Inverse kinetic theory for quantum hydrodynamic equations
  We propose a solution for the inverse kinetic theory for quantum hydrodynamic equations associated to the non-relativistic Schr\"{o}dinger equation. It is shown that an inverse kinetic equation of the form of the Vlasov equation can be non-uniquely determined under suitable mathematical prescriptions. 
Massimo Tessarotto, Marco Ellero and Piero Nicolini