Generic properties of dispersion relations for discrete periodic operators
An old problem in mathematical physics deals with the structure of the dispersion relation of the Schrödinger operator-Δ+ V(x) in Rn 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 nondegenerate Hessian (i.e., dispersion relations are graphs of Morse functions). The important notion of effective masses in solid state physics, as well as the Liouville property, Green's function asymptotics, and so on 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 even for Schrödinger operators on simple 2D-periodic two-atomic structures, showing that the genericity fails in some discrete situations. We start with establishing in a very general situation the following 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."Noticing that the known counterexample has only two free parameters, one can suspect that this might be too tight for the genericity to hold. We thus consider the maximal Z2-periodic two-atomic nearest-cell interaction graph, which has nine edges per unit cell and the discrete "Laplace-Beltrami"operator on it, which has nine free parameters. We then use methods from computational and combinatorial algebraic geometry to prove the genericity conjecture for this graph. Since the proof is non-trivial and would be much harder for more general structures, we show three different approaches to the genericity, which might be suitable in various situations. It is also proven in this case that adding more parameters indeed cannot destroy the genericity result. This allows us to list all "bad"periodic subgraphs of the one we consider and discover that in all these cases the genericity fails for "trivial"reasons only.
Do, Ngoc, Peter Kuchment, and Frank Sottile. "Generic properties of dispersion relations for discrete periodic operators." Journal of Mathematical Physics 61, no. 10 (2020): 103502.
Journal of Mathematical Physics