Title: Computational tools for the continuous spectrum of self-adjoint operators
Abstract: Ever since Joseph Fourier used sines and cosines to diagonalize the Laplacian and solve the heat equation in 1822, spectral decompositions of linear operators have empowered scientists to disentangle complex physical phenomena. However, the spectrum of a self-adjoint operator can be more sophisticated than its familiar matrix counterpart; it may contain a continuum and the operator may not be diagonalized by eigenvectors alone. In this talk, we present a tool kit of algorithms for computing spectral properties associated with the continuous spectrum of self-adjoint operators. These algorithms use the resolvent operator and careful regularization to construct high-order accurate approximations of inherently infinite-dimensional spectral properties, including smooth spectral measures, projections onto absolutely continuous subspaces, and non-normalizable eigenfunctions. We illustrate the flexibility and power of these algorithms with examples from canonical models in quantum and condensed matter physics.