For some integrable systems, such as the open Toda molecule, the spectral curve of the Lax representation becomes the graph C = {(λ, z) | z = A(λ)} of a function A(λ). Those integrable systems provide an interesting “toy model” of separation of variables. Examples of this type of integrable systems are presented along with generalizations for which A(λ) lives on a cylinder, a torus or a Riemann...

We consider the mass-subcritical Hartree equation with a homogeneous kernel, in the space of square integrable functions whose Fourier transform is integrable. We prove a global well-posedness result in this space. On the other hand, we show that the Cauchy problem is not even locally wellposed if we simply work in the space of functions whose Fourier transform is integrable. Similar results ar...