The Ground State Energy is not Always Convex in the Number of Electrons
By using Grand Canonical Optimal Transport we provide a counter-example showing that the ground state energy of electrons in an external Coulomb potential is not always a convex function of the number of electrons. This property has been conjectured to hold for decades and it plays an important role in quantum chemistry. Our counter-example involves an external potential generated by six nuclei of very small fractional charges, placed far away from each other. The ground state energy of 3 electrons is proved to be higher than the average of the energies for 2 and 4 electrons. In addition, we show that the nuclei can bind 2 or 4 electrons, but not 3. Although the conjecture remains open for real nuclei (of integer charges), our work sets some doubt on the validity of the energy convexity for general atoms and molecules. This is based on joint works with S. Di Marino and M. Lewin.