Inverse eigenvalue problem for a class of Dirac operators with discontinuous coefficient
© Mamedov and Akcay; licensee Springer. 2014
Received: 30 November 2013
Accepted: 25 April 2014
Published: 13 May 2014
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.
KeywordsDirac operator inverse problem necessary and sufficient condition
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.
The following two theorems are obtained by Huseynov and Latifova in .
- (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(9)
Theorem 2 (i) The system of eigen vector-functions () of problem (1), (2) is complete in space .
moreover, the series converges uniformly with respect to .
where () are zeros of the function and δ is a sufficiently small number.
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
where and are, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2) when .
is obtained. □
Lemma 4 For each fixed , (13) has a unique solution .
Now, we shall prove that is invertible, i.e. has a bounded inverse in .
Since the system () is complete in , we have , i.e. . For invertible in , is obtained. □
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 .
is obtained. For , from (3) we get (30). □
is obtained, i.e., (47) is valid. □
The system is minimal in and consequently by (3), the system is minimal in . Hence and we obtain (56). □
we find , and then is obtained. □
Theorem 11 For 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. □
By the given numbers () the functions and are constructed, respectively, by (14) and (15).
The function is found from (13).
is calculated by (4).
This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK).
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