Refined asymptotics of the spectral gap for the Mathieu operator

The Mathieu operator L ( y ) = − y ″ + 2 a cos ( 2 x ) y , a ∈ C , a ≠ 0 , considered with periodic or anti-periodic boundary conditions has, close to n 2 for large enough n , two periodic (if n is even) or anti-periodic (if n is odd) eigenvalues λ n − , λ n + . For fixed a , we show that λ n + − λ n − = ± 8 ( a / 4 ) n [ ( n − 1 ) ! ] 2 [ 1 − a 2 4 n 3 + O ( 1 n 4 ) ] , n → ∞ . This result extends the asymptotic formula of Harrell–Avron–Simon by providing more asymptotic terms.