ON THE LIPSCHITZ STABILITY OF INVERSE NODAL PROBLEM FOR DIRAC SYSTEM

Inverse nodal problem on Dirac operator is determination problem of the parameters in the boundary conditions, number m and potential function V by using a set of nodal points of a component of two component vector eigenfunctions as the given spectral data. In this study, we solve a stability problem using nodal set of vector eigenfunctions and show that the space of all V functions is homeomorphic to the partition set of all space of asymptotically equivalent nodal sequences induced by an equivalence relation. Moreover, we give a reconstruction formula for the potential function as a limit of a sequence of functions and associated nodal data of one component of vector eigenfunction. Our technique depends on the explicit asymptotic expressions of the nodal parameters and, it is basically similar to [1, 2] which is given for Sturm-Liouville and Hill’s operators, respectively.


Introduction
Inverse spectral problems have been a signi…cant research area in mathematical physics. Di¤erent methods have been proposed to recover coe¢ cient functions in di¤erential equations by using spectral data [3][4][5][6][7][8][9][10]. Generally, the spectral data have consisted of the eigenvalues and a corresponding sequence of norming constants, or two eigenvalue sequences. In 1988, McLaughlin showed that knowledge of nodal points can determine the potential function of Sturm-Liouville problem up to a constant [11]: This is so called inverse nodal problem. Numerical schemes were then given by Hald and McLaughlin [12] to reconstruct the density function of a vibrating string, the elastic modulus of a vibrating rod, the potential function in Sturm-Liouville problem. Independently, Shen et al. [13] studied the relation between nodal points and density function of string equation in 1988. Many results and reconstruction formulas have been derived about inverse nodal problem 342 E. YILM AZ, H. KOYUNBAKAN by several authors [14][15][16][17][18]. Here, we deal with the inverse nodal problem for Dirac system.
Dirac system is a modern presentation of the relativistic quantum mechanics of electrons intended to make new mathematical results accesible to a wider audience. It treats in some depth relativistic invariance of a quantum theory, self-adjointness and spectral theory, qualitative features of relativistic bound and scattering states and the external …eld problem in quantum electrodynamics, without neglecting the interpretational di¢ culties and limitations of the theory [19].
Inverse problems for Dirac system had been investigated by Moses [20], Prats and Toll [21], Verde [22], Gasymov and Levitan [23], and Panakhov [24]. It is well known that two spectra uniquely determine the matrix valued potential function in Dirac system [25]. In [26], eigenfunction expansions for one dimensional Dirac operator describing the motion of a particle in quantum mechanics were investigated. In addition, inverse spectral problems for weighted Dirac system were studied in [27].
One studied the properties of the eigenvalues and vector-valued eigenfunctions for the Dirac system with the same spectral parameter in the equations and the boundary conditions [28]. Sampling theory of signal analysis associated with Dirac systems, when the eigenvalue parameter appears linearly in the boundary conditions was investigated in [29]. One investigated a problem for the Dirac di¤erential operators in the case where an eigenparameter not only appears in the di¤erential equation but is also linearly contained in a boundary condition, and proved uniqueness theorems for inverse spectral problem with known collection of eigenvalues and normalizing constants or two spectra [30]. Other than these studies, there are many papers in literature (see [31][32][33][34][35]).
Inverse nodal problems for Dirac system had not been studied until the works of Yang and Huang [36]. They gave reconstruction formulas for one dimensional Dirac operator by using nodal datas. Later years, inverse nodal problem was solved for Dirac system under di¤erent boundary conditions [37,38]. Consider the Dirac system with boundary conditions ( cos + a 0 ) y 1 (0) + ( sin + b 0 ) y 2 (0) = 0; ( cos + a 1 ) y 1 ( ) + ( sin + b 1 ) y 2 ( ) = 0; where is a spectral parameter, (3) and V is a real valued, continuous function on [0; ]: Furthermore, m; a k ; b k (k = 0; 1); and are real constants: moreover 2 ; 2 [38]. Throughout the paper [38], Yang and Pivovarchik supposed that The properties of the eigenvalues and eigenfunctions of the problem (1)-(2) were studied in [28]. Under the condition (4), the eigenvalues of the problem (1)-(2) are real and algebraically simple [28]. Considering (3) in (1), we get 0 1 1 0 and thus, equation (1) is equivalent to a system of two simultaneous …rst order di¤erential equations In general, potential function of Dirac system (1) has the following form where p ik (x) (i; k = 1; 2) are real valued and continuous functions on [0; ]: For the case in which p 12 (x) = p 21 (x) = 0 and p 11 ( where m is the mass of particle, the system (5) is known in relativistic quantum theory as a stationary one dimensional Dirac system or …rst canonical form of Dirac system [4]. Let y(x; n ) = [y 1 (x; n ); y 2 (x; n )] T be two dimensional vector eigenfunction of the Dirac system (1) related to the eigenvalue = n , where T denotes transpose. Assume that x j;i n are the nodal points of i th component y i (x; n ) of the n th eigenfunction y(x; n ); where 0 < x 1;i n < x 2;i n < ::: < x n 1;i n < : In other words, y i (x j;i n ; n ) = 0: Let I j;i n = x j;i n ; x j+1;i n be the j th nodal domain; and let l j;i n = x j+1;i n x j;i n ; be the associated nodal length. For simplicity, we agree that x 0;i n = 0 and x jnj+1 i;i n = : We also de…ne the function j n;i (x) to be the largest index j i such that 0 x j;i n x for n > 0 and j n;i (x) to be the largest index j i such that 0 x x j;i n for n < 0: Thus, j i = j n;i (x) if and only if x 2 [x j;i n ; x j+1;i n ) for n > 0 and (x j+1;i n ; x j;i n ] for n < 0 [36]: Denote i = fx j;i n g; i = 1; 2: Hence, = 1 [ 2 is called the set of all nodal points of Dirac operator. This set is dense on [0; ] [38]: Throughout this study, we'll give all proofs for the …rst component of the eigenfunction. The rest of this study is arranged as follows: In remaining part of section 1, we give some properties of Dirac system and quote some important results to use in main theorems. In section 2, we obtain some reconstruction formulas for potential function under di¤erent boundary conditions. Finally, we de…ne d 0 , d Dir to prove Lipschitz stability of inverse nodal problem. Then, we express Theorem 14 in section 3. By this theorem, we prove the Lipschitz stability of inverse nodal problem for Dirac operator. Now, we need to remind some conclusions which are given by [38] to use in our main results. Lemma 1. [38] The spectrum of the problem (1)- (2) consists of eigenvalues f n g n2Z which are all real and algebraically simple behave asymptotically as and be the solution of (1) satisfying the condition y(0; ) = ( sin + b 0 ) cos + a 0 ; then, we have and where = jIm j : for 0 and 0 n 1; for < 0 and > 0: Moreover, uniformly with respect to j 2 f1; 2; :::; N ( ; )g, the nodal parameters of the problem (1)-(7) has the following asymptotic formulas, respectively for su¢ ciently large n; where n 6 = m p 2 + 2: Now, we consider the system (1) with boundary conditions u 1 (0) cos e + u 2 (0) sin e = 0; where 0 e ; e ; and m is positive in (3). It is well known that the spectrum of the system (1) with the boundary conditions (10) includes the eigenvalues e n ; n 2 Z which are all real and simple, and the sequence f e n g satis…es the classical asymptotic form [4], [36] Then, by successive approximations method, there hold for large e ; where [36] We can easily obtain e l j;2 n similarly as jnj ! 1 by using de…nition of nodal lengths and (16). Here, fx j;i n g, fe x j;i n g; i = 1; 2 and f n g; f e n g are the nodal sets and eigenvalues of the problems (1), (7) and (1), (12), respectively. where i = 1; 2 and n = n 2 for the problem (1), (7). We can express the similar theorem for the problem (1), (12).
Proof. It can be proved by similar method given in [1]. The fact that the function V is on the L 1 (0; ) space is used here.

Reconstruction of potential function by using nodal points
In this section, we will derive some reconstruction formulas of potential functions V and e V , where l j;i n ; e l j;i n and x j;i n ; e x j;i n are nodal lengths and nodal points for the problems (1), (7) and (1), (12), respectively. Here, all of our proofs and de…nitions will be given for the …rst component of eigenfunction (That is, for i = 1). where l j;i n = x j+1;i n x j;i n and e l j;i n = e x j+1;i n e x j;i n (j is odd or even): Then, F n and e F n converge to V and e V pointwisely almost everywhere, respectively and also in L 1 sense .Moreover, pointwise convergence holds for all the continuity points of V and e V : Proof. a) We will consider the reconstruction formula for the potential function of the problem (1), (7). Observe that, by Lemma 3, we have We may assume x j;1 n 6 = x: By Theorem 5 , if we take limit of both sides of (18) as n ! 1 for almost x 2 (0; ); we get for almost every x 2 (0; ), with j 1 = j n;1 (x): Proof. It can be proved by using similar process to Theorem 8.

Main Results
In this section, we solve a Lipschitz stability problem for Dirac operator. Lipschitz stability is about a continuity between two metric spaces. So, we have to …rst construct these spaces. To show continuity, we use a homeomorphism between these spaces. Stability problems were studied by many authors [2,39,40]. To solve stability problem, we give a main theorem which execute that the inverse nodal problem for Dirac system is stable with Lipschitz stability. Here and later, we denote the space of all admissible nodal sequences which converge to V by X = X k;i n ; where L k;i n = X k+1;i n X k;i n ; i = 1; 2: De…nition 10. Let N 0 = N f1g: We denote the space Dir of all potential functions of Dirac system and the space Dir of all admissible sequences by (i): Dir = fV 2 L 1 [0; ] : V is the potential function of the Dirac systemg; and Dir =The collection of the all double sequences de…ned as X = X k;1 n : k = 1; 2; :::; n; n 2 N 0 ; 0 < X 1;1 n < X 2;1 n < ::: < X n 1;1 n < for each n 2 N: (ii): Let X 2 Dir and de…ne X = fX k;1 n g where I k;1 n = X k;1 n ; X k+1;1 n : We say X is quasinodal to some V 2 L 1 (0; ) if X is an admissible sequence of nodes and satis…es (I) and (II) below: (I) X has the following asymptotics uniformly for k; as n ! 1 X k;1 n = k n 2 + O 1 n ; k = 1; 2; :::; n for the problem (1), (7). And the sequence ; converges to V in L 1 . X 2 Dir is nodal if X satis…es one of the above asymptotic behaviours.
We denote Dir as a collection of all Dirac operators and the space Dir as a collection of all admissible double sequences of nodes such that related functions are convergent in L 1 : A pseudometric d Dir on Dir will be de…ned: For convenience, we will use the notation X for the …rst component: Essentially, d Dir (X; X) is so close to De…ne d 0 X; X = lim n!1 S n X; X and d Dir X; X = lim n!1 S n X; X 1 + S n X; X : We get this metric by evaluating V V 1 in Theorem 14 This de…nition was …rst made by [1]. Since the function f (x) = x 1 + x is monotonic, we have admitting that if d 0 X; X = 1; then d Dir X; X = 1: Conversely We can easily prove this equality by using Law's method [1,2].
Lemma 13. Let X; X 2 Dir : a) d Dir is a pseudometric on Dir : b) If X and X belong to di¤ erent cases, d Dir X; X = 1: Proof. It can be proved similar way with in [1]. While realizing the conditions for being metric in the …rst proof, the space of all admissible nodal sequences is used in the second proof.
Stability problems for Sturm-Liouville and Hill's operators were studied in [1,2] respectively. Now, we prove the stability of the inverse nodal problem for Dirac operator with Lipschitz stability. The below theorem guarantees the Lipschitz stability of inverse nodal problem for Dirac operator.
Theorem 14. The metric spaces ( Dir ; k:k 1 ) and Dir1 = ; d Dir are homeomorphic to each other. Here, is the equivalence relation induced by d Dir : Furthermore where d Dir X; X < 1.
Proof. By Lemma 13 we only need to consider when X; X 2 Dir belong to same case. Without loss of generality, let X; X belong to case I. In this case, we should denote V V 1 = d 0 X; X : According to the Theorem 7, F n and F n converge to V and V ; respectively. If we use the de…nition of norm in L 1 for the functions V and V ; we have V V 1 = (n 2) n 2 m 2 2(n 2) Here, …rst and second terms can be written as Similarly, we can get V V 1 n 2 m 2 2(n 2) If we consider (21) and (22)  The proof is complete after taking the limit as n ! 1.

Conclusion
In this study, Lipschitz stability of inverse nodal problem is proved for Dirac operator by using zeros of the …rst component function of two dimensional vector eigenfunction. Proofs are made for the …rst component of the eigenfunction throughout the study. Such a way was followed, since the proofs for the second component have similar behavior. Especially, two metric spaces were de…ned and it was shown that they were homeomorphic to each other. These results are new and can be generalized.
Authors Contribution Statement All authors contributed equally to the article.

Declaration of Competing Interest
On behalf of all authors, the corresponding author states that there is no con ‡ict of interest.