Fermionic minimization problem

We review old and new results on a family of minimization problems, where the minimization space is a set of N orthonormal functions in L2 (the fermions), which interact locally. Such problems arise naturally when we want to optimize a sum of eigenvalues (Lieb-Thirring inequality). We will explain how to show the existence of minimizers for this kind of problem, and give several examples. If time allows, we will display an ad hoc problem where the N = 2 problem is well-posed, but the N = 1 problem has no minimizer.