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Properties of some elliptic Hill’s potentials

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Abstract

We study Hill’s differential equation with potential expressed by elliptic functions which arises in some problems of physics and mathematics. Analytical method can be applied to study the local properties of the potential in asymptotic regions of the parameter space. The locations of the saddle points of the potential are determined, the locations of turning points can be determined too when they are close to a saddle point. Combined with the quadratic differential associated with the differential equation, these local data give a qualitative explanation for the asymptotic eigensolutions obtained recently. A relevant topic is about the generalisation of Floquet theorem for ODE with doubly-periodic elliptic function coefficient which bears some new features compared to the case of ODE with real valued singly-periodic coefficient. Beyond the local asymptotic regions, global properties of the elliptic potential are studied using numerical method.

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Notes

  1. Taking into account the 1/2 factor, one needs to make the following change to the spectral solutions given in reference [4]: for the solution of case (A), \(\mu \) is changed to \(\mu -i/2\), that means \(i\mu \) is shifted to \(i\mu +1/2\); for the solution of case (B), \(\mu \) is changed to \(\mu +k^{\,\prime }/2\), that means \(\mu /k^{\,\prime }\) is shifted to \(\mu /k^{\,\prime }+1/2\). Notice that in [4] parameters are denoted by letters different from the ones used in this paper.

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Acknowledgements

This work is supported by a Grant from CWNU (No. 18Q068).

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  1. School of Physics and Astronomy, China West Normal University, Nanchong, 637002, China

    Wei He & Peng Su

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  1. Wei He
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Correspondence to Wei He.

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He, W., Su, P. Properties of some elliptic Hill’s potentials. Anal.Math.Phys. 14, 40 (2024). https://doi.org/10.1007/s13324-024-00897-z

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