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Discrete Dynamics in Nature and Society
Volume 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

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

Correspondence should be addressed to Qinglan Bao; moc.361@91nalgniqoab

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 [13] in Hilbert space are of interest in mathematics and physics; in Quantum Mechanics they represent observables [47]. 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 defineandInductively, for , we definewhere , and denotes the set of complex-valued functions which are absolutely continuous on all compact subintervals of . Finally we setThe 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 thatwheresatisfyingi.e.,withThen is called a -symmetric differential expression.

Let be positive a.e. on We consider the Hilbert spacewith its usual inner productFor 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 haveThus 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 haveandwhereand is defined by (11).
In fact,and by (11) we have and ,

Proof. Set , and . Then we infer thatSo 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 getHence from (15) we have and Together with some caculations we haveand has the form (11) and
Then we also haveThis completes the proof.

Following this we consider direct sum Hilbert space where .

The inner product in space is defined byand is the usual inner product in :Define two differential expressions with complex-valued coefficients byLet ; 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 haveThe 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:whereand has the form (11).

Lemma 5. Let . The number of linearly independent solutions oflying in is independent of , provided If one endpoint of is regular and the other is singular, then the inequalitieshold. 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 equationsAssume 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 byFor the solutions on can be ordered such that the matrix , is given by (3)For every we have(4)For ,we haveFor , 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 aswhere , .
Let ThenSimilarly,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 thatLet , 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 .
LetThen for we haveSimilarly, 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 . HenceThis contradicts the fact that .
Next we show that (ii) holds. We have From Theorem 7 we getSimilarly, 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 letwhereThen we have andSo 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