The Analysis and PDE seminar will take place Monday, March 5, in 740 Evans from 4:10 to 5pm.
Title: Several questions on Laplace eigenfunctions
Abstract: Let $(M,g)$ be a compact Riemannian manifold without boundary. We are interested in asymptotic properties of Laplace eigenfunctions on $M$ as the eigenvalue $\lambda$ tends to infinity. The advances of the last few years will be discussed and a survey of interesting open questions will be given.
The Analysis and PDE seminar will take place on Monday, March 12, in 740 Evans from 4:10 to 5 pm.
Title: Honeycomb Structures, Edge States, and the Strong Binding Regime
Abstract: We review recent progress on the propagation of waves for the 2D Schrödinger and Maxwell equations for media with the symmetry of a hexagonal tiling of the plane.
The Analysis and PDE seminar will take place Monday, February 12, in 740 Evans from 4:10 to 5 pm.
Title: Edge (resonant) states for 1D bi-periodic systems
Abstract: 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.