The Pauli problem and wave function lifting: reconstruction of quantum states from physical observables

In this paper we consider the problem to determine a unique quantum state from given distribution of position (density) and momentum density of particles, namely the so called Pauli problem in quantum physics. In the first part, we will review the method of wave function lifting developed in ^[4,5] to construct a complex wave function $\psi \in H^s({\mathbf{R}}^d)$, $s=1,2$, associated to given density and momentum density. The second part is focused on the dynamical version of the wave function lifting, namely we study the relation between solutions to quantum fluid models and wave functions solving the nonlinear Schrödinger equation. The uniqueness of the lifted wave function is essentially related to the structure of vacuum regions of the position density.