Asymptotics of Instability Zones of the Hill Operator with a Two Term Potential

Let γ n denote the length of the n-th zone of instability of the Hill operator Ly = −y ′′ − [4tα cos 2x + 2α 2 cos 4x]y, where α = 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 → 0 γ n = 8α n 2 n [(n − 1)!] 2 n/2 k=1 t 2 − (2k − 1) 2 (1 + O(α)) , and if α, t are fixed, then for n → ∞ γ n = 8|α/2| n [2 · 4 · · · (n − 2)] 2 cos π 2 t 1 + O log n n. Similar formulae (see Theorems 7 and 9) hold for odd n. The asymptotics for α → 0 imply interesting identities for squares of integers (see Sect. 4, Thm.8).