Rafael de la Llave, Enrico Valdinoci
Ground states and critical
points for generalized Frenkel-Kontorova models
(45K, LaTeX)

ABSTRACT.  We consider a multidimensional
model of Frenkel-Kontorova type but we allow
non-nearest neighbor interactions.
For every possible frequancy vector, we
show that there are quasi-periodic
ground states which enjoy further geometric properties.
The gound states we produce are either bigger or
smaller than the state. They are are at bounded distance from
the plane wave with the given frequency.
The comparison property above implies that the
ground states and the translations are organized into
laminations. If these leave a gap, we show
that there are critical points inside the gap which
also satisfy the comparizon properties.
In particular, given any frequency, we show
that either there is a continuous parameter of
ground states or there is a ground state and
another critical point which is not a ground state.