Analysis and PDE Seminar: Edge (resonant) states for 1D bi-periodic systems

Seminar | February 12 | 4-5 p.m. | 740 Evans Hall

 Alexis Drouot, Columbia University

 Department of Mathematics

We study the bifurcation of Dirac points under the introduction of edges in certain periodic systems. For honeycomb Schrodinger operators, Fefferman, Lee-Thorp and Weinstein showed that if introducing the edge opens an essential spectral gap near the Dirac energy, then the perturbed operator has an edge-localized eigenstate. This state is associated to the topologically protected zero-mode of a Dirac operator, which emerges from a formal multiscale approach (one of the two scales being the size of the edge).

We consider 1D models where the introduction of a large edge does not necessarily open an essential spectral gap near the Dirac energy. We approach such systems with Fredholm analytic tools and show that Dirac points bifurcate to resonant states. When the edge perturbation happens to open an essential spectral gap, this improves a previous result of Fefferman–Lee-Thorp–Weinstein by (a) proving the validity of the multiscale approach; (b) relating each eigenvalue in the gap to an eigenvalue of the above Dirac operator; (c) deriving full expansions of the associated states.

Joint work with Michael Weinstein and Charles Fefferman.