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Asymptotics of instability zones of the Hill operator with a two term potential

Plamen Djakov·Boris Mityagin
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Abstract

Let $γ_n $ denote the length of the $n$-th zone of instability of the Hill operator $Ly= -y^{\prime \prime} - [4tα\cos2x + 2 α^2 \cos 4x ] y,$ where $α\neq 0, $ and either both $α, t $ are real, or both are pure imaginary numbers. For even $n$ we prove: if $t, n $ are fixed, then, for $ α\to 0, $ $$ γ_n = | \frac{8α^n}{2^n [(n-1)!]^2} \prod_{k=1}^{n/2} (t^2 - (2k-1)^2) | (1 + O(α)), $$ and if $ α, t $ are fixed, then, for $ n \to \infty, $ $$ γ_n = \frac{8 |α/2|^n}{[2 \cdot 4 ... (n-2)]^2} | \cos (\fracπ{2} t) | [ 1 + O (\frac{\log n}{n}) ]. $$ Similar formulae (see Theorems \ref{thm2} and \ref{thm4}) hold for odd $n.$ The asymptotics for $α\to 0 $ imply interesting identities for squares of integers.

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Plamen Djakov, Boris Mityagin. 2005-09-16. Asymptotics of instability zones of the Hill operator with a two term potential. https://arxiv.org/abs/math-ph/0509034

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