Spectral gaps of the periodic Schrödinger operator when its potential is an entire function. (Q1415394)

The authors study spectral gaps of the Schrödinger operator, \(L\), given by \(Ly\equiv -y''+vy\), in the case in which \(v\) is a complex-valued entire function of period 1 satisfying \(v(x)= \sum_{m=-\infty}^{\infty}V(2m)\exp(2\pi imx)\). Defining \(d_n\) as the diameter of the triangle with vertices \(\lambda _n ^+\), \(\lambda _n ^-\) and \(\mu _n\), where \(\lambda _n ^+\) and \(\lambda _n ^-\) are the eigenvalues near \(n^2 \pi ^2\) of \(L\) with periodic (for \(n\) even) or antiperiodic (for \(n\) odd) boundary conditions on \([0,1]\) and \(\mu _n\) is the corresponding Dirichlet eigenvalue, they show that, if \(| V(n)| \exp(a| n| ^b )\in \ell ^{\infty}\) for some \(a>0\) and \(b>1\), then there exists a \(c>0\) such that \((d_n \exp(cn(\log n)^{1-1/b}))\in \ell ^{\infty}\).