In this section, we define our boundary value problem and state some of its properties. We consider the Dirac system
and transmission conditions
where ; the real-valued functions and are continuous in and and have finite limits and ; , ; and .
] 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
where ⊤ denotes the matrix transpose,
. For convenience, we put
Equation (2.1) can be written as
, which are defined on
and have finite limit
, we denote the functions
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.
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
, we obtain
Then from (2.2), (2.3) and transmission conditions, we have, respectively,
, it follows from the last three equations and (2.11
), (2.12) that
This contradicts the conditions and . Consequently, must be real. □
be the set of all
are absolutely continuous on
. Define the operator
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
) 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:
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
has a unique solution , which is an entire function of parameter λ for each fixed .
Now the functions
are defined in terms of
, respectively, as follows: The initial-value problem,
has a unique solution for each .
Similarly, the following problem also has a unique solution
Let us construct two basic solutions of equation (2.1
) as follows:
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 .
denote the Wronskian of
defined in [[26
], p.194], i.e.
It is obvious that the Wronskian
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
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 taking into account the initial conditions (2.17), (2.21) and (2.23), we obtain
Dividing both sides of (2.29) by
and by letting
, we arrive to the relation
We show that equation
has only simple roots. Assume the converse, i.e.
, equation (2.31
) has a double root
, say. Then the following two equations hold:
is real, then
. From (2.32) and (2.33),
Combining (2.34) and (2.30) with
, we obtain
contradicting the assumption . The other case, when , can be treated similarly and the proof is complete. □
denote the sequence of zeros of
are the corresponding eigenvectors of the operator
is symmetric, then it is easy to show that the following orthogonality relation holds:
is a sequence of eigen-vector-functions of (2.1)-(2.5) corresponding to the eigenvalues
. We denote by
the normalized eigenvectors of
satisfies (2.3)-(2.5), then the eigenvalues are also determined via
is another set of eigen-vector-functions which is related by
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:
the following estimates hold uniformly with respect to x
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
then from the estimates (2.47), (2.48) and (2.49), we get
which can be written as
Then, from (2.45) and (2.46), equation (2.50
) has the form
, equation (2.51
) obviously has solutions which, as is not hard to see, have the form
Inserting these values in (2.51), we find that
. Thus we obtain the following asymptotic formula for the eigenvalues:
Using the formulae (2.53), we obtain the following asymptotic formulae for the eigen-vector-functions