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

Authors: Qinglan Bao; Xiaoling Hao; Jiong Sun
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

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


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## Body

## 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 forreal  λ 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 (a1,b1) into two intervals (a1,c1) and (c1,b1) for some point c1∈(a1,b1) when both endpoints a1 and b1 are singular. In [16], Suo et al. extend the characterization in [13] to two-interval case for one endpoint of each interval (a1,b1), and (a2,b2) 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, En=((-1)rδr,n+1-s)r,s=1n, 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 0I~-I~o, I~=(δr,n+1-s)r,s=1k, which also generates self-adjoint operators. Actually these matrices can be generalized as a fixed nonsingular matrix C 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 C-symmetric differential operators is given when four endpoints a1,b1,a2,b2 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 generalC-symmetric quasidifferential expressions of even order (n=2k, k≥1) and real or complex coefficients on one-interval and two-interval cases for the convenience of the reader.Firstly, letJ=(a,b) be an interval with -∞≤a<b≤∞ and Mn(S) denote the set of n×n complex matrices with entries from a given set S.SetCn=(cr,s)1≤r,s≤n as a skew-diagonal constant matrix satisfying (1)Cn-1=-Cn=Cn∗,and let(2)ZnJ≔qr,sr,s=1n∈MnLlocJ,qr,r+1≠0a.e.J,qr,r+1-1∈LlocJ,1≤r≤n-1,qr,s=0a.e.J,2≤r+1<s≤n;qr,s∈LlocJ,s≠r+1,1≤r≤n-1.LetQ∈Zn(J). We define(3)V0≔y:J→C,yis  measurableand(4)y0≔yy∈V0.Inductively, for r=1,…,n, we define(5)Vr=y∈Vr-1:yr-1∈AClocJ,(6)yr=qr,r+1-1yr-1′-∑s=1rqr,sys-1y∈Vr,where qn,n+1≔cn,1, and ACloc(J) denotes the set of complex-valued functions which are absolutely continuous on all compact subintervals of J. Finally we set(7)My=MQy≔inyny∈Vn.The expression M=MQ is called the quasidifferential expression associated with Q. For Vn we also use the notations V(M) and D(Q).Definition 1.
LetQ∈Zn(J) and let M=MQ be defined as above. Assume that(8)Q=-Cn-1Q∗Cn,where(9)Cn=0k×kC12C210k×ksatisfying(10)Cn-1=-Cn=Cn∗,C21,C12∈MkC,i.e.,(11)Cn=0k×kC12-C12∗0k×k,with(12)cr,sc¯r,s=1,r+s=n+1.Then M=MQ is called a C-symmetric differential expression.Letw∈Lloc(J) be positive a.e. on J. We consider the Hilbert space(13)H=L2J,wwith its usual inner product(14)y,z≔∫Jyz¯wdx,y,z∈H.For the C-symmetry MQ, the Green’s formula has the form (15)∫JMyz¯-yMz¯dx=y,zb-y,zay,z∈DQ,where [y,z](b)=limt→b-[y,z](t), [y,z](a)=limt→a+[y,z](t) and the limits always exist and are finite. Here the skew-symmetric sesquilinear form [·,·] maps D(Q)×D(Q)→C.Every self-adjoint extensionT of the minimal operator TQ,0 is between the minimal operator TQ,0 and maximal operator TQ; i.e., we have(16)TQ,0⊂T=T∗⊂TQ.Thus these self-adjoint operators T are distinguished from one another only by their domains.Lemma 2 (Lagrange identity).
AssumeQ∈Zn(J) satisfies (8) and let M=MQ be the corresponding C-symmetric differential expression. Then for any y,z∈D(Q) we have(17)z¯My-yMz¯=y,z′,and(18)y,z=-1k+1∑r=0n-1cn-r,r+1zn-r-1¯yr=-1k+1∑r=1kcr,n-r+1zr-1¯yn-r-c¯r,n-r+1zn-r¯yr-1=-1k+1Z∗CnY,where(19)Y=y0y1⋮yn-1,Z=z0z1⋮zn-1,and (20)Cn=0k×kC12C210k×kis defined by (11).
In fact,(21)C12=00⋯0c1,n00⋯c2,n-10⋮⋮⋰⋮⋮0ck-1,k+2⋯00ck,k+10⋯00,and by (11) we have C21=-C12∗ and cr,s=-c¯s,r, r+s=n+1.Proof.
SetQ=(qr,s)r,s=1n, and Q+≔-Cn-1Q∗Cn=(pr,s)r,s=1n. Then we infer that(22)pr,s=∑l=1ncl,s∑j=1ncr,jq¯l,j=cr,n-r+1q¯n-s+1,n-r+1cn-s+1,s,r,s=1,2,…,n.So for 1≤r≤n-1,(23)pr,r+1=cr,n-r+1q¯n-r,n-r+1cn-r,r+1is invertible a.e. on J.
Since for2≤r+1<s≤n, r+1-s=(n-s+1)+1-(n-r+1)<0, qn-s+1,n-r+1=0, then (24)pr,s=cr,n-r+1q¯n-s+1,n-r+1cn-s+1,s=0.This concludes that Q+∈Zn(J).
SinceQ∈Zn(J) satisfies (8), i.e., Q=Q+.
Now, letf=-c¯1,nyQ[n],g=-c¯1,nzQ[n], y,z∈Vn; then, from (4) and (6) we have (25)Y′=QY+F,Z′=QZ+G,where (26)F=00⋮f,G=00⋮g.So from Q∗Cn=-CnQ, we have (27)Z∗CnY′=Z∗′CnY+Z∗Cn′Y+Z∗CnY′=Z′∗CnY+Z∗CnY′=QZ+G∗CnY+Z∗CnQY+F=Z∗Q∗+G∗CnY+Z∗CnQY+F=Z∗Q∗Cn+CnQY+G∗CnY+Z∗CnF=Z∗CnF+G∗CnY=c1,nz0¯f-c¯1,ng¯y0=-z0¯yn+zn¯y0=--inz0¯My-Mz¯y0.After integrating both sides of the above equation, we get(28)∫abz¯MQydx-∫abyMQz¯dx=-1k+1Z∗CnYab.
Hence from (15) we have (29)z¯MQy-yMQz¯=y,z′,and (30)y,z=-1k+1Z∗CnY.Together with some caculations we have(31)Z∗CnY=∑r=0n-1cn-r,r+1zn-r-1¯yr,and Cn has the form (11) and cr,s=-c¯s,r,r+s=n+1.
Then we also have(32)Z∗CnY=∑r=1kcr,n-r+1zr-1¯yn-r-c¯r,n-r+1zn-r¯yr-1.This completes the proof.Following this we consider direct sum Hilbert space(33)H=H1⊕H2,Hj=L2Jj,wj,wj>0,where Jj=(aj,bj),-∞≤aj<bj≤∞,j=1,2.The inner product in spaceH is defined by(34)y,z=∑j=12yj,zjj,y=y1,y2,z=z1,z2,and (·,·)j is the usual inner product in Hj:(35)yj,zjj=∫Jjyjz¯jwjdx,y1,z1∈H1,y2,z2∈H2.Define two differential expressions with complex-valued coefficients by(36)Mjy=MQjy≔inyQjnonJj.Let M={M1,M2}; i.e., My={M1y1,M2y2}.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.,(37)TQ=TQ1⊕TQ2,TQ,0=TQ1,0⊕TQ2,0,and(38)DQ=DTQ=DTQ1⊕DTQ2,DQ,0=DTQ,0=DTQ1,0⊕DTQ2,0.We also have the following lemma.Lemma 4 (see [8, 16]).
In the direct sum spaces, we have(39)TQ,0∗=TQ1,0∗⊕TQ2,0∗=TQ1⊕TQ2=TQ,TQ∗=TQ1∗⊕TQ2∗=TQ1,0⊕TQ2,0=TQ,0.The minimal operator TQ,0 is a closed, symmetric, densely defined operator in the Hilbert space H with deficiency index d given by d=d1+d2.It is interesting to note that Lemma2 extends to the two-interval case:(40)z¯My-yMz¯=y,z′,y,z=∑j=12yj,zjjbj-yj,zjjaj,y,z∈H,where(41)yj,zjj=-1k+1Zj∗CnYj,Yj=yj0yj1⋮yjn-1,Zj=zj0zj1⋮zjn-1,j=1,2,and Cn has the form (11).Lemma 5.
Letaj≤αj<βj≤bj. The number dj of linearly independent solutions of(42)Mjy=λjwjyonαj,βjlying in L2((αj,βj),wj) is independent of λj∈C, provided Im(λj)≠0. If one endpoint of (αj,βj) is regular and the other is singular, then the inequalities(43)k≤dj≤2k=nhold. For λ=λj∈R, the number of linearly independent solutions of (42)j=1 lying in L2((α1,β1),w1) is less than or equal to d1 and of (42)j=2 lying in L2((α2,β2),w2) is less than or equal to d2.
Letcj∈Jj=(aj,bj). If dj1 is the deficiency index on (aj,cj), dj2 is the deficiency index on (cj,bj) and dj is the deficiency index on (aj,bj), then(44)dj=dj1+dj2-n,j=1,2.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 C-symmetric differential expression. It is expressed as follows.Lemma 6 (GKN).
LetTQ,0 be the two-interval minimal operator in H and let d be the deficiency index of TQ,0. Then a linear submanifold D(T) of DQ is the domain of a self-adjoint extension T of TQ,0 if and only if there exist vectors w1,w2,···,wd in DQ satisfying the following conditions:(i)w1,w2,···,wd are linearly independent modulo DQ,0;(ii)[wi,wl]=[wi1,wl1]1(b1)-[wi1,wl1]1(a1)+[wi2,wl2]2(b2)-[wi2,wl2]2(a2)=0,i,l=1,···,d;(iii)D(T)={y={y1,y2}∈DQ:[y,wl]=[y1,wl1]1(b1)-[y1,wl1]1(a1)+[y2,wl2]2(b2)-[y2,wl2]2(a2)=0, l=1,···,d}.


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

In this section we assume thatM={MQ1,MQ2} are generated by Qj∈Zn(j)(Jj),j=1,2 satisfying (8), n=2k, k≥1, the endpoints aj and bj 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.
LetMj be a C-symmetric differential expression on (aj,bj) and let cj∈(aj,bj). Consider the equations(45)Mjy=λjwjy,j=1,2.Assume that for some λ=λj1∈R (45) has dj1 linearly independent solutions uj1,uj2,···,ujdj1 on (aj,cj) which lie in L2((aj,cj),wj) and that for some λ=λj2∈R (45) has dj2 linearly independent solutions vj1,vj2,···,vjdj2 on (cj,bj) which lie in L2((cj,bj),wj). Then, we have the following: (1)
The solutionsuj1,uj2,···,ujdj1 can be extended to Jj=(aj,bj) such that the extended functions, also denoted by uj1,uj2,···,ujdj1, satisfy ujl∈DQj(aj,bj) and ujl is identically zero in a left neighborhood of bj,l=1,···,dj1. The solutions vj1,vj2,···,vjdj2 can be extended to (aj,bj) such that the extended functions, also denoted by vj1,vj2,···,vjdj2, satisfy vjl∈DQj(aj,bj) and vjl is identically zero in a right neighborhood of aj,l=1,···,dj2.(2)
Formj=2dj1-2k the solutions uj1,uj2,···,ujdj1 on (aj,cj) can be ordered such that the mj×mj matrix Uj=([ujl1,ujl2]j(cj)),1≤l1,l2≤mj, is given by(46)Uj=-1k+1CmjT,j=1,2.Fornj=2dj2-2k the solutions vj1,vj2,···,vjdj2 on (cj,bj) can be ordered such that the nj×nj matrix Vj=([vjl1,vjl2]j(cj)),1≤l1,l2≤nj, is given by(47)Vj=-1k+1CnjT,j=1,2.(3)
For everyy={y1,y2}∈DQ we have(48)yj,ujljaj=0,forl=mj+1,···,dj1,(49)yj,vjljbj=0,forl=nj+1,···,dj2.(4)
For1≤l1,l2≤dj1,we have(50)ujl1,ujl2jaj=ujl1,ujl2jcj.For1≤l1,l2≤dj2, we have(51)vjl1,vjl2jbj=vjl1,vjl2jcj.Proof.
By Naimark Patching Lemma the solutionsuj1,uj2,···,ujdj1 can be “patched” at cj to obtain maximal domain functions in DQj(aj,bj). By another application of Naimark Patching Lemma these extended functions can be modified to be identically zero in a left neighborhood of bj,j=1,2. 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 solutionsujmj+1,···,ujdj1 and vjnj+1,···,vjdj2 are of LP (limit-point) type at aj and bj, 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 uj1,uj2,⋯,ujmj LC solutions at aj, vj1,vj2,⋯,vjnj LC solutions at bj.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 Theorem7 hold. Then (52)DQjaj,bj=DQj,0aj,bj⊕spanuj1,uj2,···,ujmj⊕spanvj1,vj2,···,vjnj.Proof.
The method of this proof is similar to the citation [16].According to Theorems7 and 9 we have our main result as follows.Theorem 10.
Let the hypotheses and notations of Theorem7 hold. Then a linear submanifold D(T)⊂DQ is the domain of a self-adjoint extension T of two-interval minimal operator TQ,0 if and only if there exist complex d×mj matrices Aj and complex d×nj matrices Bj such that the following three conditions hold:(1)
rank(A1,B1,A2,B2)=d;(2)
∑j=12‍{AjCmjAj∗-BjCnjBj∗}=0;(3)
D(T)={y={y1,y2}∈DQ:(53)∑j=12Ajyj,uj1jaj⋮yj,ujmjjaj+Bjyj,vj1jbj⋮yj,vjnjjbj=0⋮0,where (A1,B1,A2,B2) denotes the d by 4d matrix whose first d columns are those of A1, the second d columns are those of B1, etc. And Cmj,Cnj are complex matrices of the form (11).Proof (necessity).
LetD(T) be the domain of a self-adjoint extension T of TQ,0. By Lemma 6 there exist w1={w11,w12},⋯,wd={wd1,wd2}∈DQ satisfying conditions (i), (ii), (iii) of Lemma 6. By Theorem 9, each wi1 and wi2 can be uniquely written as(54)wi1=y^i1+∑l=1m1ailu1l+∑l=1n1bilv1l,wi2=y^i2+∑l=1m2cilu2l+∑l=1n2dilv2l,where y^i1∈DQ1,0,y^i2∈DQ2,0, ail,bil,cil,dil∈C,i=1,2,…,d.
Let(55)A1=-a¯ild×m1,B1=b¯ild×n1,A2=-c¯ild×m2,B2=d¯ild×n2,
Then(56)
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Next we prove thatA1, B1, A2, and B2 satisfy conditions (1) and (2) of Theorem 10.
Clearlyrank(A1,B1,A2,B2)≤d. If rank(A1,B1,A2,B2)<d, then there exist constants h1,⋯,hd, not all zero, such that(58)h1,…,hdA1,B1,A2,B2=0.
Letf={f1,f2}=∑i=1d‍h¯iwi, so f1=∑i=1d‍h¯iwi1, f2=∑i=1d‍h¯iwi2; from (54), we obtain (59)f1=∑i=1dh¯iy^i1+∑i=1d∑l=1m1h¯iailu1l+∑i=1d∑l=1n1h¯ibilv1l,f2=∑i=1dh¯iy^i2+∑i=1d∑l=1m2h¯icilu2l+∑i=1d∑l=1n2h¯idilv2l.By (58), we have (h1⋯hd)A1=(h1⋯hd)B1=(h1⋯hd)A2=(h1⋯hd)B2=0. Hence (60)f1=∑i=1dh¯iy^i1,f2=∑i=1dh¯iy^i2.So we have f1∈DQ1,0 and f2∈DQ2,0; thus, f={f1,f2}∈DQ,0. This contradicts the fact that the functions w1,w2,…,wd are linearly independent modulo DQ,0. Therefore rank(A1,B1,A2,B2)=d.
Now we verify part (2). By (54), we have (61)wi1,wl11a1=∑k=1m1aiku1k,∑s=1m1alsu1s1a1=∑k=1m1∑s=1m1aika¯lsu1k,u1s1a1,i,l=1,⋯,d.So (62)wi1,wl11a1d×dT=A1U1TA1∗=-1k+1A1Cm1A1∗,where the matrix U1 is defined in Theorem 7.
Similarly, we have(63)wi1,wl11b1d×dT=B1V1TB1∗=-1k+1B1Cn1B1∗,wi2,wl22a2d×dT=A2U2TA2∗=-1k+1A2Cm2A2∗,wi2,wl22b2d×dT=B2V2TB2∗=-1k+1B2Cn2B2∗.
Hence condition (ii) of Lemma6 becomes(64)A1Cm1A1∗-B1Cn1B1∗+A2Cm2A2∗-B2Cn2B2∗=0.(sufficiency). Let the matrices A1, B1, A2, and B2 satisfy conditions (1) and (2) of Theorem 10. We need to prove that D(T) defined by (3) is the domain of a self-adjoint extension T of TQ,0.
Let(65)A1=-a¯ild×m1,B1=b¯ild×n1,A2=-c¯ild×m2,B2=d¯ild×n2,wi1=∑l=1m1ailu1l+∑l=1n1bilv1l,wi2=∑l=1m2cilu2l+∑l=1n2dilv2l.Then for y={y1,y2}∈DQ we have(66)
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Similarly, we have(67)-A2y2,u212a2⋮y2,u2m22a2=y2,w122a2⋮y2,wd22a2,B2y2,v212b2⋮y2,z2n22b2=y2,w122b2⋮y2,wd22b2.Therefore the boundary condition (3) in Theorem 10 becomes the boundary condition (iii) in Lemma 6; i.e.,(68)y1,wi11b1-y1,wi11a1+y2,wi22b2-y2,wi22a2=0,i=1,⋯,d.
It remains to show thatwi,i=1,⋯,d satisfy conditions (i) and (ii) of Lemma 6.
Condition (i) holds. If not, then there exist constantsc1,⋯,cd, not all zero, such that (69)γ=∑i=1dciwi∈DQ,0,
i.e.,(70)γ1=∑i=1dciwi1∈DQ1,0,γ2=∑i=1dciwi2∈DQ2,0.Hence we have [γ1,y1]1(a1)=[γ1,y1]1(b1)=[γ2,y2]2(a2)=[γ2,y2]2(b2)=0, for any y={y1,y2}∈DQ. Using the notation U1 from Theorem 7,(71)0⋯0=∑l=1dclwl1,u111a1⋯∑l=1dclwl1,u1m11a1=c1⋯cdaild×m1U1.Since U1 is nonsingular, we have (c¯1⋯c¯d)A1=0.
Similarly, we have(c¯1⋯c¯d)B1=0, (c¯1⋯c¯d)A2=0, and (c¯1⋯c¯d)B2=0. Hence(72)c¯1⋯c¯dA1,B1,A2,B2=0.This contradicts the fact that rank(A1,B1,A2,B2)=d.
Next we show that (ii) holds. We have(73)wi1,wl11a1=∑s=1m1aisu1s,∑k=1m1alku1k1a1=∑s=1m1∑k=1m1aisa¯lku1s,u1k1a1.
From Theorem7 we get(74)wi1,wl11a1d×dT=A1U1TA1∗=-1k+1A1Cm1A1∗.
Similarly,(75)wi1,wl11b1d×dT=-1k+1B1Cn1B1∗,wi2,wl22a2d×dT=-1k+1A2Cm2A2∗,wi2,wl22b2d×dT=-1k+1B2Cn2B2∗.Therefore (76)wi,wld×dT=wi1,wl11b1-wi1,wl11a1+wi2,wl22b2-wi2,wl22a2T=-1k+1B1Cn1B1∗--1k+1A1Cm1A1∗+-1k+1B2Cn2B2∗--1k+1A2Cm2A2∗=0.
By Lemma6, we conclude that D(T) is a self-adjoint domain.


## 4. Special Case

In Theorem10 it is assumed that all four endpoints a1,b1,a2,b2 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 Theorem7 hold and assume that the endpoints b1 and b2 are regular. Then n1=n2=n and d=d11+d21. Then a linear submanifold D(T)⊂DQ is the domain of a self-adjoint extension T of TQ,0 if and only if there exist a complex d×m1 matrix A1 and a complex d×n matrix B1 and a complex d×m2 matrix A2 and a complex d×n matrix B2 such that the following three conditions hold:(1)
rank(A1,B1,A2,B2)=d;(2)
A1Cm1A1∗+A2Cm2A2∗=B1CnB1∗+B2CnB2∗;(3)
D(T)={y={y1,y2}∈DQ:(77)A1y1,u111a1⋮y1,u1m11a1+B1y1b1⋮y1n-1b1+A2y2,u212a2⋮y2,u2m22a2+B2y2b2⋮y2n-1b2=0⋮0.Proof.
SinceMj are regular at bj, for any y={y1,y2}∈DQ the limits (78)yjsbj=limt→bj-⁡yjst,j=1,2,exist and are finite for s=0,1,…,n-1.
Whend12=d22=n, for matrices B1,B2 determined by Theorem 10, we let(79)B1=-1kR1∗Cn,B2=-1kR2∗Cn,where(80)Rj∗=w¯1j⋯w¯1jn-1⋯⋯⋯w¯dj⋯w¯djn-1,j=1,2.Then we have (81)yj,wljjbjd×1=-1k+1R1∗Cnyjbj⋮yjn-1bj,j=1,2,wi1,wl11b1d×dT=-1k+1R1∗CnR1=-1k+1R1∗CnCn-1CnR1=-1k+1R1∗Cn-CnCnR1=-1k+1B1CnB1∗,and(82)wi2,wl22b2d×dT=-1k+1B2CnB2∗.So by Theorem 10, we may complete the proof.Remark 12.
In the minimal deficiency cased11=k,m1=0,d21=k,m2=0 the terms involving A1 and A2 in (77) drop out and Theorem 11 reduces to the self-adjoint boundary conditions at the regular endpoints b1 and b2:(83)B1y1b1⋮y1n-1b1+B2y2b2⋮y2n-1b2=0⋮0,where the n×n complex matrices B1 and B2 satisfy rank(B1,B2)=n and B1CnB1∗+B2CnB2∗=0. In this case there are no conditions required or allowed at the endpoints a1 and a2.Theorem 13.
LetMj be two C-symmetric differential expressions of order n=2k on (aj,bj),j=1,2 and wj a positive function in L(aj,bj) and assume that each endpoint is regular. Then a linear submanifold D(T) of DQ is the domain of a self-adjoint extension T of TQ,0 if and only if there exist a complex 2n×n matrix A1 and a complex 2n×n matrix B1 and a complex 2n×n matrix A2 and a complex 2n×n matrix B2 such that the following three conditions hold:(1)
rank(A1,B1,A2,B2)=2n;(2)
A1CnA1∗-B1CnB1∗+A2CnA2∗-B2CnB2∗=0;(3)
D(T)={y={y1,y2}∈DQ: (84)A1y1a1⋮y1n-1a1+B1y1b1⋮y1n-1b1+A2y2a2⋮y2n-1a2+B2y2b2⋮y2n-1b2=0⋮0.Proof.
In this cased=d1+d2=d11+d12-n+d21+d22-n=n+n-n+n+n-n=2n. And for any y={y1,y2}∈DQ the limits(85)yjsaj=limt→aj+⁡yjst;yjsbj=limt→bj-⁡yjst,j=1,2,exist and are finite for s=0,1,…,n-1.
From Lagrange identity in two-interval case (40) we have (86)∫abz¯Mydx-∫abMz¯ydx=y,zab=0,where(87)y,zab=∑j=12yj,zjjbj-yj,zjjaj=∑j=12Zj∗bjCnYjbj-Zj∗ajCnYjaj.Then (88)DT=y=y1,y2∈DQ:A1Y1a1+B1Y1b1+A2Y2a2+B2Y2b2=0is a self-adjoint domain if and only if (89)A1CnA1∗+A2CnA2∗=B1CnB1∗+B2CnB2∗.It is worthy noting that if we setCn=En, where(90)En=-1rδr,n+1-sr,s=1n,then our C-symmetric condition can be reduced to Lagrange symmetric case; therefore, we have the following well-known chracterization.Corollary 14 (see [16]).
LetMj be two Lagrange symmetric differential expressions of order n=2k on (aj,bj),j=1,2, and assume that each endpoint is regular. Then a linear submanifold D(T) of DQ is the domain of a self-adjoint extension T of TQ,0 if and only if there exist a complex 2n×n matrix A~1 and a complex 2n×n matrix A~2 and a complex 2n×n matrix A~3 and a complex 2n×n matrix A~4 such that the following three conditions hold:(1)
rank(A~1,A~2,A~3,A~4)=2n;(2)
∑k=14‍(-1)k+1A~kEnA~k∗=0;(3)
D(T)={y={y1,y2}∈DQ:(91)A~1y1a1⋮y1n-1a1+A~2y1b1⋮y1n-1b1+A~3y2a2⋮y2n-1a2+A~4y2b2⋮y2n-1b2=0⋮0.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 ofn=4 as an example. Since the conditions when 0≤d≤2 are the same as in the one-interval case, we give examples for 3≤d≤8inthefollowing.Example 1.
Assumed11=d12=3,d21=2,d22=3. Then d1=d11+d12-4=2,d2=d21+d22-4=1,d=d1+d2=3 and m1=2d11-4=2,m2=2d21-4=0,n1=2d12-4=2,n2=2d22-4=2. If C2=0c-c¯0 satisfy C2-1=-C2=C2∗, i.e., cc¯=1, let (92)A1=100100,B1=00001c,A2=O,B2=10c¯100,c∈C.Then(93)y1,u111a1+y2,v212b2=0,y1,u121a1+y2,v222b2=-c-y2,v212b2,y1,v111b1+cy1,v121b1=0,and (93) is a self-adjoint boundary condition. Furthermore we notice that there is one separated singular boundary condition at b1, one singular ‘continuity’ boundary condition and one singular jump boundary condition; these singular conditions involve the Lagrange bracket.Example 2.
Setd11=d12=d21=d22=3. Then d1=d2=2, d=4 and m1=m2=n1=n2=2. Let(94)A1,B1,A2,B2=100000-10010000c¯-10010-10000001c¯-100,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. (95)y1,u111a1=y2,v212b2,y1,u121a1-y2,v222b2=-c¯y2,v212b2,y1,v111b1=y2,u212a2,y1,v121b1-y2,u222a2=-c¯y2,u212a2,(96)c∈C.Here C2 has the form as Example 1.Example 3.
Assumed11=2,d12=4,d21=4,d22=3. Then d1=2,d2=3,d=5 and m1=0,m2=4,n1=4,n2=2. Let C2,C4 satisfy C-1=-C=C∗; i.e., C2 has the form as Example 2 and (97)C4=000c100c200-c¯200-c¯1000,cc¯=1,c1c¯1=c2c¯2=1.Then(98)y2,v212b2+cy2,v222b2=0,y1,v111b1=y2,u212a2,y1,v121b1-y2,u222a2=c¯1y2,u212a2,y1,v131b1=y2,u232a2,y1,v141b1-y2,u242a2=-c2y2,u232a2.are self-adjoint boundary conditions and there are two singular “continuity” conditions, one separated singular boundary condition at b2 and two interior coupled singular jump conditions.Example 4.
Assumed11=d22=3,d12=d21=4. Then d1=d2=3,d=6 and m1=2,m2=4,n1=4,n2=2. Then we have two-interval self-adjoint boundary conditions below: (99)y2,u212a2-y2,u222a2=0,c2y2,u232a2+c1y2,u242a2=0,y1,v111b1-y1,v121b1=0,c2y1,v131b1+c1y1,v141b1=0,y1,u111a1=y2,v212b2,y1,u121a1-y2,v222b2=-c¯y2,v212b2.Here the complex numbers c,c1,c2 are shown as Example 3.Example 5.
In this example we consider the case:d=7. Assume d11=3,d12=5,d21=5,d22=2. Then d1=4,d2=3,d=7 and m1=2,m2=6,n1=6,n2=0. Let C2,C4,C6 satisfy C-1=-C=C∗; i.e., C2,C4 has the form as Example 3 and set C6=00000c10000c20000c30000-c¯30000-c¯20000-c¯100000. The following self-adjoint boundary conditions consist of one separated singular condition at a1, three singular “continuity” conditions, and three singular jump conditions.(100)c¯y1,u111a1+y1,u121a1=0,y2,u212a2=y1,v111b1,y1,v121b1-y2,u222a2=c¯1y2,u212a2,y2,u232a2=y1,v131b1,y1,v141b1-y2,u242a2=-c¯3y2,u232a2,y2,u252a2=y1,v151b1,y1,v161b1-y2,u262a2=-c2y2,u252a2.Example 6.
In this example we setC4=000-i00i00i00-i000. Assume d11=d12=4,d21=d22=4. Then d1=d2=4,d=8 and m1=m2=4,n1=n2=4. The following boundary conditions feature separated self-adjoint boundary conditions at all four endpoints:(101)y1,u111a1-iy1,u121a1=0,iy1,u131a1-y1,u141a1=0,y1,v111b1-iy1,v121b1=0,iy1,v131b1-y1,v141b1=0,y2,u212a2-iy2,u222a2=0,iy2,u232a2-y2,u242a2=0,y2,v212b2-iy2,v222b2=0,iy2,v232b2-y2,v242b2=0.


## 6. Conclusion

This paper characterize all self-adjoint domains for two-interval even orderC-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.



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