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.
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 where or with boundary conditions and transmission conditions at the inner point Here and later on, is a complex eigenvalue parameter; the functions are continuous on which have finite limits . , , are real numbers and .
2. Operator Formulation of the Problem
For convenience, we will assume that . To formulate a theoretic approach to problem (1)–(6), we define the Hilbert space with an inner product where stands for the transpose and . The constant is defined by Let be set of all , such that are absolutely continuous on , and , , have finite limits, . Now define the operator by Hence, we can rewrite the problem (1)–(6) in the operator form as Obviously, the operator 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 .
Lemma 1. The is dense in .
Proof. It is easily seen that there is no nonzero vector such that for every , . This implies , where . Therefore, is dense in .
Theorem 2. The operator is symmetric.
Proof. For each from the inner product (7) and the integration by parts, we have Since and satisfy the same boundary condition (4) at , From transmission condition (6), it follows that Furthermore, Now substituting (13), (14), and (15) in (12), we obtain
Since the operator is symmetric, the following orthogonality relation is valid.
Corollary 3. All the eigenvalues of the system (1)–(6) are real and to every eigenvalue , there corresponds a vector-valued eigenfunction ,. Moreover, vector-valued eigenfunctions belonging to different eigenvalues are orthogonal in the sense of
Remark 4. The vector-valued eigenfunctions stated in Corollary 3 are not orthogonal in the usual sense in the Hilbert space .
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 on satisfying boundary condition (4) and transmission condition (6). For each , is a vector-valued entire function of .
Proof. From the classical theory of differential equations (see [10]), since the Dirac system
with the initial conditions
is continuous on the interval , this system has a unique solution which is an entire function of on .
Now consider the Dirac system of differential equations
and nonstandard initial conditions contain eigenparameter
Let us denote solutions of (20) by in the case . It is clear that the vector-valued function is written as
From the initial conditions (21), we obtain constants and . Then, inserting these values into (22) and using some basic trigonometric identities, we arrive at
By applying the method of variation of the constants as in [11, page 243], we find the following system of integral equations:
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 for defined as
where and are defined in (23). It is obvious that each of is an entire function of for every .
Set
where , and let , , , , . Then,
where the norm can be any convenient norm in , but for the sake of presentation, we used . Furthermore, let , , and in closed contour ; then
Similarly,
and so generally,
Now, consider the infinite series
The th partial sum of this series is ; that is,
Therefore, the sequence converges if and only if series (31) does so. In view of (30), it follows that series (31) is uniformly convergent with respect to on and in the closed contour . Let the sum of series (31) be ; that is,
and so, (32) gives
Finally, we will show next that the limit function satisfies (20). For this, we need to find . From (33),
For the first term on the right-hand side of (35), if we take in (25), then
now from (25) and the fact that is a solution of the homogeneous system, we have
For the second term on the right-hand side of (35), it follows from (25) and (26) that
and its derivative is
In this equation
By using (39) and (40), the second term on the right-hand side of (35) becomes
Substituting (37) and (41) into (35) gives
so that satisfies (20) on . It also clearly satisfies the boundary conditions (21). As a result, the vector-valued function defined by
satisfies the Dirac system (1), (4), and (6).
Theorem 6. For any , the Dirac system has a solution on satisfying the boundary condition (5) and transmission condition (6). For each , is a vector-valued entire function of .
Proof. The proof of this theorem is similar to that of Theorem 5 and hence is omitted.
4. The Eigenvalues of the Problem
We know from [11, page 194] that the Wronskians , do not depend on . They depend only on , and let . However, it follows from (6) that Hence, we get Here we defined a function .
Let the solutions and of (1)–(6) be defined by the initial conditions for some Therefore, any solution of (1)–(6) may be represented as
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 , , , :
and let us denote the determinant of this matrix by ; then for every ,
Theorem 7. The eigenvalues of the problem (1)–(6) are the zeros of the function .
Proof. Let for any . Then, it follows from (51) that the Wronskian of and is zero, so that is a constant multiple of , say
It follows that also fulfils the boundary condition (5) and, therefore, is a vector-valued eigenfunction of the problem (1)–(6) for eigenvalue .
Conversely, let be a vector-valued eigenfunction corresponding to eigenvalue , but . Then, from (51), at least one of the pair of the functions and would be linearly independent. Therefore, can be expressed as
where at least one of the constants , , , is not zero. Since is a vector-valued eigenfunction corresponding to eigenvalue 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 , and from (51), which yields a contradiction to the assumption that . This completes the proof.
Since is an entire function of and the eigenvalues of the problem (1)–(6) consist of the zeros of , 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.