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.
Mots clés : polaron, quantum crystal, binding inequalities, hvz theorem, choquard-pekar equation
@article{COCV_2013__19_3_629_0,
author = {Lewin, Mathieu and Rougerie, Nicolas},
title = {On the binding of polarons in a mean-field quantum crystal},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {629--656},
publisher = {EDP-Sciences},
volume = {19},
number = {3},
year = {2013},
doi = {10.1051/cocv/2012025},
mrnumber = {3092354},
zbl = {1291.35248},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2012025/}
}
TY - JOUR AU - Lewin, Mathieu AU - Rougerie, Nicolas TI - On the binding of polarons in a mean-field quantum crystal JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 629 EP - 656 VL - 19 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012025/ DO - 10.1051/cocv/2012025 LA - en ID - COCV_2013__19_3_629_0 ER -
%0 Journal Article %A Lewin, Mathieu %A Rougerie, Nicolas %T On the binding of polarons in a mean-field quantum crystal %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 629-656 %V 19 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012025/ %R 10.1051/cocv/2012025 %G en %F COCV_2013__19_3_629_0
Lewin, Mathieu; Rougerie, Nicolas. On the binding of polarons in a mean-field quantum crystal. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 629-656. doi : 10.1051/cocv/2012025. http://www.numdam.org/articles/10.1051/cocv/2012025/
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