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

Abstract

Let ${\gamma }_n$ denote the length of the $n$-th zone of instability of the Hill operator $Ly=-y^{\prime \prime }-[4t\alpha \cos 2x+2{\alpha }^2\cos 4x]y,$ where $\alpha \ne 0,$ 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 $n\to \infty ,$ $${\gamma }_n=\frac{8|\alpha /2{|}^n}{[2\cdot \mathrm{4...}(n-2){]}^2}|\cos (\frac{\pi }{2}t)|[1+O(\frac{\log n}{n})].$$ 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.

arXiv comment39 pages