Beyond Eigenvalues: Computational Tools for the Continuous Spectrum by Andrew Horning, MIT

Beyond Eigenvalues: Computational Tools for the Continuous Spectrum by Andrew Horning, MIT, Thursday January 18, 2:30 – 3:30pm, North Science Building 015, Wachenheim

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 and eigenvalues. Until now, computational techniques for the continuous spectrum have typically focused on narrow classes of operators with known analytical structure or relied on heuristic approximations with poorly understood convergence properties.

In this talk, we present a tool kit of new algorithms for computing key quantities associated with the continuous spectrum of self-adjoint operators. These algorithms use the resolvent to construct high-order accurate approximations of inherently infinite-dimensional spectral properties, including smooth spectral measures, projections onto absolutely continuous subspaces, and non-normalizable modes. The algorithms are embarrassingly parallelizable and capable of leveraging state-of-the-art software for the resolvent of differential, integral, and lattice operators. Their flexibility and power are illustrated with applications in quantum and condensed matter physics. We also highlight several exciting new developments in reachability analysis and data-driven modeling, where resolvent-based ideas are poised to make an impact.