In this section, we define our boundary value problem and state some of its properties. We consider the Dirac system
(2.1)
(2.2)
(2.3)
and transmission conditions
(2.4)
(2.5)
where ; the real-valued functions and are continuous in and and have finite limits and ; , ; and .
In [24] the authors discussed problem (2.1)-(2.5) but with the condition instead of (2.3). To formulate a theoretic approach to problem (2.1)-(2.5), we define the Hilbert space with an inner product, see [19, 20],
(2.6)
where ⊤ denotes the matrix transpose,
, , . For convenience, we put
(2.7)
Equation (2.1) can be written as
(2.8)
where
(2.9)
For functions , which are defined on and have finite limit , by and , we denote the functions
(2.10)
which are defined on and , respectively.
In the following lemma, we prove that the eigenvalues of problem (2.1)-(2.5) are real.
Lemma 2.1 The eigenvalues of problem (2.1)-(2.5) are real.
Proof Assume the contrary that is a nonreal eigenvalue of problem (2.1)-(2.5). Let be a corresponding (non-trivial) eigenfunction. By (2.1), we have
Integrating the above equation through and , we obtain
(2.11)
(2.12)
Then from (2.2), (2.3) and transmission conditions, we have, respectively,
and
Since , it follows from the last three equations and (2.11), (2.12) that
(2.13)
This contradicts the conditions and . Consequently, must be real. □
Let be the set of all such that , are absolutely continuous on , , and , , , . Define the operator by
(2.14)
Thus, the operator is symmetric in . Indeed, for ,
The operator and the eigenvalue problem (2.1)-(2.5) have the same eigenvalues. Therefore they are equivalent in terms of this aspect.
Lemma 2.2 Let λ and μ be two different eigenvalues of problem (2.1)-(2.5). Then the corresponding eigenfunctions and of this problem satisfy the following equality:
(2.15)
Proof Equation (2.15) follows immediately from the orthogonality of the corresponding eigenelements:
□
Now, we construct a special fundamental system of solutions of equation (2.1) for λ not being an eigenvalue. Let us consider the next initial value problem:
(2.16)
(2.17)
By virtue of Theorem 1.1 in [25], 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 [25], we see that the problem
(2.18)
(2.19)
has a unique solution , which is an entire function of parameter λ for each fixed .
Now the functions and are defined in terms of and , , respectively, as follows: The initial-value problem,
(2.20)
(2.21)
has a unique solution for each .
Similarly, the following problem also has a unique solution :
(2.22)
(2.23)
Let us construct two basic solutions of equation (2.1) as follows:
where
(2.24)
(2.25)
Then
(2.26)
By virtue of equations (2.21) and (2.23), these solutions satisfy both transmission conditions (2.4) and (2.5). These functions are entire in λ for all .
Let denote the Wronskian of and defined in [[26], p.194], i.e.,
It is obvious that the Wronskian
(2.27)
are independent of and are entire functions. Taking into account (2.21) and (2.23), a short calculation gives
for each .
Corollary 2.3 The zeros of the functions and coincide.
Then we may take into consideration the characteristic function as
(2.28)
In the following lemma, we show that all eigenvalues of problem (2.1)-(2.5) are simple.
Lemma 2.4 All eigenvalues of problem (2.1)-(2.5) are just zeros of the function . Moreover, every zero of has multiplicity one.
Proof Since the functions and satisfy the boundary condition (2.2) and both transmission conditions (2.4) and (2.5), to find the eigenvalues of the (2.1)-(2.5), we have to insert the functions and in the boundary condition (2.3) and find the roots of this equation.
By (2.1) we obtain for , ,
Integrating the above equation through and , and taking into account the initial conditions (2.17), (2.21) and (2.23), we obtain
(2.29)
Dividing both sides of (2.29) by and by letting , we arrive to the relation
(2.30)
We show that equation
(2.31)
has only simple roots. Assume the converse, i.e., equation (2.31) has a double root , say. Then the following two equations hold:
(2.32)
(2.33)
Since and is real, then . Let . From (2.32) and (2.33),
(2.34)
Combining (2.34) and (2.30) with , we obtain
(2.35)
contradicting the assumption . The other case, when , can be treated similarly and the proof is complete. □
Let denote the sequence of zeros of . Then
(2.36)
are the corresponding eigenvectors of the operator . Since is symmetric, then it is easy to show that the following orthogonality relation holds:
(2.37)
Here is a sequence of eigen-vector-functions of (2.1)-(2.5) corresponding to the eigenvalues . We denote by the normalized eigenvectors of , i.e.,
(2.38)
Since satisfies (2.3)-(2.5), then the eigenvalues are also determined via
(2.39)
Therefore is another set of eigen-vector-functions which is related by with
(2.40)
where are non-zero constants, since all eigenvalues are simple. Since the eigenvalues are all real, we can take the eigen-vector-functions to be real-valued.
Now we derive the asymptotic formulae of the eigenvalues and the eigen-vector-functions . We transform equations (2.1), (2.17), (2.21) and (2.24) into the integral equations, see [26], as follows:
(2.41)
(2.42)
(2.43)
(2.44)
For the following estimates hold uniformly with respect to x, , cf. [[25], p.55],
(2.45)
(2.46)
(2.47)
(2.48)
Now we find an asymptotic formula of the eigenvalues. Since the eigenvalues of the boundary value problem (2.1)-(2.5) coincide with the roots of the equation
(2.49)
then from the estimates (2.47), (2.48) and (2.49), we get
which can be written as
(2.50)
Then, from (2.45) and (2.46), equation (2.50) has the form
(2.51)
For large , equation (2.51) obviously has solutions which, as is not hard to see, have the form
(2.52)
Inserting these values in (2.51), we find that , i.e., . Thus we obtain the following asymptotic formula for the eigenvalues:
(2.53)
Using the formulae (2.53), we obtain the following asymptotic formulae for the eigen-vector-functions :
(2.54)
where
(2.55)