Computing spectra of linear operators

Bernard Deconinck
Department of Applied Mathematics
University of Washington

Many problems in pure and applied mathematics may be reduced to that of determining the spectrum of a linear operator. This is the case for the linear stability analysis of equilibrium solutions of finite or infinite-dimensional evolution systems, and for the forward scattering problem associated with any integrable system. In this talk, I will show how a method which goes back to Hill (1886) may be used to compute spectra of linear operators with periodic coefficients. It may be extended to problems on an infinite domain. The method is algorithmic in nature and as such its only competitors are finite-difference methods. Hill's method converges exponentially, due to its spectral origins. It also incorporates Floquet theory, allowing for the determination of the entire spectrum, as opposed to isolated elements of it. I will illustrate the method using a variety of examples.