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Investigation of the spectrum and the Jost solutions of discrete Dirac system on the whole axis
Journal of Inequalities and Applications volume 2014, Article number: 73 (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.
Let us consider the BVP generated by the Sturm-Liouville equation,
and the boundary condition
where q is a real valued function and λ is a spectral parameter. The bounded solution of (1) satisfying the condition
will be denoted by . The solution satisfies the integral equation
It has been shown that, under the condition
the solution has the integral representation
where the function is defined by q. The function is analytic with respect to λ in , continuous in , and
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.
Let us consider the system
where are vector sequences, , , , and are real sequences, , for all , and λ is a spectral parameter.
If for all , and , then the system (8) reduces to
where Δ is the forward difference operator, i.e.,
The system (9) is the discrete analogue of the well-known Dirac system
(, Chapter 2). Therefore the system (9) is called a discrete Dirac system.
Various problems of spectral analysis of self-adjoint difference equations have been investigated in detail [6, 7]. But all of them give an exponential type Jost solution of the difference equations. In this paper, we find a polynomial type Jost solution of (9) with the boundary condition
which is analytic in .
2 Jost solutions of (9)
We will assume that the real sequences , satisfy
If for all and
from (9), we get
It is clear that
are the solutions of (9).
Now we find the solutions , , and , , of (9) for , satisfying the condition
respectively, where .
Theorem 1 Under the condition (11) for and , (9) has the solutions and having the representations
Proof Substituting the vector-valued functions f and g defined by (17) and (18) in (9), taking , , we get
where . For and , we obtain
Also for and , we get
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). □
The solutions f and g are called Jost solutions of (9). Using the equalities for and () given in Theorem 1, we find
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.
Theorem 2 The following asymptotics hold:
Proof From (17), we obtain
Using (20) and (24), we get
where C is constant. So we have by (25)
In a manner similar to (26), we obtain
From (26) and (27), we get (22). Also from (18), we obtain
Using (21) and (28), we have
where C is constant. So, we get by (29)
Similarly, we can obtain
From (30) and (31), we obtain (23). □
3 Continuous and discrete spectrum of the BVP (9)
Let denote the Hilbert space of all complex vector sequences
with the norm
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.
Proof Let denote the operator generated in by the BVP
We also define the operator in by the following:
It is clear that , and we can easily prove that
It follows from (11) that the operator is compact in .
By the Weyl Theorem  of a compact perturbation, we get
The Wronskian of the solutions
of (8) is defined by
If we define , then F is analytic in D. Since the operator L is self-adjoint, the eigenvalues of L is real. From the definition of the eigenvalues we obtain
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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The authors declare that they have no competing interests.
Both authors completed the paper together. Both authors read and approved the final manuscript.
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Aygar, Y., Olgun, M. Investigation of the spectrum and the Jost solutions of discrete Dirac system on the whole axis. J Inequal Appl 2014, 73 (2014). https://doi.org/10.1186/1029-242X-2014-73
- Boundary Value Problem
- Continuous Spectrum
- Real Eigenvalue
- Vector Sequence
- Real Sequence