Research Article | Open Access
Dirac System with Discontinuities at Two Points
We deal with a regular Dirac system which has discontinuities at two points and contains eigenparameter in a boundary condition and two transmission conditions. We investigate asymptotic behaviour of eigenvalues and corresponding eigenfunctions of this Dirac system and construct Green’s function.
We consider the Dirac system where , ; ; the real valued functions and are continuous in , and and have finite limits , , , ; , ; , , ; , and .
Boundary-value problems with transmission (or discontinuity) conditions inside the interval often appear in mathematical physics, mechanics, electronics, geophysics, and other branches of natural sciences (see [1, 2]). References [3–5] are examples of works with boundary conditions depending linearly on an eigenparameter and transmission conditions at the point of discontinuity for Sturm Liouville problem.
The basic and comprehensive results about Dirac operators were given in . The oscillation property and asymptotic formulas for the eigenvalues of Dirac systems were given in  and the derivative sampling theorem to compute eigenvalues of Dirac systems was used in [8, 9]. Continuous Dirac systems have been investigated in these works. Works in direction of Dirac systems with an internal point of discontinuity are few (see [10–13]). Sampling theories associated with discontinuous Dirac systems were investigated in [11, 12] (also the problem in  contains eigenparameter in a boundary condition). Direct and inverse problems for Dirac operators with discontinuity inside an interval were studied in . Dirac operators with eigenparameter dependent on both boundary conditions and one of the discontinuity conditions were investigated in . In these discontinuous works, Dirac systems had only one point of discontinuity.
We examine Dirac system which has two points of discontinuity and contains at the same time an eigenparameter in a boundary condition and two transmission conditions in this paper. Dirac system which has two points of discontinuity does not exist as far as we know. Our investigation will be a good example for discontinuous Dirac systems. Firstly, we give some spectral properties of eigenvalues and eigenfunctions and then obtain asymptotic formulas for eigenvalues and corresponding eigenfunctions. Finally, we construct Green's function of the problem (1)–(7).
2. Spectral Properties
For functions , which are defined on and have finite limit and , by , , and , we denote the functions which are defined on , , and , respectively.
Proof. Assume the contrary that is a nonreal eigenvalue of problem (1)–(7). Let be a corresponding (nontrivial) eigenfunction. By (1), we have
Integrating the above equation through , , and , we obtain
Then from boundary conditions (2)-(3) and transmission conditions (4)–(7), we have, respectively,
Since , it follows from (13)–(16) and (12) that
This contradicts the conditions and . Consequently, must be real.
One must note that, since the eigenfunctions are real valued, the left hand side in the last expression coincides with the inner product in .
Now one will construct a special fundamental system of solutions of (1). By virtue of Theorem 1.1 in , one will define two solutions of (1), where as follows. Let one consider the next initial value problem:
By virtue of Theorem 1.1 in , this problem has a unique solution , which is an entire function of for each fixed . Similarly, employing the same method as in the proof of Theorem 1.1 in , one sees that the problem has a unique solution , which is an entire function of parameter for each fixed .
Now the functions and are defined in terms of and , , , respectively, as follows. The initial value problem has a unique solution for each , and the initial value problem has a unique solution for each .
Similarly, the functions and are defined in terms of and , , , respectively, as follows. The initial value problem has a unique solution for each , and the initial value problem has a unique solution for each .
Let denote the Wronskians of and defined by
Corollary 3. The zeros of the functions , , and coincide.
Then, one may take into consideration the characteristic function as
Proof. Since the functions and satisfy the boundary condition (2) and transmission conditions (4)–(7), to find the eigenvalues of problem (1)–(7), we have to insert the functions and in the boundary condition (3) and find the roots of this equation.
By (1), we obtain, for , ,
Integrating the above equation through , , and and taking into account the initial conditions (23), (26), (28), (30), and (32), we obtain
Dividing both sides of (38) by and letting , we reach the relation
We show that equation has only simple roots. Assume the converse; that is, (40) has a double root . Then the following two equations hold:
Since and is real, then , from (41),
Combining (42) and (39), with , we obtain contradicting the assumption .
3. Asymptotic Approximate Formulas
For , the solutions and have the following asymptotic representation uniformly with respect to , , cf. : where .
By substituting the obtained asymptotic formulas for and in the definitions of , we can establish the following theorem.
Theorem 5. Let Then the characteristic function has the following asymptotic representation:
By putting and in (46), one derives that where
Lemma 6. Let be the set of real numbers satisfying the inequalities and the set of complex numbers. If , then the roots of the equation have the form where is a bounded sequence .
Proof. By using Lemma 6, has an infinite number of roots with asymptotic expression where . By applying well-known Rouche's theorem, which assets that if and are analytic inside and on a closed contour , and on , then and have the same number of zeros inside , and then we obtain
4. Green’s Function
Let be a continuous vector-valued function. Now, we derive Green's function of problem (1)–(7). Consider the inhomogeneous eigenvalue problem consisting of the differential system, and boundary conditions (2), (3) and transmission conditions (4)–(7) with are not an eigenvalue of problem (1)–(7).
Now we can represent the general solution of (56) in the following form:
We applied the standard method of variation of the constants to (57); thus, the functions and satisfy the linear system of equations
Since is not an eigenvalue and , each of the linear systems in (58) has a unique solution which leads to where are arbitrary constants and
Then (61) can be written as where