Multi-marginal optimal transport and potential optimization problems for Schrodinger operator

In the part I: Optimal Transport and Density Functional Theory, we investigate methods to compute the ground state energy for a stationary Schrodinger Equation -∆+V with singular potential V. This is related to Quantum Mechanics or, more precisely, to a computational quantum mechanical modelling, named Density Functional Theory (DFT), which investigates the electronic structure of many-body systems. In a sort of semi-classical regime, the problem to compute the ground state energy for a electronic Schr ̈odinger equation can be seen as a generalized version of a mass-transportation problem. In the part II: Optimal Potentials for Schrodinger Operators, we are dealing with stationary Schrodinger operators −∆ + V. Our goal is to study some optimization problems where an optimal potential V ≥ 0 has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.