Reviews in Mathematical Physics

We consider Schrödinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may not even be differentiable. The well-posedness of the Cauchy problem is proved in the frame of the modulation spaces, and results of micro-local propagation of singularities are given in terms of Gabor wave front sets.

We consider the one-dimensional discrete Schrödinger operator on ℤ, and study the relation between the generalized eigenstates and the asymptotic expansion of the resolvent for the threshold 0. We decompose the generalized zero eigenspace into subspaces, some of which correspond to the bound states or the resonance states, only by their growth properties at infinity, and precisely describe the first few leading coefficients in the expansion using these subspaces. The generalized zero eigenspace we consider is the largest possible one, consisting of all solutions to the eigenequation. For the resolvent expansion, we implement the expansion scheme of Jensen–Nenciu [Rev. Math. Phys. 13 (2001) 717–754] and [Rev. Math. Phys. 16 (2004) 675–677] in its full generality.

Using a new method [19], it is possible to derive mean field equations from the microscopic N body Schrödinger evolution of interacting particles without using BBGKY hierarchies. This method also allows for error estimates and can be generalized to systems with external fields, of which both points are relevant from a physics perspective.

Recently, this method was used to derive the Hartree equation for singular interactions [11] and the Gross–Pitaevskii equation without positivity condition on the interaction [17] where one had to restrict the scaling behavior of the interaction.

Assuming positivity of the interaction, this paper deals with more general scalings including the so-called Gross–Pitaevskii scaling.