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Speaker: Stephen Shipman, Louisiana State University
Title: The Fermi variety in the spectral theory of periodic tight-binding QM models
Abstract: In a tight-binding model for an electron in a periodic medium, the Hamiltonian is approximated by a periodic operator A on a graph. The central object in the spectral theory of A is the Bloch variety, which is the locus of pairs of energy (eigenvalue of A) and momentum (eigenvalue of the shift group) admitted by the medium. A constant-energy cross section is called the Fermi variety in momentum space. These are algebraic varieties whose properties have significance in the spectral theory of A. We concentrate on the reducibility of the Fermi variety and its role in constructing defect states in the continuum (embedded eigenvalues). For special graphs, notably AB-stacked graphene, a more subtle matrix factorization of the Hamiltonian results in localization of the defect. I will present some interesting directions around this work concerning the role of commutative algebra in spectral theory. (Work with Daniel Massatt, Ilya Vekhter, and Justin Wilson; and Frank Sottile.)