Direct and inverse spectral problems for discrete Sturm-Liouville problem with generalized function potential
- Bayram Bala1,
- Abdullah Kablan1 and
- Manaf Dzh Manafov2Email author
https://doi.org/10.1186/s13662-016-0898-z
© Bala et al. 2016
Received: 14 March 2016
Accepted: 14 June 2016
Published: 30 June 2016
Abstract
In this work, we study the inverse problem for difference equations which are constructed by the Sturm-Liouville equations with generalized function potential from the generalized spectral function (GSF). Some formulas are given in order to obtain the matrix J, which need not be symmetric, by using the GSF and the structure of the GSF is studied.
Keywords
MSC
1 Introduction
The definitions and some properties of GSF are given in [1–6]. The inverse problem for the infinite Jacobi matrices from the GSF was investigated in [3–6], see also [7]. The inverse spectral problem for \(N\times N\) tridiagonal symmetric matrix has been studied in [8] and the inverse spectral problem with spectral parameter in the initial conditions has been studied in [9]. The goal of this paper is to study the almost symmetric matrix J of the form (1.1). Almost symmetric here means that the entries above and below the main diagonal are the same except the entries \(a_{M}\) and \(c_{M}\).
2 Generalized spectral function
Lemma 1
Proof
Theorem 1
Proof
If the functional Ω is applied to both sides of these equations, and using (2.15)-(2.19), we obtain for \(B_{mn}\) the following boundary value problems:
3 Inverse problem from the generalized spectral function
In this section, we solve the inverse spectral problem of reconstructing the matrix J by its GSF and we give the structure of GSF. The inverse spectral problem may be stated as follows: determine the reconstruction procedure to construct the matrix J from a given GSF and find the necessary and sufficient conditions for a linear functional Ω on \(\mathbb{C} _{2N} [ \lambda ] \), to be the GSF for some matrix J of the form (1.1). For the investigation of necessary and sufficient conditions for a given linear functional to be the GSF, we will refer to Theorems 2 and 3 in [8]. In this paper, we only find the formulas to construct the matrix J.
As a result of all discussions above, we write the procedure to construct the matrix in (1.1). In turn, in order to find the entries \(a_{n}\), \(b_{n}\), \(c_{n}\), \(d_{n}\) of the required matrix J, it suffices to know only the quantities \(\gamma _{n}\), \(\chi _{nk}\). Given the linear functional Ω which satisfies the conditions of Theorem 2 in [8] on \(\mathbb{C}_{2N} [ \lambda ] \), we can use (3.7) to find the quantities \(t_{l}\) and write down the inhomogeneous system of linear algebraic equations (3.8) with the unknowns \(\chi _{n0}, \chi _{n1},\ldots,\chi _{n,n-1}\), for every fixed \(n\in \{ 1,2,\ldots,N \} \). After solving this system uniquely and using (3.10), we find the quantities \(\gamma _{n}\) and so we obtain \(a_{n}\), \(b_{n}\), \(c_{n}\), \(d_{n}\), recalling (3.2), (3.3). Therefore, we can construct the matrix J.
In the following theorem, we will show that the GSF of Jhas a special form and we will give a structure of the GSF. Let J be a matrix which has the form (1.1) and Ω be the GSF of J. Here we characterize the structure of Ω.
Theorem 2
Proof
Now, we shall work out two examples to illustrate our formulas. In the first example, in order to determine \(\chi _{n,k}\) and \(\gamma _{n}\), we will use (3.8)-(3.10).
Example 1
In the following example, by using Theorem 3 in [8], it can be shown that the necessary and sufficient conditions for a given linear functional Ω to be the GSF hold and the matrix J can be constructed from (3.14)-(3.18).
Example 2
Notes
Declarations
Acknowledgements
The first author is thankful to The Scientific and Technological Research Council of Turkey (TUBITAK) for their support with the Ph.D. scholarship. We thank both referees for their useful suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Marchenko, VA: Expansion in eigenfunctions of non-selfadjoint singular second order differential operators. Mat. Sb. 52, 739-788 (1960) (in Russian) MathSciNetGoogle Scholar
- Rofe-Beketov, FS: Expansion in eigenfunctions of infinite systems of differential equations in the non-selfadjoint and selfadjoint cases. Mat. Sb. 51, 293-342 (1960) (in Russian) MathSciNetGoogle Scholar
- Guseinov, GS: Determination of an infinite non-selfadjoint Jacobi matrix from its generalized spectral function. Mat. Zametki 23, 237-248 (1978). English transl.: Math. Notes 23, 130–136 (1978) MathSciNetGoogle Scholar
- Guseinov, GS: The inverse problem from the generalized spectral matrix for a second order non-selfadjoint difference equation on the axis. Izv. Akad. Nauk Azerb. SSR Ser. Fiz.-Tekhn. Mat. Nauk 5, 16-22 (1978) (in Russian) MATHGoogle Scholar
- Kishakevich, YL: Spectral function of Marchenko type for a difference operator of an even order. Mat. Zametki 11, 437-446 (1972). English transl.: Math. Notes 11, 266–271 (1972) MathSciNetMATHGoogle Scholar
- Kishakevich, YL: On an inverse problem for non-selfadjoint difference operators. Mat. Zametki 11, 661-668 (1972). English transl.: Math. Notes 11, 402-406 (1972) MathSciNetGoogle Scholar
- Bohner, M, Koyunbakan, H: Inverse problems for the Sturm-Liouville difference equations. Filomat 30(5), 1297-1304 (2016) View ArticleGoogle Scholar
- Guseinov, GS: Inverse spectral problems for tridiagonal N by N complex Hamiltonians. SIGMA 5(18), 28 (2009) MathSciNetMATHGoogle Scholar
- Manafov, MD, Bala, B: Inverse spectral problems for tridiagonal N by N complex Hamiltonians with spectral parameter in the initial conditions. Adıyaman Univ. Fen Bilimleri Dergisi 3(1), 20-27 (2013) Google Scholar
- Akhmedova, EN, Huseynov, HM: On eigenvalues and eigenfunctions of one class of Sturm-Liouville operators with discontinuous coefficients. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 23(4), 7-18 (2003) MathSciNetMATHGoogle Scholar
- Akhmedova, EN, Huseynov, HM: On inverse problem for Sturm-Liouville operator with discontinuous coefficients. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. 10(1), 3-9 (2010) Google Scholar