Critical Points Of Discrete Periodic Operators

It is believed that the dispersion  relation of a Schrodinger operator with
a  periodic potential  has non-degenerate  critical points.   In work  with
Kuchment and  Do, we considered this  for discrete operators on  a periodic
graph G, for then the dispersion  relation is an algebraic hypersurface. We
showed how tools from computational algebraic geometry may be used to study
the  dispersion relation.   A first  step was  to determine  the number  of
critical points for a particular graph.

With  Matthew Faust,  we use  combinatorial algebraic  geometry to  give an
upper bound  for the number  of critical points,  and also a  criterion for
when  that  bound is  obtained.   The  dispersion  relation has  a  natural
compactification  in  a  toric  variety, and  the  criterion  concerns  the
smoothness of  the dispersion relation  at the toric infinity.   This toric
compactification enables  other questions from  physics to be  expressed in
terms of algebraic geometry,