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, for  general
values of the potential and  interaction strengths.  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, for a given  periodic graph G, this may be  established from a single
numerical  verification, if  we knew  the  number of  critical points  for
general values of the parameters.

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.