On the binding of polarons in a mean-field quantum crystal

We consider a multi-polaron model obtained by coupling the many-body Schr\"odinger equation for N interacting electrons with the energy functional of a mean-field crystal with a localized defect, obtaining a highly non linear many-body problem. The physical picture is that the electrons constitute a charge defect in an otherwise perfect periodic crystal. A remarkable feature of such a system is the possibility to form a bound state of electrons via their interaction with the polarizable background. We prove first that a single polaron always binds, i.e. the energy functional has a minimizer for N=1. Then we discuss the case of multi-polarons containing two electrons or more. We show that their existence is guaranteed when certain quantized binding inequalities of HVZ type are satisfied.