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

AbstractLet γn denote the length of the nth zone of instability of the Hill operator Ly=−y″−[4tαcos2x+2α2cos4x]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→0γn=|8αn2n[(n−1)!]2∏k=1n/2(t2−(2k−1)2)|(1+O(α)), and if α, t are fixed, then for n→∞γn=8|α/2|n[2⋅4⋯(n−2)]2|cos(π2t)|[1+O(lognn)]. The asymptotics for α→0, for n=2m, imply the following identities for squares of integers:∑∏s=1k(m2−is2)=∑1⩽j1<⋯<jk⩽m∏s=1k(2js−1)2, where 1⩽k⩽m, and the left sum is over all indices i1,…,ik such that−m<i1<⋯<ik