Investigation of the Spectrum of Nonself-Adjoint Discontinuous Sturm-Liouville Operator

Year 2024, Volume: 12 Issue: 3, 119 - 130

Özge Akçay Nida Palamut Koşar

https://doi.org/10.36753/mathenot.1410536

Abstract

In this paper, we study nonself-adjoint Sturm-Liouville operator containing both the discontinuous coefficient and discontinuity conditions at some point on the positive half-line. The eigenvalues and the spectral singularities of this problem are examined and it is proved that this problem has a finite number of spectral singularities and eigenvalues with finite multiplicities under two different additional conditions. Furthermore, the principal functions corresponding to the eigenvalues and the spectral singularities of this operator are determined.

Keywords

Discontinuous coefficient, Discontinuity conditions, Eigenvalues and spectral singularities, Nonself-adjoint Sturm-Liouville operator, Principal functions

References

• [1] Gomilko, A., Pivovarchik, V.: On basis properties of a part of eigenfunctions of the problem of vibrations of a smooth inhomogeneous string damped at the midpoint. Mathematische Nachrichten. 245, 72-93 (2002).
• [2] Lapwood, F. R., Usami, T.: Free Oscillations of the Earth. Cambridge University Press, Cambridge (UK), 1981.
• [3] Shepelsky, D. G.: The inverse problem of reconstruction of medium’s conductivity in a class of discontinuous and increasing functions. Advances in Soviet Mathematics. 19, 209-231 (1994).
• [4] Willis, C.: Inverse problems for torsional modes. Geophysical Journal of the Royal Astronomical Society. 78, 847-853 (1984).
• [5] Naimark, M. A.: Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of the second order on a semi-axis. American Mathematical Society Translations Series. 16, 103-193 (1960).
• [6] Naimark, M. A.: Lineer Differential Operators II. Frederick Ungar Publishing Co., New York, 1968.
• [7] Schwartz, J. T.: Some non-self-adjoint operators. Communications on Pure and Applied Mathematics. 13, 609-639 (1960).
• [8] Marchenko, V. A.: Sturm-Liouville operators and applications. AMS Chelsea Publishing, Providence, Rhode Island, 2011.
• [9] Pavlov, B. S.: The non-self-adjoint Schrödinger operator. In spectral theory and wave processes, Topics in mathematical physics. Springer. 1, 87-114 (1967).
• [10] Adıvar, M., Akbulut, A.: Non-self-adjoint boundary-value problem with discontinuous density function. Mathematical Methods in the Applied Sciences. 33(11), 1306-1316 (2010).
• [11] Pavlov, B. S.: On the spectral theory of non-selfadjoint differential operators. Doklady Akademii Nauk SSSR. 146(6), 1267-1270 (1962).
• [12] Lyantse, V. E.: On a differential equation with spectral singularities. I. Matematicheskii Sbornik. 106(4), 521-561 (1964).
• [13] Lyantse, V. E.: On a differential equation with spectral singularities. II. Matematicheskii Sbornik. 107(1), 47-103 (1964).
• [14] Adıvar, M., Bairamov, E.: Spectral singularities of the nonhomogenous Sturm-Liouville equations. Applied Mathematics Letters. 15, 825-832 (2002).
• [15] Bairamov, E., Arpat, E. K., Mutlu, G.: Spectral properties of non-selfadjoint Sturm-Liouville operator with operator coefficient. Journal of Mathematical Analysis and Applications. 456(1), 293-306 (2017).
• [16] Gasymov, M. G., Maksudov, F. G.: The principal part of the resolvent of non-selfadjoint operators in neighborhood of spectral singularities. Functional Analysis and Its Applications. 6, 185-192 (1972).
• [17] Maksudov, F. G., Allakhverdiev, B. P.: Spectral analysis of a new class of non-selfadjoint operators with continuous and point spectrum. Sov. Math. Dokl. 30, 566-569 (1984).
• [18] Mutlu, G.: Spectral analysis of non-selfadjoint matrix Schrödinger equation on the half-line with general boundary condition at the origin. Tbilisi Mathematical Journal. 8, 227-236 (2021).
• [19] Mutlu, G., Arpat, E. K.: Spectral properties of non-selfadjoint Sturm-Liouville operator equation on the real axis. Hacettepe Journal of Mathematics and Statistics. 49(5), 1686-1694 (2020).
• [20] Yoku¸s, N., Co¸skun, N.: A note on the matrix Sturm-Liouville operators with principal functions. Mathematical Methods in the Applied Sciences. 42(16), 5362-5370 (2019).
• [21] Yokuş, N., Köprüba¸sı, T.: Spectrum of the Sturm-Liouville operators with boundary conditions polynomially dependent on the spectral parameter. Journal of Inequalities and Applications. 2015, 42 (2015).
• [22] Yokuş, N., Arpat, E. K.: Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics. 68(2), 1316-1334 (2019).
• [23] Bairamov, E., Erdal, ˙I., Yardımcı, S.: Spectral properties of an impulsive Sturm-Liouville operator. Journal of Inequalities and Applications. 2018, 191 (2018).
• [24] Akçay, Ö.: On the investigation of a discontinuous Sturm-Liouville operator of scattering theory. Mathematical Communications. 27(1), 33-45 (2022).
• [25] Privalov, I. I.: The Boundary Properties of Analytic Functions. Hochshulbücher für Mathematics. 25. VEB Deutscher Verlag, 1956.
• [26] Dolzhenko, E. P.: Boundary value uniqueness theorems for analytic functions. Mathematical Notes. 26, 437-442 (1979).