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

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.