Asymptotic behavior of eigenvalues and eigenfunctions in a non-self-adjoint problem for the Schrödinger operator with involution

Akbope Beisebayeva; Abdizhahan M. Sarsenbi; Akbope Beisebayeva; Abdizhahan M. Sarsenbi

doi:10.3934/math.2026892

AIMS Mathematics

2026, Volume 11, Issue 7: 22052-22068. doi: 10.3934/math.2026892

Previous Article Next Article

Research article

Asymptotic behavior of eigenvalues and eigenfunctions in a non-self-adjoint problem for the Schrödinger operator with involution

Department of Mathematics, M.Auezov South Kazakhstan University, Tauke Khan 5, Shymkent 160012, Kazakhstan

Received: 25 March 2026 Revised: 04 May 2026 Accepted: 12 May 2026 Published: 23 July 2026
MSC : 34K08, 35S16, 39B22

In this paper, the spectral properties of non-self-adjoint boundary value problems for linear second-order differential equations with involution and variable complex-valued coefficients are investigated. The existence of linearly independent solutions is established, and the form of the general solution of linear second-order differential equations with involution and variable complex-valued coefficients is presented. Asymptotic formulas for linearly independent solutions of the equations under consideration are obtained. Using the asymptotics of linearly independent solutions, asymptotic formulas for eigenvalues and eigenfunctions of a non-self-adjoint spectral problem for a second-order differential equation with involution are proved. Estimates for the norms of derivatives of eigenfunctions are also established.
Citation: Akbope Beisebayeva, Abdizhahan M. Sarsenbi. Asymptotic behavior of eigenvalues and eigenfunctions in a non-self-adjoint problem for the Schrödinger operator with involution[J]. AIMS Mathematics, 2026, 11(7): 22052-22068. doi: 10.3934/math.2026892

Related Papers:

Abstract

In this paper, the spectral properties of non-self-adjoint boundary value problems for linear second-order differential equations with involution and variable complex-valued coefficients are investigated. The existence of linearly independent solutions is established, and the form of the general solution of linear second-order differential equations with involution and variable complex-valued coefficients is presented. Asymptotic formulas for linearly independent solutions of the equations under consideration are obtained. Using the asymptotics of linearly independent solutions, asymptotic formulas for eigenvalues and eigenfunctions of a non-self-adjoint spectral problem for a second-order differential equation with involution are proved. Estimates for the norms of derivatives of eigenfunctions are also established.

References

[1]	J. Wiener, Generalized solutions of functional differential equations, World Scientific, 1993.
[2]	A. Cabada, F. A. F. Tojo, Differential equations with involutions, New York: Atlantis Press, 2015.
[3]	D. Przeworska-Rolewicz, Equations with transformed argument, Elsevier-PWN, Amsterdam, Warszawa, 1973.
[4]	A. Signorini, Sul moto di un punto soggelto a resistenza idraulica e forza di richiamo, Atti R. Ist. Veneto di Sc. Lett. Ed. Arti., 73 (1913), 803–858.
[5]	J. P. Van Zandt, Oscillating systems damped by resistance proportional to the square of the velocity, Phys. Rev., 10 (1917), 415–431.
[6]	E. Cartan, H. Cartan, Note sur la generation des oscillations entretenues, Ann. des PTT, 14 (1925), 1196–1207.
[7]	F. Tricomi, Integrazione di un'equazione differenziale presentatasi in Elettrotecnica, Ann. Sc. Norm. Sup. di Pisa, 2 (1933), 1–20.
[8]	G. Duffing, Erzwungene schwingungen bei veranderlicher eigenfrequenz und ihre technische bedeutung, Berlin, 1918.
[9]	L. H. Thomas, The calculation of atomic fields, Mathematical Proceedings of the Cambridge Philosophical Society, 23 (1927), 542–548. https://doi.org/10.1017/S0305004100011683 doi: 10.1017/S0305004100011683
[10]	S. F. Lacroix, Traite du calcul differential et du calcul integral, Paris, 1797.
[11]	K. L. Cooke, Differential-difference equations, New York: Academic Press, 1963.
[12]	D. N. Zubarev, Statistical mechanics of nonequilibrium processes (in Russian), Fizmatlit, Moscow, 2002.
[13]	N. M. Gersevanov, Iterative calculus and its applications (in Russian), Moscow, 1950.
[14]	M. Kirane, M. A. Sadybekov, A. A. Sarsenbi, On an inverse problem of reconstructing a subdiffusion process from nonlocal data, Math. Methods Appl. Sci., 42 (2019), 2043–2052. https://doi.org/10.1002/mma.5498 doi: 10.1002/mma.5498
[15]	E. Mussirepova, A. A. Sarsenbi, A. M. Sarsenbi, Solvability of mixed problems for the wave equation with reflection of the argument, Math. Methods Appl. Sci., 45 (2022), 11262–11271. https://doi.org/10.1002/mma.8448 doi: 10.1002/mma.8448
[16]	E. Mussirepova, A. Sarsenbi, A. Sarsenbi, The inverse problem for the heat equation with reflection of the argument and with a complex coefficient, Bound. Value Probl., 2022 (2022), 99. https://doi.org/10.1186/s13661-022-01675-1 doi: 10.1186/s13661-022-01675-1
[17]	A. Štikonas, E. Şen, Asymptotic analysis of Sturm-Liouville problem with Dirichlet and nonlocal two-point boundary conditions, Math. Model. Anal., 28 (2023), 308–329. https://doi.org/10.3846/mma.2023.17617 doi: 10.3846/mma.2023.17617
[18]	E. Şen, A. Štikonas, Asymptotic distribution of eigenvalues and eigenfunctions of a nonlocal boundary value problem, Math. Model. Anal., 26 (2021), 253–266. https://doi.org/10.3846/mma.2021.13056 doi: 10.3846/mma.2021.13056
[19]	N. S. Imanbaev, Distribution of eigenvalues of a third-order differential operator with strongly regular boundary conditions, AIP Conf. Proc., 1997 (2021), 020027. https://doi.org/10.1063/1.5049021 doi: 10.1063/1.5049021
[20]	N. S. Imanbaev, M. A. Sadybekov, Characteristic determinant of the spectral problem for the ordinary differential operator with the boundary load, AIP Conf. Proc., 1611 (2014), 261–265. https://doi.org/10.1063/1.4893844 doi: 10.1063/1.4893844
[21]	N. S. Imanbaev, On stability of basis property of root vectors system of the Sturm-Liouville operator with an integral perturbation of conditions in nonstrongly regular Samarskii-Ionkin type problems, Int. J. Differ. Equ., 2015 (2015), 641481. https://doi.org/10.1155/2015/641481 doi: 10.1155/2015/641481
[22]	A. M. Sarsenbi, Unconditional bases related to a nonclassical second-order differential operator, Diff. Equat., 46 (2010), 509–514. https://doi.org/10.1134/S0012266110040051 doi: 10.1134/S0012266110040051
[23]	M. A. Sadybekov, A. M. Sarsenbi, Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution, Diff. Equat., 48 (2012), 1112–1118. https://doi.org/10.1134/S001226611208006X doi: 10.1134/S001226611208006X
[24]	L. V. Kritskov, A. M. Sarsenbi, Equiconvergence property for spectral expansions related to perturbations of the operator –$u''$ (–x) with initial data, Filomat, 32 (2018), 1069–1078.
[25]	A. A. Sarsenbi, B. K. Turmetov, Basis property of a system of eigenfunctions of a second-order differential operator with involution, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29 (2019), 183–196. https://doi.org/10.20537/vm190204 doi: 10.20537/vm190204
[26]	B. Turmetov, V. Karachik, On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution, AIMS Math., 9 (2024), 6832–6849. https://doi.org/10.3934/math.2024333 doi: 10.3934/math.2024333
[27]	F. Dekhkonov, B. Turmetov, On one time-optimal control problem for a parabolic equation with involution in a bounded domain, AIMS Math., 10 (2025), 20531–20549. https://doi.org/10.3934/math.2025916 doi: 10.3934/math.2025916
[28]	A. Sarsenbi, A. Sarsenbi, Boundary value problems for a second-order differential equation with involution in the second derivative and their solvability, AIMS Math., 8 (2023), 26275–26289. https://doi.org/10.3934/math.20231340 doi: 10.3934/math.20231340
[29]	S. G. Georgiev, A. Boukarou, K. Zennir, K. Bouhali, Classical solutions with new topological properties for n-dimensional physical models: Schrödinger and wave equations, J. Nonlinear Convex Anal., 26 (2025), 2723 –2740.
[30]	S. G. Georgiev, S. M. Mirgani, K. Zennir, K. Bouhali, On nonnegative classical solutions of coupled Euler–Poisson–Darboux–Tricomi equations with nonlinear sources, AIMS Math., 11 (2026), 5557–5574. https://doi.org/10.3934/math.2026229 doi: 10.3934/math.2026229
[31]	S. B. Norkin, Differential equations of the second order with retarded argument: some problems of the theory of vibrations of systems with retardation, 1972.
[32]	G. Sansone, Ordinary differential equations (in Russian), Vol. 1, Moscow: IL, 1953.
[33]	M. A. Naimark, Linear differential operators, Ungar: New York, NY, USA, 1968.
[34]	V. A. Il'in, L. V. Kritskov, Properties of spectral expansions corresponding to non-self-adjoint differential operators, J. Math. Sci., 116 (2003), 3489–3550. https://doi.org/10.1023/A:1024180807502 doi: 10.1023/A:1024180807502

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)