On a boundary value problem with symmetric double well potential and spectral parameter in the boundary condition

Year 2024, Volume: 42 Issue: 2, 572 - 577, 30.04.2024

Elif Başkaya

Abstract

The asymptotic expansion of the eigenvalue of Sturm-Liouville problem is presented. The problem has a symmetric double well potential that is continuous, symmetrical to both the midpoint and quarter point of the related interval and non-increasing on the quarter interval.

Keywords

The Boundary Value Problem, Symmetric Double Well Potential, Eigenvalue Parameter İn The Boundary Condition, Asymptotic Expansion

References

• REFERENCES
• [1] Ak T, Karakoc SBG, Biswas A. Numerical scheme to dispersive shallow water waves, J Comput Theor Nanosci 2016;13:7084–7092. [CrossRef]
• [2] Ali KK, Karakoc SBG, Rezazadeh H. (2020) Optical soliton solutions of the fractional perturbed nonlinear Schrodinger equation. TWMS J Appl Eng Math 2020;10:930939. [CrossRef]
• [3] Başkaya E. Asymptotics of eigenvalues for Sturm-Liouville problem with eigenparameter-dependent boundary conditions. New Trends Math Sci 2018;6:247257. [CrossRef]
• [4] Başkaya E. Asymptotics of eigenvalues for Sturm-Liouville problem including eigenparameter-dependent boundary conditions with integrable potential. New Trends Math Sci 2018;6:3947. [CrossRef]
• [5] Başkaya E. Asymptotics of eigenvalues for Sturm-Liouville problem including quadratic eigenvalue in the boundary condition. New Trends Math Sci 2018;6:7682. [CrossRef]
• [6] Başkaya E. Periodic and semi-periodic eigenvalues of Hill’ s equation with symmetric double well potential, TWMS J Appl Eng Math 2020;10:346352.
• [7] Başkaya E. Asymptotics of eigenvalues for regular Sturm-Liouville problems with spectral parameter-dependent boundary conditions and symmetric single well potential. Turk J Math Comput Sci 2021;13:4450. [CrossRef]
• [8] Coşkun H, Başkaya E. Asymptotics of eigenvalues of regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition for integrable potential. Math Scand 2010;107:209223. [CrossRef]
• [9] Coşkun H, Bayram N. Asymptotics of eigenvalues for regular Sturm–Liouville problems with eigenvalue parameter in the boundary condition. J Math Anal Appl 2005;306:548566. [CrossRef]
• [10] Coşkun H, Kabataş A. Asymptotic approximations of eigenfunctions for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition for integrable potential. Math Scand 2013;113:143160. [CrossRef]
• [11] Coşkun H, Kabataş A. Green’s function of regular Sturm-Liouville problem having eigenparameter in one boundary condition. Turk J Math Comput Sci 2016;4:1–9.
• [12] Coşkun H, Kabataş A, Başkaya E. On Green’s function for boundary value problem with eigenvalue dependent quadratic boundary condition. Bound Value Probl 2017;71. [CrossRef]
• [13] Coşkun H, Başkaya E, Kabataş A. Instability intervals for Hill’s equation with symmetric single well potential. Ukr Math J 2019;71:977983. [CrossRef]
• [14] Fulton C. Two point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc R Soc Edinb A: Math 1977;77:293308. [CrossRef]
• [15] Fulton C. Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions. Proc R Soc Edinb A: Math 1980;87:134. [CrossRef]
• [16] Guliyev NJ. Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, J Math Phys 2019;60:063501. [CrossRef]
• [17] Haaser NB, Sullivian JA. Real analysis (Dover Books on Mathematics). 1st ed. New York: Dover Publications;1991.
• [18] Hinton DB. Eigenfunction expansions for a singular eigenvalue problem with eigenparameter in the boundary condition. SIAM J Math Anal 1981;12:572584. [CrossRef]
• [19] Huang MJ. The first instability interval for Hill equations with symmetric single well potentials. Proc Am Math Soc 1997;125:775778. [CrossRef]
• [20] Kabataş A. Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential. Ukr Mathl J 2022;74:218231. [CrossRef]
• [21] Kabataş A. On eigenfunctions of Hill' s equation with symmetric double well potential. Commun Fac Sci Univ Ankara Ser A1 Math Stat 2022;71:634649. [CrossRef]
• [22] Kabataş A. One boundary value problem including a spectral parameter in all boundary conditions. Opusc Math 2023;43:651661. [CrossRef]
• [23] Karakoc SBG, Saha A, Sucu D. A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: Generalized Korteweg-de Vries equation. Chin J Phys 2020;68:605617. [CrossRef]
• [24] Karakoc SBG, Neilan M. A C0 finite element method for the biharmonic problem without extrinsic penalization. Numer Methods Partial Differ Equ 2014;30:12541278. [CrossRef]
• [25] Roncaratti LF, Aquilanti V. Whittaker–Hill equation, Ince polynomials and molecular torsional modes. Int J Quantum Chem 2010;110:716730. [CrossRef]
• [26] Walter J. Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math Z 1973;133:301312. [CrossRef]