An inverse problem for Hankel operators and turbulent solutions of the cubic Szegő equation on the line

Abstract: We construct inverse spectral theory for finite rank Hankel operators acting on the Hardy space of the upper half-plane. A particular feature of our theory is that we completely characterise the set of spectral data. As an application of this theory, we prove the genericity of turbulent solutions of the cubic Szeg\H{o} equation on the real line.