Spectral properties of the finite system of Klein-Gordon S-wave equations with condition depends on spectral parameter

Year 2025, Volume: 54 Issue: 4, 1300 - 1307, 29.08.2025

Elgiz Bayram , Esra Kır Arpat

https://doi.org/10.15672/hujms.1393132

Abstract

The spectral characteristics of the operator $L$ are studied where $L$ is defined within the Hilbert space $L_{2}(\mathbb{R}_{+}, \mathbb{C}^{V})$ given by a finite system of Klein-Gordon type differential equations and boundary condition depends on spectral parameter. The research of the Klein-Gordon type operator continues to be an important topic for researchers due to the range of applicability of them in numerous branches of mathematics and quantum physics. Contrary to the previous works, we take the potential as complex valued and generalize the problem to the matrix Klein-Gordon operator case. The spectrum is derived by determining the Jost function and resolvent operator of the prescribed operator. Further, we provide the conditions that must be met for the certain quantitative properties of the spectrum.

Keywords

eigenvalues , spectral singularities , Klein-Gordon S-wave equation

References

• [1] M. Advar and E. Bairamov, Spectral properties of non-selfadjoint difference operators, J. Math. Anal. Appl. 261 (2), 461-478, 2001.
• [2] M. Advar and E. Bairamov, Difference equations of second order with spectral singularities, J. Math. Anal. Appl. 277 (2), 714-721, 2003.
• [3] Z.S. Agranovich and V.A. Marchenko, The inverse problem of scattering theory, Gordon and Breach, New York, 1963.
• [4] E.K. Arpat, An eingenfunction expansion of the non-selfadjoint Sturm-Liouville operator with a singular potential, J. Math. Chem. 51 (8), 2196-2213, 2013.
• [5] E. Bairamov and A.O. Çelebi, Spectral properties of the Klein-Gordon s-wave equation with complex potential, Indian J. Pure Appl. Math. 28 (6), 813-824, 1997.
• [6] E. Bairamov and A.O. Çelebi, Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators, Q. J. Math., Oxf. Second Ser. 50 (200), 371-384, 1999.
• [7] E. Bairamov, Ö. Çakar and A.M. Krall, An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities, J. Dier. Equ. 151 (2), 268-289, 1999.
• [8] E. Bairamov, Ö. Çakar and A.M. Krall, Non-selfadjoint difference operators and Jacobi matrices with spectral singularities, Math. Nachr. 229, 5-14, 2001.
• [9] E. Bairamov, Ö. Çakar and A.M. Krall, Spectral properties including spectral singularities of a quadratic pencil of Schrödinger operators on the whole real axis, Quaest. Math. 26 (1), 15-30, 2003.
• [10] E. Bairamov, Ö. Çakar and C. Yank, Spectral singularities of the Klein-Gordon s-wave equation, Indian J. Pure Appl. Math. 32 (6), 851-857, 2001.
• [11] E. Bairamov and Ö. Karaman, Spectral singularities of the Klein-Gordon s-wave equations with integral boundary conditions, Acta Math. Hungar. 97 (1-2), 121-131, 2002.
• [12] E.P. Dolzhenko, Boundary value uniqueness theorems for analytic functions, Math. Notes 25, 437-442, 1979.
• [13] E. Guariglia and S. Silvestrov, Fractional-Wavelet Analysis of Positive definite Distributions and Wavelets on D’(C), Engrg. Math. II. Springer Proc. Math. Stat. 179, 337-353, 2016.
• [14] A. Hussain, S. Haq and M. Uddin, Numerical solution of Klein–Gordon and sine- Gordon equations by meshless method of lines, Eng. Anal. Bound. Elem. 37 (11), 1351-1366, 2013.
• [15] A.M. Krall, E. Bairamov and Ö. Çakar, Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition, J. Dier. Equ. 151 (2), 252-267, 1999.
• [16] V.E. Lyance, A differential operator with spectral singularities I, Trans. Amer. Math. Soc. Ser. 2 60, 185-225, 1967.
• [17] V.E. Lyance, A differential operator with spectral singularities II, Trans. Amer. Math. Soc. Ser. 2 60, 227-283, 1967.
• [18] F.G. Maksudov and B.P. Allakhverdiev, Spectral analysis of a new class of nonselfadjoint operators with continuous and point spectrum, Sov. Math. Dokl. 30, 566- 569, 1984.
• [19] M.A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of the second order on a semi-axis, Trans. Amer. Math. Soc. Ser. 2 16, 103-193, 1960.
• [20] M.A. Ragusa, Elliptic boundary value problem in vanishing mean oscillation hypothesis, Comment. Math. Univ. Carol. 40, 651-663, 1999.
• [21] M.A. Ragusa and A. Tachikawa, Boundary regularity of minimizers of p (x)-energy functionals, Ann. Inst. Henri Poincare (C) Anal. Non Linéaire 33 (2), 451-476, 2016.
• [22] J.T. Schwartz, Some nonselfadjoint operators, Comm. Pure Appl. Math. 13, 609-639, 1960.
• [23] G.B. Tunca, Spectral expansion of a non-selfadjoint differential operator on the whole axis, J. Math. Anal. Appl. 252 (1), 278-297, 2000.