Abstract

Computing spectra of operators is fundamental in the sciences, with wide-ranging applications in condensed-matter physics, quantum mechanics and chemistry, statistical mechanics, etc. While there are algorithms that in certain cases converge to the spectrum (e.g. Bloch’s theorem for periodic operators), no general procedure is known that (a) always converges, (b) provides bounds on the errors of approximation, and© provides approximate eigenvectors. This may lead to incorrect simulations. It has been an open problem since the 1950s to decide whether such reliable methods exist at all. We affirmatively resolve this question, and the algorithms provided are optimal, realizing the boundary of what digital computers can achieve. The algorithms work for discrete operators and operators over the continuum such as PDEs. Moreover, they are easy to implement and parallelize, offer fundamental speed-ups, and allow problems that before, regardless of computing power, were out of reach. Results are demonstrated on difficult problems such as the spectra of quasicrystals and non-Hermitian phase transitions in optics. This algorithm is part of a wider programme on determining what is computationally possible and optimal for spectral properties in infinite-dimensional spaces. If time permits, we will also discuss extensions to compute other spectral properties such as measures. The main paper for this talk can be found here https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.250201 and more details on this programme can be found here http://www.damtp.cam.ac.uk/user/mjc249/home.html