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

Mathieu Lewin (1) , Nicolas Rougerie (2)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 LPM2C - Laboratoire de physique et modélisation des milieux condensés

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 two electrons or more. We show that their existence is guaranteed when certain quantized binding inequalities of HVZ type are satisfied.

Mots clés: polaron; quantum crystal; binding inequalities; HVZ theorem; Choquard-Pekar equation

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Mathieu Lewin, Nicolas Rougerie. On the binding of polarons in a mean-field quantum crystal. ESAIM: Control, Optimisation and Calculus of Variations, 2013, 19 (3), pp.629--656. ⟨10.1051/cocv/2012025⟩. ⟨hal-00673045v3⟩