On the Hill Discriminant of Lamé's Differential Equation

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2024
• 1. Verfasser: Volkmer, Hans
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2024
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/212174
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: On the Hill Discriminant of Lamé's Differential Equation. Hans Volkmer. SIGMA 20 (2024), 021, 9 pages

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 , and two additional parameters and . 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 (, , ). 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 = 1 can be solved by hypergeometric functions because then the elliptic function 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 (, , ) when the modulus tends to 1.