Geodesic
On the Hill Discriminant of Lamé's Differential Equation
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

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Lamé's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function $\mathrm{sn}$ 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 $\mathrm{sn}$ 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 $1$.
Keywords: asymptotic expansion, stability.
Mots-clés : Lamé's equation, Hill's discriminant 
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Hans Volkmer. On the Hill Discriminant of Lamé's Differential Equation. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a20/

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