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

Abstract

Let γ n denote the length of the nth zone of instability of the Hill operator L y = − y ″ − [ 4 t α cos 2 x + 2 α 2 cos 4 x ] 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 = | 8 α n 2 n [ ( n − 1 ) ! ] 2 ∏ k = 1 n / 2 ( t 2 − ( 2 k − 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 ) ] . The asymptotics for α → 0 , for n = 2 m , imply the following identities for squares of integers: ∑ ∏ s = 1 k ( m 2 − i s 2 ) = ∑ 1 ⩽ j 1 < ⋯ < j k ⩽ m ∏ s = 1 k ( 2 j s − 1 ) 2 , where 1 ⩽ k ⩽ m , and the left sum is over all indices i 1 , … , i k such that − m < i 1 < ⋯ < i k < m , | i s − i r | ⩾ 2 if s ≠ r . Similar formulae (see Theorems 7–9) hold for odd n.