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

Published online by Cambridge University Press:  28 March 2013

Mathieu Lewin  and
Nicolas Rougerie
Mathieu Lewin
Affiliation:
UniversitéGrenoble 1 and CNRS, LPMMC (UMR 5493), B.P. 166, 38 042 Grenoble, France. [email protected]
Nicolas Rougerie
Affiliation:
CNRS and Department of Mathematics (UMR 8088), University of Cergy-Pontoise, 95 000 Cergy-Pontoise, France; [email protected]
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Abstract

We consider a multi-polaron model obtained by coupling the many-body Schrödinger equationfor N interacting electrons with the energy functional of a mean-fieldcrystal with a localized defect, obtaining a highly non linear many-body problem. Thephysical picture is that the electrons constitute a charge defect in an otherwise perfectperiodic crystal. A remarkable feature of such a system is the possibility to form a boundstate of electrons via their interaction with the polarizable background. We prove firstthat a single polaron always binds, i.e. the energy functional has aminimizer for N = 1. Then we discuss the case of multi-polaronscontaining N ≥ 2 electrons. We show that their existence is guaranteedwhen certain quantized binding inequalities of HVZ type are satisfied.

Keywords

Polaronquantum crystalbinding inequalitiesHVZ theoremChoquard-Pekar equation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 629 - 656
Copyright
© EDP Sciences, SMAI, 2013

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