Investigation of the spectrum and the Jost solutions of discrete Dirac system on the whole axis
© Aygar and Olgun; licensee Springer. 2014
Received: 8 October 2013
Accepted: 23 January 2014
Published: 13 February 2014
We consider the boundary value problem (BVP) for the discrete Dirac equations
where and , are real sequences, and λ is an eigenparameter. We find a polynomial type Jost solution of this BVP. Then we investigate the analytical properties and asymptotic behavior of the Jost solution. Using the Weyl compact perturbation theorem, we prove that a self-adjoint discrete Dirac system has a continuous spectrum filling the segment . We also prove that the Dirac system has a finite number of real eigenvalues.
The functions and are called the Jost solution and Jost function of the BVP (1) and (2), respectively. These functions play an important role in the solution of inverse problems of the quantum scattering theory [1–4]. In particular, the scattering data of the BVP (1) and (2) is defined in terms of Jost solution and Jost function.
where are vector sequences, , , , and are real sequences, , for all , and λ is a spectral parameter.
(, Chapter 2). Therefore the system (9) is called a discrete Dirac system.
which is analytic in .
2 Jost solutions of (9)
are the solutions of (9).
respectively, where .
Due to the condition (11), the series in the definition of and () are absolutely convergent. Therefore, and () can uniquely be defined by and (), i.e., the system (9) for , , have the solutions given by (17) and given by (18). □
by induction, where is the integer part of and is a constant.
Using (20), (21), and the definitions of f and g, we obtain (15) and (16). Also the Jost solutions have an analytic continuation from to . Because of (11) and (20), we see that the series and are uniformly convergent in D. Similarly from (11) and (21), we see that the series and are uniformly convergent in D.
From (30) and (31), we obtain (23). □
3 Continuous and discrete spectrum of the BVP (9)
We also define the operator L generated in by (9). The operator L is self-adjoint.
Theorem 3 If the condition (11) holds, then , where denotes the continuous spectrum of L.
It follows from (11) that the operator is compact in .
where denotes the set of all eigenvalues of L.
Definition 1 The multiplicity of a zero of the function is called the multiplicity of the corresponding eigenvalue of L.
Theorem 4 Under the condition (11) the operator L has a finite number of real eigenvalues in D.
Proof To prove the theorem, we have to show that the function has a finite number of real zeros in D. The cluster points of the zeros of the analytic function F could be −i, 0 and i. Since L is a self-adjoint bounded operator its eigenvalues should be different from infinity and as z is ‘0’, the eigenvalue λ is infinity, we cannot consider ‘0’ as a zero of the function F. Also, for z is ±i, the eigenvalue λ is ±2 and D is bounded. But, as we know, ±2 are elements of the continuous spectrum of the operator L. On the other hand from the operator theory, the eigenvalues of the self-adjoint operator are not the elements of the continuous spectrum of that operator. Therefore, from the Bolzano Weierstrass Theorem the set of zeros of the function F in D are finite i.e., the operator L has a finite number of eigenvalues. □
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