Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 021, 9 pages      arXiv:2306.12539
Contribution to the Special Issue on Asymptotics and Applications of Special Functions in Memory of Richard Paris

On the Hill Discriminant of Lamé's Differential Equation

Hans Volkmer
Department of Mathematical Sciences, University of Wisconsin - Milwaukee, USA

Received July 25, 2023, in final form March 08, 2024; Published online March 16, 2024

Lamé's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function ${\rm sn}$ depending on the modulus $k$, and two additional parameters $h$ and $\nu$. This differential equation appears in several applications, for example, the motion of coupled particles in a periodic potential. Stability and existence of periodic solutions of Lamé's equations is determined by the value of its Hill discriminant $D(h,\nu,k)$. The Hill discriminant is compared to an explicitly known quantity including explicit error bounds. This result is derived from the observation that Lamé's equation with $k=1$ can be solved by hypergeometric functions because then the elliptic function ${\rm sn}$ reduces to the hyperbolic tangent function. A connection relation between hypergeometric functions then allows the approximation of the Hill discriminant by a simple expression. In particular, one obtains an asymptotic approximation of $D(h,\nu,k)$ when the modulus $k$ tends to $1$.

Key words: Lamé's equation; Hill's discriminant; asymptotic expansion; stability.

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