a left neighborhood of . The solutions can be extended to such that the extended functions, also denoted by , satisfy and is identically zero in a right neighborhood of .(2)For the solutions on can be ordered such that the matrix , is given by

 For the solutions on can be ordered such that the matrix , is given by

(3)For every we have

(4)For ,we have

 For , we have

Proof. By Naimark Patching Lemma the solutions can be “patched” at to obtain maximal domain functions in . By another application of Naimark Patching Lemma these extended functions can be modified to be identically zero in a left neighborhood of . By using the similar method, we can proof the latter part of (1). Parts (2) and (3) are established by Corollary 6 in [13] for complex case. Part (4) follows from Corollary 3.8 in [15].

Remark 8. We call that the solutions and are of LP (limit-point) type at and , respectively, which satisfy conditions (3) of Theorem 7. The LP solutions play an important role in studies on distribution of continuous spectrum (see [15]). These solutions play no role in the formulation of the self-adjoint boundary conditions. But the LC (limit-circle) case requires boundary conditions to determine self-adjoint extensions. For this reason we call LC solutions at , LC solutions at .

Next we give the decomposition of the maximal domain and the characterization of all self-adjoint domains.

Theorem 9. Let the hypotheses and notations of Theorem 7 hold. Then

Proof. The method of this proof is similar to the citation [16].

According to Theorems 7 and 9 we have our main result as follows.

Theorem 10. Let the hypotheses and notations of Theorem 7 hold. Then a linear submanifold is the domain of a self-adjoint extension of two-interval minimal operator if and only if there exist complex matrices and complex matrices such that the following three conditions hold:

(1);(2);(3)

where denotes the by matrix whose first columns are those of , the second columns are those of , etc. And are complex matrices of the form (11).

Proof (necessity). Let be the domain of a self-adjoint extension of . By Lemma 6 there exist satisfying conditions (i), (ii), (iii) of Lemma 6. By Theorem 9, each and can be uniquely written as

where , .
Let

Then

Similarly,

Hence the boundary condition (iii) of Lemma 6 is equivalent to part (3) of Theorem 10.
Next we prove that , , , and satisfy conditions (1) and (2) of Theorem 10.
Clearly . If , then there exist constants , not all zero, such that

Let , so , ; from (54), we obtain

By (58), we have . Hence

So we have and ; thus, This contradicts the fact that the functions are linearly independent modulo . Therefore .
Now we verify part (2). By (54), we have

So

where the matrix is defined in Theorem 7.
Similarly, we have

Hence condition (ii) of Lemma 6 becomes

(sufficiency). Let the matrices , , , and satisfy conditions (1) and (2) of Theorem 10. We need to prove that defined by (3) is the domain of a self-adjoint extension of .
Let

Then for we have

Similarly, we have

Therefore the boundary condition (3) in Theorem 10 becomes the boundary condition (iii) in Lemma 6; i.e.,

It remains to show that satisfy conditions (i) and (ii) of Lemma 6.
Condition (i) holds. If not, then there exist constants , not all zero, such that

i.e.,

Hence we have , for any . Using the notation from Theorem 7,

Since is nonsingular, we have .
Similarly, we have , , and . Hence

This contradicts the fact that .
Next we show that (ii) holds. We have

From Theorem 7 we get

Similarly,

Therefore

By Lemma 6, we conclude that is a self-adjoint domain.

4. Special Case

In Theorem 10 it is assumed that all four endpoints are singular. It can be specialized to the results when at least one endpoint is regular. We state several cases here for the convenience of the reader.

Theorem 11. Let the hypotheses and notations of Theorem 7 hold and assume that the endpoints and are regular. Then and . Then a linear submanifold is the domain of a self-adjoint extension of if and only if there exist a complex matrix and a complex matrix and a complex matrix and a complex matrix such that the following three conditions hold:

(1);(2);(3):

Proof. Since are regular at , for any the limits

exist and are finite for .
When , for matrices determined by Theorem 10, we let

where

Then we have

and

So by Theorem 10, we may complete the proof.

Remark 12. In the minimal deficiency case the terms involving and in (77) drop out and Theorem 11 reduces to the self-adjoint boundary conditions at the regular endpoints and :

where the complex matrices and satisfy and . In this case there are no conditions required or allowed at the endpoints and .

Theorem 13. Let be two -symmetric differential expressions of order on and a positive function in and assume that each endpoint is regular. Then a linear submanifold of is the domain of a self-adjoint extension of if and only if there exist a complex matrix and a complex matrix and a complex matrix and a complex matrix such that the following three conditions hold:

(1);(2);(3):

Proof. In this case And for any the limits

exist and are finite for .
From Lagrange identity in two-interval case (40) we have

where

Then

is a self-adjoint domain if and only if

It is worthy noting that if we set , where

then our -symmetric condition can be reduced to Lagrange symmetric case; therefore, we have the following well-known chracterization.

Corollary 14 (see [16]). Let be two Lagrange symmetric differential expressions of order on , and assume that each endpoint is regular. Then a linear submanifold of is the domain of a self-adjoint extension of if and only if there exist a complex matrix and a complex matrix and a complex matrix and a complex matrix such that the following three conditions hold:

(1);(2);(3):

This corollary is the part IV of theorem 4.12 in paper [16].

5. Examples

A number of examples are given here to account for the main theorems. These examples include the interactions between the singular endpoints: interactions which generate self-adjoint operators. The self-adjoint interactions involve jump discontinuous of the Lagrange bracket of solutions at singular interior points. Here, let us take the case of as an example. Since the conditions when are the same as in the one-interval case, we give examples for .

Example 1. Assume . Then and . If satisfy , i.e., , let

Then

and (93) is a self-adjoint boundary condition. Furthermore we notice that there is one separated singular boundary condition at , one singular ‘continuity’ boundary condition and one singular jump boundary condition; these singular conditions involve the Lagrange bracket.

Example 2. Set . Then , and . Let

then (94) satisfies condition (1) and (2) in Theorem 10. Therefore the following conditions are self-adjoint boundary conditions and all of them involve interactions between singular endpoints, i.e., interactions between Lagrange brackets.

Here has the form as Example 1.

Example 3. Assume Then and . Let satisfy ; i.e., has the form as Example 2 and

Then

are self-adjoint boundary conditions and there are two singular “continuity” conditions, one separated singular boundary condition at and two interior coupled singular jump conditions.

Example 4. Assume . Then and . Then we have two-interval self-adjoint boundary conditions below:

Here the complex numbers are shown as Example 3.

Example 5. In this example we consider the case: Assume Then and . Let satisfy ; i.e., has the form as Example 3 and set . The following self-adjoint boundary conditions consist of one separated singular condition at , three singular “continuity” conditions, and three singular jump conditions.

Example 6. In this example we set Assume . Then and . The following boundary conditions feature separated self-adjoint boundary conditions at all four endpoints:

6. Conclusion

This paper characterize all self-adjoint domains for two-interval even order -symmetric differential operators in direct sum spaces, where both endpoints in each interval are singular, and there is not any singular point in each interval. And this characterization can be reduced to the regular case. Moreover the characterization in this paper is generalization of previous results for Lagrange symmetric case. So our work is valued.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Nature Science Foundation of China (No. 11561050) and Natural Science Foundation of Inner Mongolia (No. 2018MS01021).

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