Resumen de Spectral triangles of Schrödinger operators with complex potentials

Plamen Djakov, Boris Mityagin

• Consider the Schrödinger operator Ly=−y′′+v(x)y with a complex-valued potential v of period 1, v(x)=∑∞m=−∞V(2m)exp(2πimx). Let λ+n,λ−n and μn be the eigenvalues of L that are close to π2n2, respectively, with periodic (for n even), antiperiodic (for n odd), and Dirichelet boundary conditions on [0,1], and let Δn be the diameter of the spectral triangle with vertices λ+n,λ−n,μn. We prove the following statement: If ∑∞n=1Δ2n(1+n)2se2anb<∞,s≥0,a>0,b∈(0,1), then v(x) is a Gevrey function, and moreover ∑∞m=−∞|V(2m)|2(1+|m|)2se2a|m|b<∞.