On eigenfunctions of Hill's equation with symmetric double well potential

Year 2022, Volume: 71 Issue: 3, 634 - 649, 30.09.2022

Ayşe Kabataş

https://doi.org/10.31801/cfsuasmas.974409

Cited By: 2

Abstract

Throughout this paper the asymptotic approximations for eigen- functions of eigenvalue problems associated with Hill’s equation satisfying periodic and semi-periodic boundary conditions are derived when the potential is symmetric double well. These approximations are used to determine the Green’s functions of the related problems. Then, the obtained results are adapted to the Whittaker-Hill equation which has the symmetric double well potential and is widely investigated in the literature.

Keywords

Hill's equation, symmetric double well potential, periodic and semi-periodic eigenfunctions, Green's functions, asymptotics

References

• Arscott, F. M., The Whittaker-Hill equation and the wave equation in paraboloidal coordinates, Proc. R. Soc. Edinb. A: Math., 67(4) (1967), 265–276. https://doi.org/10.1017/S008045410000813X
• Bartuccelli, M., Gentile, G., Wright, J. A., On a class of Hill’s equations having explicit solutions, Appl. Math. Lett., 26(10) (2013), 1026–1030. https://doi.org/10.1016/j.aml.2013.05.005
• Başkaya, E., On the gaps of Neumann eigenvalues for Hill’s equation with symmetric double well potential, Tbil. Math. J., 8 (2021), 139-145.
• Başkaya, E., Periodic and semi-periodic eigenvalues of Hill’s equation with symmetric double well potential, TWMS J. of Apl. Eng. Math., 10(2) (2020), 346–352.
• Bender, C. M., Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.
• Bognar, G., Periodic and antiperiodic eigenvalues for half-linear version of Hill’s equation, International Journal of Mathematical Models and Methods in Applied Sciences, 2(1) (2007), 33–37.
• Cabada, A., Cid, J. A., On comparison principles for the periodic Hill’s equation, J. London Math. Soc., 86(1) (2012), 272–290. https://doi.org/10.1112/jlms/jds001
• Cabada, A., Cid, J. A., Lopez-Somoza, L., Green’s functions and spectral theory for the Hill’s equation, Appl. Math. Comput., 286 (2016), 88–105. https://doi.org/10.1016/j.amc.2016.03.039
• Cabada, A., Cid, J. A., Lopez-Somoza, L., Maximum Principles for the Hill’s Equation, Academic Press, 2018.
• Casperson, L. W., Solvable Hill equation, Phys. Rev. A., 30 (1984), 2749– 2751. https://doi.org/10.1103/PhysRevA.30.2749
• Coşkun, H., Başkaya, E., Kabataş, A., Instability intervals for Hill’s equation with symmetric single well potential, Ukr. Math. J., 71(6) (2019), 977–983. DOI: 10.1007/s11253-019-01692-x
• 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., 113 (2013), 143-160. https://doi.org/10.7146/math.scand.a-15486
• Coşkun, H., Kabataş, A., Green’s function of regular Sturm-Liouville problem having eigenparameter in one boundary condition, Turk. J. Math. Comput. Sci., 4 (2016), 1-9.
• Coşkun, H., Kabataş, A., Başkaya, E., On Green’s function for boundary value problem with eigenvalue dependent quadratic boundary condition, Bound. Value Probl., (2017). http://dx.doi.org/10.1186/s13661-017-0802-0
• Coşkun, H., On the spectrum of a second order periodic differential equation, Rocky Mountain J. Math., 33 (2003), 1261–1277. DOI: 10.1216/rmjm/1181075461
• Coşkun, H., Some inverse results for Hill’s equation, J. Math. Anal. Appl., 276(2) (2002), 833–844. https://doi.org/10.1016/S0022-247X(02)00454-7
• Eastham, M. S. P., Theory of Ordinary Differential Equations, Van Nostrand Reinhold, 1970.
• Eastham, M. S. P., The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh and London, 1973.
• Flaschka, H., On the inverse problem for Hill’s operator, Arch. Rational. Mech. Anal., 59 (1975), 293–304.
• Fulton, C. T., Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinb. A: Math., 77(3-4) (1977), 293–308. https://doi.org/10.1017/S030821050002521X
• Hill, G. W., On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and the moon, Acta Math., 8 (1886), 1–36. DOI: 10.1007/BF02417081
• Huang, M. J., The first instability interval for Hill equations with symmetric single well potentials, Proc. Amer. Math. Soc., 125(3) (1997), 775–778.
• Kabataş, A., Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential, Ukrains’kyi Matematychnyi Zhurnal, 74(2) (2022), 191-203. https://doi.org/10.37863/umzh.v74i2.6246
• Magnus, W., Winkler, S., Hill’s Equation, Dover Publications, New York, 2004.
• Mathews, J. , Walker, R. L., Mathematical Methods of Physics, W. A. Benjamin, New York, 1965.
• McKean, H. P., Van Moerbeke, P., The spectrum of Hill’s equation, Invent. Math., 30 (1975), 217–274.
• Moriguchi, H., An improvement of the WKBJ Method in the presence of turning points and the asymptotic solutions of a class of Hill equations, J. Phys. Soc. Jpn., 14(12) (1959), 1771-1796. https://doi.org/10.1143/JPSJ.14.1771
• Pokutnyi, A. A., Periodic solutions of the Hill equation, J. Math. Sci., 197(1) (2014), 114–121. http://dx.doi.org/10.1007/s10958-014-1707-4
• Roncaratti, L. F., Aquilanti, V., Whittaker-Hill equation, Ince polynomials, molecular torsional modes, Int. J. Quantum Chem., 110(3) (2010), 716-730. http://dx.doi.org/10.1002/qua.22255
• Say, F., First-order general differential equation for multi-level asymptotics at higher levels and recurrence relationship of singulants, Comptes Rendus. Matematique, 359(10) (2021), 1267-1278. https://doi.org/10.5802/crmath.264
• Say, F., Late-order terms of second order ODEs in terms of pre-factors, Hacet. J. Math. Stat., 50(2) (2021), 342-350. https://doi.org/10.15672/hujms.657267
• Slater, J. C., The design of linear accelerators, Rev. Mod. Phys., 20(3) (1948), 473–518. https://doi.org/10.1103/RevModPhys.20.473