Spectral gaps of the periodic Schrodinger operator when its potential is an entire function

Consider the Schrodinger equation -y" + v(x)y = lambday with periodic complex-valued potential, of period 1, v(x) = Sigma(m=-infinity)(infinity) V(2m)exp(2piimx). Let lambda(n)(+), lambda(n)(-), and mu(n) be the eigenvalues of L that are close to pi(2)n(2), respectively with periodic (for n even), antiperiodic (for n odd), and Dirichlet boundary conditions on [0, 1], and let an be the diameter of the spectral triangle with vertices lambda(n)(+), lambda(n)(-), mu(n). We study the relationship between the rate of decay of "gap sequence" (d(n)) and the rate of decay of the sequence of Fourier coefficients (V(m)) in the case v(x) is an entire function (then (V(m)) decays superexponentially). It is proven that if \V(n)\exp(a\n\(b)) is an element of l(infinity) with a > 0 and b > 1 then (d(n) exp(c n log n)(1-1/b))) is an element of l(infinity) for some c > 0. A special example shows that up to a change of types a, c, this statement is sharp. (C) 2003 Elsevier Inc. All rights reserved.