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Texas A&M University
Mathematics

Mathematical Physics and Harmonic Analysis Seminar

Date: March 22, 2019

Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: P. Kuchment, Texas A&M

  

Title: Non-degeneracy of spectral edges for periodic operators (Joint work with Ngoc Do (U. Arizona) and F. Sottile (TAMU))

Abstract: An old problem in mathematical physics deals with the structure of the dispersion relation of the Schroedinger 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. In particular, the notion of effective masses hinges upon this conjecture. The progress in proving this conjecture has been rather 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 so called tight binding model). However, counterexamples exist that show that the genericity can fail in discrete situation. The authors consider the case of a general periodic discrete operator depending polynomially on some parameters and prove that the non-degeneracy of extrema either fails for all values of parameters, or holds for all values except of a proper algebraic subset. Thus, finding a single point in the parameter space where the non-degeneracy holds implies that it holds generically.