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Texas A&M University
Mathematics

Mathematical Physics and Harmonic Analysis Seminar

Date: October 25, 2019

Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: P. Kuchment, TAMU

  

Title: On generic non-degeneracy of spectral edges. Discrete case

Abstract: This is joint work with Frank Sottile (TAMU) and Ngoc T. Do (Missouri State U.)

An old problem in mathematical physics deals with the structure of the dispersion relation of the Schrodinger 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 showing that the genericity can fail in some discrete situations do exist.

In our work, we consider the case of a general periodic discrete operator depending polynomially on some parameters. We prove that the non-degeneracy of extrema either fails or holds for all but a proper algebraic subset of values of parameters. Thus, a random choice of a point in the parameter space will give the correct answer "with probability one". A specific example of a diatomic Z^2-periodic structure is also considered, which provides a cornucopia of examples for both alternatives, as well as a different approach to the genericity problem.