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

¹ UniversitéGrenoble 1 and CNRS, LPMMC (UMR 5493), B.P. 166, 38 042 Grenoble, France
nicolas.rougerie@grenoble.cnrs.fr
² CNRS and Department of Mathematics (UMR 8088), University of Cergy-Pontoise, 95 000 Cergy-Pontoise, France
mathieu.lewin@math.cnrs.fr

Received: 24 February 2012
Revised: 14 June 2012

Abstract

We consider a multi-polaron model obtained by coupling the many-body Schrödinger 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 N ≥ 2 electrons. We show that their existence is guaranteed when certain quantized binding inequalities of HVZ type are satisfied.