Spectral triangles of Schrödinger operators with complex potentials

Abstract

Consider the Schrödinger operator $Ly = -y^{\prime\prime} + v(x)y$ with a complex-valued potential v of period $1, \ v(x) = \sum_{m=-\infty}^\infty V(2m) \exp(2\pi i mx) .$ 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 Dirichelet boundary conditions on [0,1], and let $\Delta_n$ be the diameter of the spectral triangle with vertices $\lambda_n^+, \lambda_n^-, \mu_n .$ We prove the following statement: If $\sum_{n=1}^\infty \Delta_n^2 (1+n)^{2s} e^{2a n^b }< \infty , \quad s\geq 0, \; a>0, \; b\in (0,1),$ then v(x) is a Gevrey function, and moreover $\sum_{m=-\infty}^\infty |V(2m) |^2 (1+|m|)^{2s} e^{2a |m|^b } < \infty .$