In this paper, the inverse problem of recovering the coefficient of a Dirac operator is studied from the sequences of eigenvalues and normalizing numbers. The theorem on the necessary and sufficient conditions for the solvability of this inverse problem is proved and a solution algorithm of the inverse problem is given.
MSC: 34A55, 34L40.
Keywords:Dirac operator; inverse problem; necessary and sufficient condition
In this paper, we consider the boundary value problem generated by the system of Dirac equations on the finite interval :
with boundary conditions
, are real valued functions, , , λ is a spectral parameter,
The inverse problem for the Dirac operator with separable boundary conditions was completely solved by two spectra in [1,2]. The reconstruction of the potential from one spectrum and norming constants was investigated in . For the Dirac operator, the inverse periodic and antiperiodic boundary value problems were given in [4-6]. Using the Weyl-Titschmarsh function, the direct and inverse problems for a Dirac type-system were developed in [7,8]. Uniqueness of the inverse problem for the Dirac operator with a discontinuous coefficient by the Weyl function was studied in  and discontinuity conditions inside an interval were worked out in [10,11]. The inverse problem for weighted Dirac equations was obtained in . The reconstruction of the potential by the spectral function was given in . For the Dirac operator with peculiarity, the inverse problem was found in . Inverse nodal problems for the Dirac operator were examined in [15,16]. In the case of potentials that belong entrywise to , for some , the inverse spectral problem for the Dirac operator was studied in , and in this work, not only the Gelfand-Levitan-Marchenko method but also the Krein method  was used. In the positive half line, the inverse scattering problem for the Dirac operator with discontinuous coefficient was analyzed in . Besides, in a finite interval, for Sturm-Liouville operator inverse problem has widely been developed (see [20-22]). The inverse problem of the Sturm-Liouville operator with discontinuous coefficient was worked out in [23,24] and discontinuous conditions inside an interval were obtained in . In the mathematical and physical literature, the direct and inverse problems for the Dirac operator are widespread, so there are numerous investigations as regards the Dirac operator. Therefore, we can mention the studies concerned with a discontinuity, which is close to our topic, in the references list.
In this paper, our aim is to solve the inverse problem for the Dirac operator with a piecewise continuous coefficient on a finite interval. Let and ( ) be, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2). The quantities ( ) are called spectral data. We can state the inverse problem for a system of Dirac equations in the following way: knowing the spectral data ( ) to indicate a method of determining the potential and to find necessary and sufficient conditions for ( ) to be the spectral data of a problem (1), (2). In this paper, this problem is completely solved.
We give a brief account of the contents of this paper in the following section.
Let be solution of the system (1) satisfying the initial conditions
The solution has an integral representation  as follows:
is a quadratic matrix function and is the solution of the problem
Equation (4) gives the relation between the kernel and the coefficient of (1). Let be solutions of the system (1) satisfying the initial conditions
The characteristic function of the problem (1), (2) is
where is the Wronskian of the solutions and and independent of . The zeros of the characteristic function coincide with the eigenvalues of the boundary value problem (1), (2). The functions and are eigenfunctions and there exists a sequence such that
Denote the normalizing numbers by
The following relation is valid:
where . In fact, since and are solutions of the problem (1), (2), we get
Multiplying the equations by , , , , respectively, adding them together, integrating from 0 to π and using the condition (2),
is found. From (6) as , we obtain
The following two theorems are obtained by Huseynov and Latifova in .
Theorem 1 (i) The boundary value problem (1), (2) has a countable set of simple eigenvalues ( ) where
(ii) The eigen vector-functions of problem (1), (2) can be represented in the form
(iii) The normalizing numbers of problem (1), (2) have the form
Theorem 2 (i) The system of eigen vector-functions ( ) of problem (1), (2) is complete in space .
(ii) Let be an absolutely continuous vector-function on the segment and . Then
moreover, the series converges uniformly with respect to .
(iii) For series (10) converges in ; moreover, the Parseval equality holds:
From , the following inequality holds:
where is a positive number and this inequality is valid in the domain
where ( ) are zeros of the function and δ is a sufficiently small number.
In Section 3, the fundamental equation
is derived by using the method by Gelfand-Levitan-Marchenko, where
In Section 4, we show that the fundamental equation has a unique solution and the boundary value problem (1), (2) can be uniquely determined from the spectral data. In Section 5, the result is obtained from Lemma 6 that the function defined by (3) satisfies the equation
where is the solution of the fundamental equation. In Lemma 7, using the fundamental equation, the Parseval equality
is found. We demonstrate by using Lemma 6, Lemma 9, and Lemma 10 that ( ) are spectral data of the boundary value problem (1), (2). Then necessary and sufficient conditions for the solvability of problem (1), (2) are obtained in Theorem 11. Finally, we give an algorithm of the construction of the function by the spectral data ( ).
Note that throughout this paper, denotes the transposed matrix of ϕ.
3 Fundamental equation
Theorem 3For each fixed the kernel from the representation (3) satisfies the following equation:
where and are, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2) when .
Proof According to (3) we have
It follows from (3) and (16) that
Using the last two equalities, we obtain
It is easily found by using (14) and (15) that
Let . Then according to the expansion formula (10) in Theorem 2, we obtain uniformly on
From (18), we find
It follows from (3) that
Taking into account (21) and expansion formula (10) in Theorem 2, we get
Substituting , we obtain
Now, we calculate
Using (7) and the residue theorem, we get
where is oriented counter-clockwise, N is a sufficiently large number. Taking into account the asymptotic formulas as
and the relations (, Lemma 1.3.1)
it follows from (12) and (24) that
Thus, using (17), (19), (20), (22) (23), and (25), we find
Since can be chosen arbitrarily,
is obtained. □
Lemma 4For each fixed , (13) has a unique solution .
Proof When , (13) can be rewritten as
Now, we shall prove that is invertible, i.e. has a bounded inverse in .
Consider the equation , . From this and (24),
We show that
Thus, the operator is invertible in . Therefore the fundamental equation (13) is equivalent to
and is completely continuous in . Then it is sufficient to prove that the equation
has only the trivial solution . Let be a non-trivial solution of (27). Then
It follows from (14) that
Using (21), we get
Substituting into the last two integrals, we obtain
Using the Parseval equality,
it follows from (28) that
Since the system ( ) is complete in , we have , i.e. . For invertible in , is obtained. □
Theorem 5Let and be two boundary value problems and
Proof According to (14) and (15), and . Then, from the fundamental equation (13), we have . It follows from (4) that a.e. on . □
5 Reconstruction by spectral data
Let the real numbers ( ) of the form (8) and (9) be given. Using these numbers, we construct the functions and by (14) and (15) and determine from the fundamental equation (13).
Now, let us construct the function by (3) and the function by (4). From , and have a derivative in both variables and these derivatives belong to .
Lemma 6The following relations hold:
Proof Differentiating to x and y, (13), respectively, we get
It follows from (14) and (15) that
and using the fundamental equation (13), we obtain
Multiplying (31) on the left by B and we get
and multiplying (32) on the right by B and we have
Adding (36) and (37) and using (34), we find
From (33), we get
Integrating by parts and from (35)
is obtained. Substituting (40) into (38) and dividing by , we have
Multiplying (13) on the left by in the form of (4) and adding to (41)
is obtained. Setting
we can rewrite (42) as follows:
According to Lemma 4, the homogeneous equation (43) has only the trivial solution, i.e.
Differentiating (3) and multiplying on the left by B, we have
On the other hand, multiplying (3) on the left by and then integrating by parts and using (35), we find
It follows from (45) and (46) that
Taking into account (4) and (44),
is obtained. For , from (3) we get (30). □
Lemma 7For each function , the following relation holds:
Proof It follows from (3) and (21) that
Using the expression
the fundamental equation (13) is transformed into the following form:
From (48), we get
and for the kernel we have the identity
and using (48) it is transformed into the following form:
Similarly, in view of (50), we have
According to (52),
It follows from (49) and (51) that
From (18) and the Parseval equality we obtain
Taking into account (54), we have
whence, by (52) and (53),
is obtained, i.e., (47) is valid. □
Corollary 8For any function and , the following relation holds:
Lemma 9The relation
Proof (1) Let . Consider the series
Using Lemma 6 and integrating by parts, we get
Applying the asymptotic formulas in Theorem 1, is found. Consequently the series (57) converges absolutely and uniformly on . According to (55) and (58), we have
Since is arbitrary, is obtained, i.e.
(2) Fix and assume . Then, by virtue of (59),
The system is minimal in and consequently by (3), the system is minimal in . Hence and we obtain (56). □
Lemma 10For all the equality
Proof It is easily found that
According to (56), we get
We shall prove that for any n, . Assume the contrary, i.e. there exists m such that . Then for , it follows from (60) that . On the other hand, since as
. This contradicts the condition , . Hence, for any n. From (60), we have
Thus, we get , for any n. Since
we find , and then is obtained. □
Theorem 11For the sequences ( ) to be the spectral data for a certain boundary value problem of the form (1), (2) with , it is necessary and sufficient that the relations (8) and (9) hold.
Proof Necessity of the problem is proved in . Let us prove the sufficiency. Let the real numbers ( ) of the form (8) and (9) be given. It follows from Lemma 6, Lemma 9, and Lemma 10 that the numbers ( ) are spectral data for the constructed boundary value problem . The theorem is proved. □
The algorithm of the construction of the function by the spectral data ( ) follows from the proof of the theorem:
(1) By the given numbers ( ) the functions and are constructed, respectively, by (14) and (15).
(2) The function is found from (13).
(3) is calculated by (4).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK).
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