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

TL;DR

This paper derives asymptotic formulas for the lengths of instability zones of the Hill operator with a specific two-term potential, revealing detailed behavior as parameters vary and uncovering identities related to squares of integers.

Contribution

It provides new asymptotic expressions for instability zones of the Hill operator with a two-term potential, extending understanding of spectral properties in this context.

Findings

Asymptotic formula for even n as alpha approaches zero

Asymptotic formula for large n with fixed alpha and t

Identities involving squares of integers derived from asymptotics

Abstract

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