Refined asymptotics of the spectral gap for the Mathieu operator

Abstract

The Mathieu operator L(y) = -y '' + 2a cos(2x)y, a is an element of C, a not equal 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 lambda(+)(n) - lambda(-)(n). For fixed a, we show that gimel(+)(n) - gimel(-)(n) = +/- 8(a/4)(n)/left perpendicular(n - 1)!right perpendicular(2) [1 - a(2)/4n(3) + o(1/n(4))], n -> infinity. This result extends the asymptotic formula of Harrell-Avron-Simon by providing more asymptotic terms.