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.  A
consequence is a dichotomy; either  almost all parameters have all critical
points  non-degenerate or  almost all  parameters give  degenerate critical
points, and we  showed how tools from computational  algebraic geometry may
be used to study the dispersion relation.

With Matthew Faust,  we use ideas from combinatorial  algebraic geometry to
give  an  upper  bound  for  the  number  of  critical  points  at  generic
parameters, 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 toric
infinity.