# A Dirac System with Transmission Condition and Eigenparameter in Boundary Condition

Authors: Abdullah Kablan; Tülay Özden
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper deals with a Dirac system with transmission condition and eigenparameter in boundary condition. We give an operator-theoretic formulation of the problem then investigate the existence of the solution. Some spectral properties of the problem are studied.


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## Body

## 1. Introduction

After Walter [1] had given an operator-theoretic formulation of eigenvalue problems with eigenvalue parameter in the boundary conditions, Fulton [2, 3] has carried over the methods of Titchmarsh [4, chapter 1] to this problem. Then, a large amount of the mathematical literature was devoted to these subjects during the last twenty years. We will mention some of the papers published at least twenty years ago, but of course there are many other interesting and important papers published more recently, which are not referred to here. The existence of solution and some spectral properties of Sturm-Liouville problem with eigenparameter-dependent boundary conditions and also with transmission conditions at one or more inner points of considered finite interval has been studied by Mukhtarov and Tunç [5]; see also [6, 7]. A Dirac system when the eigenparameter appears in boundary conditions has been studied by Kerimov [8]. In [9], an inverse problem for the Dirac system with eigenvalue-dependent boundary conditions and transmission condition is investigated.The aim of the present paper is to study a Dirac system with transmission condition and eigenparameter in boundary condition. For this, we follow the method in [5]. We consider the Dirac system
(1)ℓ(u)=Au′(x)-P(x)u(x)=λu(x),
where
(2)A=(01-10),P(x)=(p1(x)00p2(x)),u(x)=(u1(x)u2(x)),
or
(3)u2′(x)-p1(x)u1(x)=λu1(x),u1′(x)+p2(x)u2(x)=-λu2(x),x∈[a,c)∪(c,b],
with boundary conditions
(4)sinαu1(a)-cosαu2(a)=0,(5)b1u1(b)-a1u2(b)+λ(sinβu1(b)-cosβu2(b))=0,
and transmission conditions at the inner point x=c(6)u1(c-0)=γu1(c+0),u2(c-0)=γ-1u2(c+0).
Here and later on, λ is a complex eigenvalue parameter; the functions pi(x)(i=1,2) are continuous on [a,c)∪(c,b] which have finite limits pi(±c)=limx→±cpi(x)(i=1,2). a1, b1, γ are real numbers and α,β∈[0,π).


## 2. Operator Formulation of the Problem

For convenience, we will assume that|a1|+|b1|≠0,γ≠0. To formulate a theoretic approach to problem (1)–(6), we define the Hilbert space ℍ=L2[a,c)∪L2(c,b]⊕ℂσ with an inner product
(7)〈U,V〉ℍ=∫ac‍uT(x)v¯(x)dx+∫cb‍uT(x)v¯(x)dx+1σu~v~¯,
where T stands for the transpose and
(8)U=(u(x)u~),V=(v(x)v~)∈ℍ,u(x)=(u1(x)u2(x)),v(x)=(v1(x)v2(x))∈H,ui(x),vi(x)∈L2[a,c)∪L2(c,b],(i=1,2),u~,v~∈ℂ. The constant σ is defined by
(9)σ∶=det(b1a1sinβcosβ)>0.
Let dom(A)⊆ℍ be set of all U=(u(x)u^)∈ℍ, such that u1(x),u2(x) are absolutely continuous on [a,c)∪(c,b], u^=sinβu1(b)-cosβu2(b) and ℓ(u)∈ℍ, sinαu1(a)-cosαu2(a)=0, u1(±c),u2(±c) have finite limits, u~=b1u1(b)-a1u2(b). Now define the operator A:dom(A)→ℍ by
(10)A(u(x)u^)=(ℓ(u)-u~).
Hence, we can rewrite the problem (1)–(6) in the operator form as
(11)AU=λU.
Obviously, the operator A and the Dirac system (1)–(6) have the same eigenvalues. Also the eigenvectors of (1)–(6) coincide with the first two components of the corresponding eigenelement of the operator A.Lemma 1.
Thedom(A) is dense in ℍ.Proof.
It is easily seen that there is no nonzero vectorF=(f(x),f^)∈ℍ such that for every U=(u(x),u^)∈dom(A), 〈F,U〉ℍ=0. This implies dom(A)⊥={Θ}, where Θ=(0,0,0). Therefore, dom(A) is dense in ℍ.Theorem 2.
The operatorA is symmetric.Proof.
For eachU,V∈dom(A) from the inner product (7) and the integration by parts, we have
(12)〈AU,V〉ℍ=∫ac‍(u2′-p1u1)v¯1dx-∫ac‍(u1′+p2u2)v¯2dx+∫cb‍(u2′-p1u1)v¯1dx-∫cb‍(u1′+p2u2)v¯2dx-1σu~v^¯=[u2v¯1-u1v¯2]ac-0+[u2v¯1-u1v¯2]c+0b-∫ac‍u2v¯1′dx-∫ac‍p1u1v¯1dx+∫ac‍u1v¯2′dx-∫ac‍p2u2v¯2dx-∫cb‍u2v¯1′dx-∫cb‍p1u1v¯1dx+∫cb‍u1v¯2′dx-∫cb‍p2u2v¯2dx-1σu~v^¯=[u2(c-0)v¯1(c-0)-u1(c-0)v¯2(c-0)]-[u2(a)v¯1(a)-u1(a)v¯2(a)]+[u2(b)v¯1(b)-u1(b)v¯2(b)]-[u2(c+0)v¯1(c+0)-u1(c+0)v¯2(c+0)]-∫ac‍u2(v¯1′+p2v¯2)dx+∫ac‍u1(v¯2′-p1v¯1)dx-∫cb‍u2(v¯1′+p2v¯2)dx+∫cb‍u1(v¯2′-p1v¯1)dx-1σ(b1u1(b)-a1u2(b))×(sinβv¯1(b)-cosβv¯2(b)).
Since U and V satisfy the same boundary condition (4) at x=a,
(13)u2(a)v¯1(a)=u1(a)v¯2(a).
From transmission condition (6), it follows that
(14)u2(c-0)v¯1(c-0)=u2(c+0)v¯1(c+0),u1(c-0)v¯2(c-0)=u1(c+0)v¯2(c+0).
Furthermore,
(15)[u2(b)v¯1(b)-u1(b)v¯2(b)]-1σ(b1u1(b)-a1u2(b))×(sinβv¯1(b)-cosβv¯2(b))=-1σ(sinβu1(b)-cosβu2(b))(b1v¯1(b)-a1v¯2(b))=-1σu^v~¯.
Now substituting (13), (14), and (15) in (12), we obtain
(16)〈AU,V〉ℍ=〈U,AV〉ℍ.Since the operatorA is symmetric, the following orthogonality relation is valid.Corollary 3.
All the eigenvalues of the system (1)–(6) are real and to every eigenvalue λn, there corresponds a vector-valued eigenfunction unT(x,λn)=(u1n(x,λn),u2n(x,λn)). Moreover, vector-valued eigenfunctions belonging to different eigenvalues are orthogonal in the sense of
(17)〈un,um〉ℍ=∫ac‍unTu¯mdx+∫cb‍unTu¯mdx-1σu~nu^¯m=0.Remark 4.
The vector-valued eigenfunctions stated in Corollary3 are not orthogonal in the usual sense in the Hilbert space L2[a,b].


## 3. Existence of Solutions

In this section, we study the existence of the solution of the Dirac system (1) with boundary conditions (4) and transmission condition (6).Theorem 5.
The Dirac system (1) has a solution Φ(x,λ) on [a,b] satisfying boundary condition (4) and transmission condition (6). For each x, Φ(x,λ) is a vector-valued entire function of λ.Proof.
From the classical theory of differential equations (see [10]), since the Dirac system
(18)Au′(x)-P(x)u(x)=λu(x),x∈[a,c)
with the initial conditions
(19)u1(a)=cosα,u2(a)=sinα
is continuous on the interval [a,c), this system has a unique solution Φ1(x,λ)=(Φ11(x,λ),Φ21(x,λ))T which is an entire function of λ on [a,c).
Now consider the Dirac system of differential equations(20)u2′(x)-p1(x)u1(x)=λu1(x),u1′(x)+p2(x)u2(x)=-λu2(x),x∈(c,b],
and nonstandard initial conditions contain eigenparameter(21)u1(c+0)=γ-1Φ11(c-0,λ),u2(c+0)=γΦ21(c-0,λ).
Let us denote solutions of (20) by u0(x,λ)=(u10(x,λ),u20(x,λ))T in the case p1(x)=p2(x)≡0. It is clear that the vector-valued function u0(x,λ) is written as
(22)u10(x,λ)=c1cosλx+c2sinλx,u20(x,λ)=-c1sinλx+c2cosλx.
From the initial conditions (21), we obtain constants c1 and c2. Then, inserting these values into (22) and using some basic trigonometric identities, we arrive at(23)u0(x,λ)=(u10(x,λ)u20(x,λ))=(γ-1Φ11(c-0,λ)cosλ(x-(c+0))+γΦ21(c-0,λ)sinλ(x-(c+0))γ1-1Φ11(c-0,λ)sinλ(x-(c+0))+γΦ21(c-0,λ)cosλ(x-(c+0))).
By applying the method of variation of the constants as in [11, page 243], we find the following system of integral equations:(24)u(x,λ)=(u1(x,λ)u2(x,λ))=(u10(x,λ)+∫cx‍{p1(s)u1(x,λ)sinλ(s-x)-p2(s)u2(x,λ)cosλ(s-x)}dsu20(x,λ)+∫cx‍{p1(s)u1(x,λ)cosλ(s-x)+p2(s)u2(x,λ)sinλ(s-x)}ds).
In what follows, we use the method of successive approximations, which is helpful in constructing a solution of the integral equation system (24). This method requires a sequence of functions {un(x,λ)} for n=1,2,… defined as(25)un(x,λ)=(u1n(x,λ)u2n(x,λ))=(u10(x,λ)+∫cx‍{p1(s)u1n-1sinλ(s-x)-p2(s)u2n-1cosλ(s-x)}dsu20(x,λ)+∫cx‍{p1(s)u1n-1cosλ(s-x)+p2(s)u2n-1sinλ(s-x)}ds),
whereu10(x,λ) and u20(x,λ) are defined in (23). It is obvious that each of un(x,λ) is an entire function of λ for every x∈(c,b].
Set(26)zn(x,λ)=un(x,λ)-un-1(x,λ),where znT(x,λ)=(z1n(x,λ),z2n(x,λ)), and let M1=maxx∈(c,b]|p1(x)|, M2=maxx∈(c,b]|p2(x)|, M=max(M1,M2), N1(λ)=maxx∈(c,b]|u10(x,λ)|, N2(λ)=maxx∈(c,b]|u20(x,λ)|. Then,
(27)∥z1(x,λ)∥≤∫cx|p1(s)u10sinλ(s-x)-p2(s)u20cosλ(s-x)|ds+∫cx|p1(s)u10cosλ(s-x)+p2(s)u20sinλ(s-x)|ds≤2M(N1(λ)+N2(λ))(x-c),where the norm ∥·∥ can be any convenient norm in ℍ, but for the sake of presentation, we used 1-norm. Furthermore, let N1=max|λ|≤RN1(λ), N2=max|λ|≤RN2(λ), and NR=max(N1,N2) in closed contour {λ∈ℂ:|λ|≤R}; then
(28)∥z1(x,λ)∥≤2MNR(x-c).Similarly,
(29)∥z2(x,λ)∥≤∫cx|p1(s)(u11-u10)sinλ(s-x)-p2(s)(u21-u20)cosλ(s-x)|ds+∫cx|p1(s)(u11-u10)cosλ(s-x)+p2(s)(u21-u20)sinλ(s-x)|ds≤22M2NR(x-c)22,and so generally,
(30)∥zn(x,λ)∥≤2nMnNR(x-c)nn!.Now, consider the infinite series
(31)u0(x,λ)+∑k=1∞zk(x,λ).The nth partial sum of this series is un(x,λ); that is,
(32)un(x,λ)=u0(x,λ)+∑k=1nzk(x,λ).Therefore, the sequence {un(x,λ)} converges if and only if series (31) does so. In view of (30), it follows that series (31) is uniformly convergent with respect to x on (c,b] and λ in the closed contour {λ∈ℂ:|λ|≤R}. Let the sum of series (31) be Φ2(x,λ)=(Φ12(x,λ),Φ22(x,λ))T; that is,
(33)Φ2(x,λ)=u0(x,λ)+∑k=1∞zk(x,λ),and so, (32) gives
(34)limn→∞un(x,λ)=Φ2(x,λ).
Finally, we will show next that the limit functionΦ2(x,λ) satisfies (20). For this, we need to find Φ2′(x,λ). From (33),
(35)Φ2′(x,λ)=(Φ12′(x,λ)Φ22′(x,λ))=(u11′(x,λ)u21′(x,λ))+∑k=2∞(z1k′(x,λ)z2k′(x,λ)).
For the first term on the right-hand side of (35), if we take n=1 in (25), then
(36)(u11u21)=(u10u20)+∫cx‍(p1(s)sinλ(s-x)-p2(s)cosλ(s-x)p1(s)cosλ(s-x)p2(s)sinλ(s-x))×(u10u20)ds,(u11′u21′)=(u10′u20′)+∫cx‍(-λp1(s)cosλ(s-x)-λp2(s)sinλ(s-x)λp1(s)sinλ(s-x)-λp2(s)cosλ(s-x))×(u10u20)ds+(0-p2(x)p1(x)0)(u10u20);
now from (25) and the fact that (u10,u20)T is a solution of the homogeneous system, we have
(37)(u11′u21′)=(0-λλ0)(u11u21)+(0-p2(x)p1(x)0)(u10u20).
For the second term on the right-hand side of (35), it follows from (25) and (26) that
(38)(z1kz2k)=∫cx‍(p1(s)sinλ(s-x)-p2(s)cosλ(s-x)p1(s)cosλ(s-x)p2(s)sinλ(s-x))×(z1k-1z2k-1)ds
and its derivative is
(39)(z1k′z2k′)=∫cx‍(-λp1(s)cosλ(s-x)-λp2(s)sinλ(s-x)λp1(s)sinλ(s-x)-λp2(s)cosλ(s-x))×(z1k-1z2k-1)ds+(0-p2(x)p1(x)0)(z1k-1z2k-1).
In this equation
(40)∫cx‍(-λp1(s)cosλ(s-x)-λp2(s)sinλ(s-x)λp1(s)sinλ(s-x)-λp2(s)cosλ(s-x))×(z1k-1z2k-1)ds=(0-λλ0)(z1kz2k).
By using (39) and (40), the second term on the right-hand side of (35) becomes
(41)∑n=2∞(z1k′(x,λ)z2k′(x,λ))=(0-λλ0)∑k=2∞(z1kz2k)+(0-p2(x)p1(x)0)∑k=2∞(z1k-1z2k-1)=(0-λλ0)[∑k=1∞(z1kz2k)-(z11z21)]+(0-p2(x)p1(x)0)∑k=1∞(z1kz2k).
Substituting (37) and (41) into (35) gives
(42)(Φ12′(x,λ)Φ22′(x,λ))=(0-λλ0)[(u11u21)-(z11z21)]+(0-p2(x)p1(x)0)(u10u20)+(0-λλ0)∑k=1∞(z1kz2k)+(0-p2(x)p1(x)0)∑k=1∞(z1kz2k)=(0-λλ0)(u10u20)+(0-p2(x)p1(x)0)(u10u20)+(0-λ-p2(x)λ+p1(x)0)∑k=1∞(z1kz2k)=(0-λ-p2(x)λ+p1(x)0)×[(u10u20)+∑k=1∞(z1kz2k)]=(0-λ-p2(x)λ+p1(x)0)(Φ12(x,λ)Φ22(x,λ))
so that Φ2(x,λ) satisfies (20) on (c,b]. It also clearly satisfies the boundary conditions (21). As a result, the vector-valued function Φ(x,λ) defined by
(43)Φ(x,λ)={Φ1T(x,λ)=(Φ11,Φ21),x∈[a,c),Φ2T(x,λ)=(Φ12,Φ22),x∈(c,b]
satisfies the Dirac system (1), (4), and (6).Theorem 6.
For anyλ∈ℂ, the Dirac system
(44)u2′(x)-p1(x)u1(x)=λu1(x),u1′(x)+p2(x)u2(x)=-λu2(x)
has a solution
(45)Ψ(x,λ)={Ψ1T(x,λ)=(Ψ11,Ψ21),x∈[a,c),Ψ2T(x,λ)=(Ψ12,Ψ22),x∈(c,b]
on [a,c)∪(c,b] satisfying the boundary condition (5) and transmission condition (6). For each x∈[a,c)∪(c,b], Ψ(x,λ) is a vector-valued entire function of λ.Proof.
The proof of this theorem is similar to that of Theorem5 and hence is omitted.


## 4. The Eigenvalues of the Problem

We know from [11, page 194] that the Wronskians W(Φi,Ψi), (i=1,2) do not depend on x∈[a,c)∪(c,b]. They depend only on λ, and let W(Φi(x,λ),Ψi(x,λ))=:ωi(λ)(i=1,2). However, it follows from (6) that
(46)ω1(λ)=W(Φ1,Ψ1)=|Φ11(x,λ)Φ21(x,λ)Ψ11(x,λ)Ψ21(x,λ)|=Φ11(c-0,λ)Ψ21(c-0,λ)-Φ21(c-0,λ)Ψ11(c-0,λ)=γ-1Φ12(c+0,λ)γΨ22(c+0,λ)-γΦ22(c+0,λ)γ-1Ψ12(c+0,λ)=|Φ12(x,λ)Φ22(x,λ)Ψ12(x,λ)Ψ22(x,λ)|=W(Φ2,Ψ2)=ω2(λ).
Hence, we get
(47)ω1(λ)=ω2(λ)∶=ω(λ).
Here we defined a function ω(λ).Let the solutionsΦ(x,λ) and Ψ(x,λ) of (1)–(6) be defined by the initial conditions for some α,β∈[0,π)(48)Φ11(a,λ)=cosα,Φ21(a,λ)=sinα,Ψ12(b,λ)=a1+λcosβ,Ψ22(b,λ)=b1+λsinβ.
Therefore, any solution of (1)–(6) may be represented as(49)u(x,λ)={u1T(x,λ)=(c1Φ11+c2Ψ11,c1Φ21+c2Ψ21),x∈[a,c)u2T(x,λ)=(c3Φ12+c4Ψ12,c3Φ22+c4Ψ22),x∈(c,b].Applying conditions (4), (5), and (6) to solution (49) and considering the initial values (48), we obtain the following coefficients matrix of linear system equations of the variables c1, c2, c3, c4:(50)[0ω1(λ)0000ω2(λ)0Φ11(c-0,λ)Ψ11(c-0,λ)-γΦ12(c+0,λ)-γΨ12(c+0,λ)Φ21(c-0,λ)Ψ21(c-0,λ)-γ-1Φ22(c+0,λ)-γ-1Ψ22(c+0,λ)],and let us denote the determinant of this matrix byW(λ); then for every λ∈ℂ,
(51)W(λ)=-ω1(λ)ω2(λ)|Φ11(c-0,λ)Ψ11(c-0,λ)Φ21(c-0,λ)Ψ21(c-0,λ)|=-ω12(λ)ω2(λ)=-ω3(λ).Theorem 7.
The eigenvalues of the problem (1)–(6) are the zeros of the function ω(λ).Proof.
Letω(λn)=0 for any λ=λn. Then, it follows from (51) that the Wronskian of Φ2(x,λn) and Ψ2(x,λn) is zero, so that Ψ2(x,λn) is a constant multiple of Φ2(x,λn), say
(52)Ψ2(x,λn)=kΦ2(x,λn),x∈(c,b].
It follows that Ψ(x,λn) also fulfils the boundary condition (5) and, therefore, is a vector-valued eigenfunction of the problem (1)–(6) for eigenvalue λn.
Conversely, letun(x,λn) be a vector-valued eigenfunction corresponding to eigenvalue λn, but ω(λn)≠0. Then, from (51), at least one of the pair of the functions (Φ1T,Φ2T) and (Ψ1T,Ψ2T) would be linearly independent. Therefore, un(x,λn) can be expressed as
(53)un(x,λn)={C1Φ1T(x,λn)+C2Ψ1T(x,λn),x∈[a,c),D1Φ2T(x,λn)+D2Ψ2T(x,λn),x∈(c,b],
where at least one of the constants C1, C2, D1, D2 is not zero. Since un(x,λn) is a vector-valued eigenfunction corresponding to eigenvalue λn by substitution in conditions (4)–(6), we obtain a system of linear, homogeneous equations and the determinant of this system is zero. This means that W(λn)=0, and from (51), ω(λn)=0 which yields a contradiction to the assumption that ω(λn)≠0. This completes the proof.Sinceω(λ) is an entire function of λ and the eigenvalues of the problem (1)–(6) consist of the zeros of ω(λ), we have the next theorem.Theorem 8.
The Dirac system (1)–(6) has at most denumerably many eigenvalues, and these eigenvalues have no finite limit point.



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