CMX Lunch Seminar

Traditional methods for solving infinite-dimensional eigenproblems usually follow the discretize-then-solve paradigm. Discretize first, and then solve the matrix eigenproblem. The discretize-then-solve paradigm can be tricky for infinite-dimensional eigenproblems as the spectrum of matrix discretizations may not converge to the spectrum of the operator. Moreover, it is impossible to fully capture the continuous part of the spectrum with a finite-sized matrix eigenproblem. In this talk, we will discuss an alternative solve-then-discretize paradigm for infinite-dimensional eigenproblems that is rigorously justified. To compute the discrete spectrum and pseudospectra, we will discuss infinite-dimensional analogues of contour-based eigensolvers and randomized linear algebra. For the continuous spectra, we will show how to calculate a smoothed version of the so-called spectral measure. I will demonstrate that our techniques avoid spectral pollution on a magnetic tight-binding model of graphene.