2019/06/28 by Matthew J. Colbrook, Bogdan Roman, Anders C. Hansen · 1 citation
Physics and Astronomy · Materials Science · #Quantum Mechanics and Non-Hermitian Physics #Quasicrystal Structures and Properties #Quantum chaos and dynamical systems
paper · doi:10.1103/physrevlett.122.250201
openalex publication_date 2019/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/01
Computing the spectra of operators is a fundamental problem 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, no general procedure is known that (a) always converges, (b) provides bounds on the errors of approximation, and (c) 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. 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.