Abstract

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.