On the Hill Discriminant of Lamé's Differential Equation

Abstract

Lamé's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function sn, depending on the modulus 𝑘, and two additional parameters 𝘩 and 𝑣. 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 are determined by the value of its Hill discriminant 𝐷(𝘩, 𝑣, 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 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, 𝑣, 𝑘) when the modulus 𝑘 tends to 1.