Generic properties of dispersion relations for discrete periodic operators

Ngoc T. Do, Peter Kuchment, Frank Sottile

Published 2019-10-15Version 1

An old problem in mathematical physics deals with the structure of the dispersion relation of the Schr\"odinger operator $-\Delta+V(x)$ in $\R^n$ with periodic potential near the edges of the spectrum, i.e.\ near extrema of the dispersion relation. A well known and widely believed conjecture says that generically (with respect to perturbations of the periodic potential) the extrema are attained by a single branch of the dispersion relation, are isolated, and have non-degenerate Hessian (i.e., dispersion relations are graphs of Morse functions). In particular, the important notion of effective masses hinges upon this property. The progress in proving this conjecture has been slow. It is natural to try to look at discrete problems, where the dispersion relation is (in appropriate coordinates) an algebraic, rather than analytic, variety. Moreover, such models are often used for computation in solid state physics (the tight binding model). Alas, counterexamples exist showing that the genericity fails in some discrete situations. We consider a general periodic discrete operator depending polynomially on some parameters. We prove the natural dichotomy: the non-degeneracy of extrema either fails or holds in the complement of a proper algebraic subset of the parameters. Thus, a random choice of a point in the parameter space gives the correct answer ``with probability one.'' We then use methods from computational and combinatorial algebraic geometry to prove the genericity conjecture for a particular diatomic $\Z^2$-periodic structure with many free parameters. Here several new approaches to the genericity problem introduced and many examples of both alternatives are provided.