Computing the spectra of differential operators

Spectral methods for solving differential 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 differential eigenproblems as the spectrum of matrix discretizations may not converge to the spectrum of the differential 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 differential eigenproblems. To compute the discrete spectrum, we will discuss a continuous analogue of FEAST by approximating the action of the resolvent operator. For the continuous spectra, we will use a Cauchy-like integral to calculate a smoothed version of the so-called spectral measure. This is joint work with Matthew Colbrook and Andrew Horning.