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INVERSE PROBLEM FOR A CLASS OF DIRAC OPERATORS BY THE WEYL FUNCTION.

1. INTRODUCTION

Let

[mathematical expression not reproducible]

be the well-known Pauli-matrices which has these properties: [[sigma].sup.2.sub.i] = I, (I is 2 x 2 identity matrix) [[sigma].sup.*.sub.i] = [[sigma].sub.i] (self-adjointness) i = 1, 2, 3 and for i [not equal to] j, [[sigma].sub.i][[sigma].sub.j] = -[[sigma].sub.j][[sigma].sub.i] (anticommutativity).

We consider the following boundary value problem generated by the canonical Dirac system

(1.1) By' + [OMEGA](x) y = [lambda] [rho](x) y, 0 < x < [pi]

with boundary conditions

(1 2) [mathematical expression not reproducible],

where

[mathematical expression not reproducible],

p(x), q(x) are real measurable functions, p(x) [member of] [L.sub.2](0, [pi]), q(x) [member of] [L.sub.2](0, [pi]), [lambda] is a spectral parameter,

[mathematical expression not reproducible],

and 1 [not equal to] [alpha] > 0. Let us define [k.sub.1] = [b.sub.1][b.sub.4] - [b.sub.2][b.sub.3] > 0 and [k.sub.2] = [c.sub.1][c.sub.4] - [c.sub.2][c.sub.3] > 0. The main aim of this paper is to solve the inverse problem for the boundary value problem (1.1), (1.2) by Weyl function on a finite interval.

The inverse problem and the spectral properties of Dirac operators were investigated in detail by many authors [1, 2, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 25, 26, 27, 28, 29, 30, 31, 32]. The inverse spectral problems according to two spectra was solved in [10]. Using Weyl-Titchmarsh function, direct and inverse problems for Dirac type-system were studied in [8, 9, 26]. Solution of the inverse quasiperiodic problem for Dirac system was given in [25]. For weighted Dirac system, inverse spectral problems was examined in [28]. Reconstruction of Dirac operator from nodal data was carried out in [30]. Necessary and sufficient conditions for the solution of Dirac operators with discontinuous coefficient was obtained in [20]. Inverse problem for interior spectral data of the Dirac operator was given in [23]. For Dirac operator, Ambarzumian-type theorems were proved in [14, 31]. On a positive half line, inverse scattering problem for a system of Dirac equations of order 2n was completely solved in [12] and when boundary condition contained spectral parameter, for Dirac operator, inverse scattering problem was worked in [6, 21]. Spectral boundary value problem in a 3 dimensional bounded domain for the Dirac system was studied in [2]. The applications of Dirac differential equations system has been widespread in various areas of physics, such as [3, 4, 24, 27].

This paper is organized as follows: in section 2, the operator formulation of problem (1.1), (1.2) and some spectral properties of the operator are given. In section 3, asymptotic formula of eigenvalues of the problem (1.1), (1.2) is examined. In section 4, Weyl solution, Weyl function are defined and uniqueness theorem for inverse problem according to Weyl function is proved.

2. OPERATOR FORMULATION AND SOME SPECTRAL PROPERTIES

An inner product in Hilbert space [H.sub.[rho]] = [L.sub.2,[rho]](0, [pi]; [C.sup.2]) [direct sum] [C.sup.2] is given by

(2.1) [mathematical expression not reproducible],

where

[mathematical expression not reproducible].

Let us define the operator L:

[mathematical expression not reproducible]

with domain

[mathematical expression not reproducible]

where

l(y) = 1/[rho](x) {By' + [OMEGA](x)y}.

Consequently, the boundary value problem (1.1), (1.2) is equivalent to the operator equation LY = [lambda]Y.

Lemma 2.1. (i) The eigenvector functions corresponding to different eigenvalues are orthogonal.

(ii) The eigenvalues of the operator L are real valued.

Let [mathematical expression not reproducible] be solutions of the system (1.1) satisfying the initial conditions

[mathematical expression not reproducible].

The characteristic function of the problem (1.1), (1.2) is defined by

(2.2) [DELTA]([lambda]) = W[[phi](x, [lambda]), [psi](x, [lambda])] = [[phi].sub.2](x, [lambda])[[psi].sub.1](x, [lambda]) - [[phi].sub.1](x, [lambda])[[psi].sub.2](x, [lambda]),

where W[[phi](x, [lambda]), [psi](x, [lambda])] is Wronskian of the vector solutions [phi](x, [lambda]) and [psi](x, [lambda]). The Wronskian does not depend on x. It follows from (2.2) that

[DELTA]([lambda]) = [b.sub.2][[psi].sub.1] (0, [lambda]) + [b.sub.1][[psi].sub.2] (0, [lambda]) - [lambda]([b.sub.4][[psi].sub.1] (0, [lambda]) + [b.sub.3][[psi].sub.2](0, [lambda])) = [U.sub.1]([psi])

or

[DELTA]([lambda]) = -[c.sub.1][[phi].sub.2] ([pi], [lambda]) - [c.sub.2][[phi].sub.1] ([pi], [lambda]) - [lambda]([c.sub.3][[phi].sub.2] ([pi], [lambda]) + [c.sub.4][[phi].sub.1] ([pi, [lambda])) = -[U.sub.2]([phi]).

Lemma 2.2. The zeros [[lambda].sub.n] of characteristic function coincide with the eigenvalues of the boundary value problem (1.1), (1.2). The function [phi](x, [[lambda].sub.n]) and [psi](x, [[lambda].sub.n] are eigenfunctions and there exist a sequence [[beta].sub.n] such that

[psi](x, [[lambda].sub.n]) = [[beta].sub.n][phi](x, [[lambda].sub.n]), [[beta].sub.n] [not equal to] 0.

Proof. This lemma is proved by a similar way in [7] (see Theorem 1.1.1).

Norming constants are defined as follows:

(2.3) [mathematical expression not reproducible].

Lemma 2.3. The following relation is valid:

[[alpha].sub.n][[beta].sub.n] = [??]([[lambda].sub.n]),

where [??]([lambda]) = d/d[lambda] [DELTA]([lambda]).

Proof. Since [phi](x, [lambda]) and [psi](x, [lambda]) are solutions of this problem, we have

[mathematical expression not reproducible].

Multiplying the equations by [[phi]'.sub.1](x, [[lambda].sub.n]), [[phi]'.sub.2](x, [[lambda].sub.n]), - [[psi]'.sub.1](x, [lambda]), - [[psi]'.sub.2](x, [lambda]) respectively and adding them together, we get

[mathematical expression not reproducible].

Integrating it from 0 to [pi],

[mathematical expression not reproducible]

is found. Now, we add

[mathematical expression not reproducible]

in the both sides of last equation and use the boundary condition (1.2). It follows that

[mathematical expression not reproducible].

According to Lemma 2.2, since [psi](x, [[lambda].sub.n]) = [[beta].sub.n][phi](x, [[lambda].sub.n]), as [lambda] [right arrow] [[lambda].sub.n], we obtain

[[beta].sub.n][[alpha].sub.n] = [??]([[lambda].sub.n])

3. ASYMPTOTIC FORMULA OF EIGENVALUES

Lemma 3.1. The solution [mathematical expression not reproducible] has the following integral representation

(3.1) [mathematical expression not reproducible],

(3.2) [mathematical expression not reproducible],

where

[mathematical expression not reproducible],

and [mathematical expression not reproducible].

Proof. To obtain the form of [phi](x, [lambda]), we use the integral representation for the solution of equation (1.1) [17]. This representation is not operator transformation and as follows: Assume that

[[integral].sup.[pi].sub.0] [parallel][OMEGA](x)[parallel] dx < +[infinity]

is satisfied for Euclidean norm of matrix function [OMEGA](x). Then the integral representation of the solution of equation (1.1) satisfying the initial condition E(0, [lambda]) = I, (I is unite matrix) can be represented

E(x, [lambda]) = [e.sup.-[lambda]B[mu](x)] + [[integral].sup.[mu](x).sub.- [mu](x)] K(x, t) [e.sup.-[lambda]Bt] dt,

where

[mathematical expression not reproducible],

and for a kernel K(x, t) the inequality

[[integral].sup.[mu](x).sub.-[mu](x)] [parallel]K(x, t)[parallel] dt [less than or equal to] [e.sup.[sigma](x)] - 1, [sigma](x) = [[integral].sup.x.sub.0] [parallel][OMEGA](s)[parallel] ds

holds. Moreover, if [DELTA](x) is differentiable, then K(x, t) satisfy the following relations

[mathematical expression not reproducible]

Now, to find [phi](x, [lambda]), we will use [mathematical expression not reproducible]. From the expression of E(x, [lambda])

(3.3) [mathematical expression not reproducible]

can be written. Then

[mathematical expression not reproducible].

Similar to

[mathematical expression not reproducible].

Putting these equalities into (3.3), we obtain (3.1) and (3.2). Moreover, as [absolute value of ([lambda])] [right arrow] [infinity] uniformly in x [member of] [0, [pi]], the following asymptotic formulas hold:

(3.4) [[phi].sub.1](x, [lambda]) = [lambda]([b.sub.3] cos [lambda][mu] (x) + [b.sub.4] sin [lambda][mu] (x)) + O ([e.sup.[absolute value of (Im[lambda])][mu](x)]),

(3.5) [[phi].sub.2](x, [lambda]) = [lambda]([b.sub.3] cos [lambda][mu] (x) + [b.sub.4] sin [lambda][mu] (x)) + O ([e.sup.[absolute value of (Im[lambda])][mu](x)]).

In fact, integrating by parts the integrals involved in (3.1) and (3.2) and also from [absolute value of (sin [lambda][mu] (x))] [less than or equal to] ([e.sup.[absolute value of (Im[lambda])][mu](x)] and [absolute value of (cos [lambda][mu](x))] [less than or equal to] ([e.sup.[absolute value of (Im[lambda])][mu](x)], the asymptotic formulas (3.4) and (3.5) are found.

Lemma 3.2. The eigenvalues [[lambda].sub.n], (n [member of] Z) of the boundary value problem (1.1), (1.2) are in the form

[[lambda].sub.n] = [[??].sub.n] + [[epsilon].sub.n],

where

[[??].sub.n] = [n + 1/[pi] arctan ([c.sub.3][b.sub.4] - [c.sub.4][b.sub.3]/[b.sub.3][c.sub.3] + [c.sub.4][b.sub.4])] [pu]/[mu](pi)

and {[[epsilon].sub.n]} [member of] [l.sub.2]. Moreover, the eigenvalues are simple.

Proof. Substituting asymptotic formulas (3.4) and (3.5) into the expression (2.2), we have

(3.6) [DELTA]([lambda]) = [[lambda].sup.2][chi] ([lambda]) + O ([absolute value of ([lambda])] ([e.sup.[absolute value of (Im[lambda])][mu](x)]),

where

[chi]([lambda]) = [c.sub.3][b.sub.4] cos [[lambda].sub.[mu]] ([pi]) - [b.sub.3][c.sub.3] sin [lambda][mu] ([pi]) - [c.sub.4][b.sub.3] cos [lambda][mu] ([pi]) - [b.sub.4][c.sub.4] sin [lambda][mu] ([pi]).

Denote

[G.sub.[delta]] := {[lambda] : [absolute value of ([lambda] - [[??].sup.n])] [greater than or equal to] [delta], n = 0, [+ or -] 1, [+ or -] 2, ...},

where [delta] is a sufficiently small positive number. For [lambda] [member of] [G.sub.[delta],

(3.7) [absolute value of ([chi] ([lambda]))] [greater than or equal to] [C.sub.[delta]] exp([absolute value of (Im[lambda])] [mu]([pi]))

is valid, where [C.sub.[delta]] is a positive number. This inequality is similarly obtained as in

[22, Lemma 1.3.2]. On the other hand, there exists a constant C > 0 such that

(3.8) [absolute value of ([DELTA]([lambda]) - [[lambda].sup.2][chi] ([lambda]))] [less than or equal to] C [absolute value of ([lambda])] ([e.sup.[absolute value of (Im[lambda])][mu](x)].

Therefore on infinitely expanding contours

[[GAMMA].sub.n] := {[lambda] : [absolute value of ([lambda])] = [[??].sub.n] + [pi]/2[mu]([pi]), n = 0, [+ or -] 1, [+ or -] 2, ...},

for sufficiently large n, using (3.7) and (3.8) we get

[absolute value of ([DELTA]([lambda]) - [[lambda].sup.2][chi]([lambda]))] < [[absolute value of ([lambda])].sup.2] [absolute value of ([chi] ([lambda]))], [lambda] [member of] [[GAMMA].sub.n].

Applying the Rouche theorem, it is obtained that the number of zeros of the function {[DELTA]([lambda]) - [[lambda].sup.2][chi] ([lambda])} + [[lambda].sup.2][chi] ([lambda]) = [DELTA]([lambda]) inside the counter [[GAMMA].sub.n] coincides with the number of zeros of function [[lambda].sup.2][chi]([lambda]). Moreover, using the Rouche theorem, there exist only one zero [[lambda].sub.n] of the function [DELTA]([lambda]) in the circle [[gamma].sub.n]([delta]) = {[lambda] : [absolute value of ([lambda] - [[??].sub.n])] < [delta]} is concluded. Since [delta] > 0 is arbitrary, we have

(3.9) [mathematical expression not reproducible].

Substituting (3.9) into (3.6), we get sin [[epsilon].sub.n][mu]([pi]) = O(1/n). It follows that [[epsilon].sub.n] =

O(1/n). Thus [[epsilon].sub.n] [member of] [l.sub.2] is found. Moreover, the eigenvalues are simple. In fact, since [[alpha].sub.n][[beta].sub.n] = [??]([[lambda].sub.n]) and [[alpha].sub.n] [not equal to] 0, [[beta].sub.n] [not equal to] 0, we get [??]([[lambda].sub.n]) [not equal to] 0.

4. UNIQUENESS THEOREM BY WEYL FUNCTION

In this section, we define Weyl function and Weyl solution. Uniqueness theorem for inverse problem according to Weyl function is proved.

Denote by [mathematical expression not reproducible] the solution of the system (1.1), satisfying the conditions

[mathematical expression not reproducible].

The function [PHI](x, [lambda]) is called Weyl solution of the problem (1.1), (1.2). Let the function [mathematical expression not reproducible] be the solution of system (1.1), satisfying the initial condition

[C.sub.1](0, [lambda]) = - [b.sub.3]/[k.sub.1], [C.sub.2](0, [lambda]) = [b.sub.4]/[k.sub.1].

As in Lemma 3.1, it is obtained that [mathematical expression not reproducible] has the following integral representation

[mathematical expression not reproducible],

where [mathematical expression not reproducible]. The solution [psi](x, [lambda]) can be shown that

(4.1) = [psi](x, [lambda])/[DELTA]([lambda]) = C(x, [lambda]) - ([b.sub.4][[psi].sub.1](0, [lambda]) + [b.sub.3][[psi].sub.2](0, [lambda]))/[k.sub.1][DELTA]([lambda]) [phi](x, [lambda]).

Denote

(4.2) M([lambda]) := - ([b.sub.4][[psi].sub.1](0, [lambda]) + [b.sub.3][[psi].sub.2](0, [lambda]))/[k.sub.1][DELTA]([lambda]).

It is obvious that

(4.3) [PHI](x, [lambda]) = C(x, [lambda]) + M([lambda])[phi](x, [lambda]).

The function

M([lambda]) = - ([b.sub.4][[PHI].sub.1](0, [lambda]) + [b.sub.3][[PHI].sub.2](0, [lambda]))/[k.sub.1]

is called the Weyl function of the boundary value problem (1.1), (1.2). The Weyl solution and Weyl function are meromorphic functions having simple poles at points [[lambda].sub.n] eigenvalues of problem (1.1), (1.2). It is obtained from (4.1) and (4.3) that

(4.4) [PHI](x, [lambda]) = [psi](x, [lambda])/[DELTA]([lambda]).

Theorem 4.1. For the Weyl function M([lambda]), the following representation holds:

(4.5) M([lambda]) = [[infinity][paragraph].summation over (n=-[infinity])] 1/[[alpha].sub.n]([lambda] - [[lambda].sub.n]).

Proof. Since

[mathematical expression not reproducible],

we can rewrite the Weyl function (4.2) as follows

M([lambda]) = ([c.sub.1] + [lambda][c.sub.3]) [C.sub.2]([pi], [lambda]) + ([c.sub.2] + [lambda][c.sub.4]) [C.sub.1]([pi], [lambda])/[DELTA]([lambda]).

Using the expression of solution C(x, [lambda]) and taking into account

(4.6) [absolute value of ([DELTA]([lambda]))] [greater than or equal to] [[absolute value of ([lambda])].sup.2] [C.sub.[delta]] exp([absolute value of (Im[lambda])] [mu]([pi])),

we have

(4.7) [mathematical expression not reproducible].

Since [psi](x, [[lambda].sub.n]) = [[beta].sub.n][phi](x, [[lambda].sub.n]),

[[beta].sub.n] = - [b.sub.4][[psi].sub.1] (0, [[lambda].sub.n]) + [b.sub.3][[psi].sub.2](0, [[lambda].sub.n])/[k.sub.1].

Then, we get

(4.8) [mathematical expression not reproducible].

Consider the following contour integral

[mathematical expression not reproducible],

where

[[GAMMA].sub.N] = {[lambda] : [absolute value of ([lambda])] = (N + 1/[pi] arctan ([c.sub.3][b.sub.4] - [c.sub.4][b.sub.3]/[b.sub.3][c.sub.3] + [c.sub.4][b.sub.4])) [pu]/[mu]([pi]) + [pi]/2[mu]([pi])}.

It follows from (4.7) that [lim.sub.N[right arrow][infinity]] [I.sub.N]([lambda]) = 0. On the other hand, applying Residue theorem and the residue (4.8),

[mathematical expression not reproducible]

is found. Thus, as N [right arrow] [infinity]

M([lambda]) = [[infinity].summation over (n=-[infinity])] 1/[[alpha].sub.n]([lambda] - [[lambda].sub.n])

is obtained.

Now, we seek inverse problem of the reconstruction of the problem (1.1), (1.2) by Weyl function M ([lambda]) and spectral data {[[lambda].sub.n], [[lambda].sub.n]}, (n [member of] Z). Along with problem (1.1), (1.2), we consider a boundary value problem of the same form, but with another potential function [??](x). Let's agree to that if some symbol s denotes an object relating to the problem (1.1), (1.2), then [??] will denote an object, relating to the boundary value problem with the function [??](x).

Theorem 4.2. If M([lambda]) = [??]([lambda]), then [OMEGA](x) = [??](x), i.e. the boundary value problem (1.1), (1.2) is uniquely determined by the Weyl function.

Proof. We describe the matrix P(x, [lambda]) = [[[P.sub.ij](x, [lambda])].sub.i,j=1,2] with the formula

(4.9) [mathematical expression not reproducible].

The Wronskian of the solutions [mathematical expression not reproducible] is

(4.10) [mathematical expression not reproducible].

Using (4.9) and (4.10), we calculate

(4.11) [mathematical expression not reproducible]

and

(4.12) [mathematical expression not reproducible].

Taking into account (4.4), (4.10) and (4.11),

[mathematical expression not reproducible]

are found. Using (4.6), we obtain

(4.13) [mathematical expression not reproducible].

Substituting (4.3) into (4.11), we have

[mathematical expression not reproducible] .

Hence, if M([lambda]) [equivalent to] [??]([lambda]), [P.sub.ij][(x, [lambda]).sub.i,j=1,2] are entire functions with respect to [lambda] for every fixed x. Then from (4.13), we find

[P.sub.11](x, [lambda]) [equivalent to] 1, [P.sub.12](x, [lambda]) [equivalent to] 0,

[P.sub.21](x, [lambda]) [equivalent to] 0, [P.sub.22](x, [lambda]) [equivalent to] 1.

Substituting these identities into (4.12),

[mathematical expression not reproducible]

are obtained for all x and [lambda], so [OMEGA](x) [equivalent to] [??](x).

According to (4.5), the specification of the Weyl function M([lambda]) is equivalent to the specification of the spectral data {[[lambda].sub.n], [[alpha].sub.n]}, n [member of] Z. That is, if [mathematical expression not reproducible] for all n [member of] Z, M([lambda]) = [??]([lambda]) is obtained. It follows from Theorem 4.2 that [OMEGA](x) = [??](x). We have thus proved the following theorem:

Theorem 4.3. The boundary value problem (1.1), (1.2) is uniquely determined by spectral data {[[lambda].sub.n], [[alpha].sub.n]}, n [member of] Z.

Received April 11, 2016

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KHANLAR R. MAMEDOV AND OZGE AKCAY

Mathematics Department, Mersin University, 33343, Mersin, Turkey; Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141, Baku, Azerbaijan. Mathematics Department, Mersin University, 33343, Mersin, Turkey.
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