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How to Compute Spectra with Error Control.

Published version
Peer-reviewed

Type

Article

Change log

Authors

Colbrook, Matthew J 
Roman, Bogdan 
Hansen, Anders C 

Abstract

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.

Description

Keywords

0202 Atomic, Molecular, Nuclear, Particle and Plasma Physics

Journal Title

Physical Review Letters

Conference Name

Journal ISSN

1079-7114
1079-7114

Volume Title

122

Publisher

American Physical Society

Rights

All rights reserved
Sponsorship
EPSRC (1804238)
Engineering and Physical Sciences Research Council (EP/N014588/1)
Engineering and Physical Sciences Research Council (EP/L003457/1)
Engineering and Physical Sciences Research Council (EP/L016516/1)
Engineering and Physical Sciences Research Council (EP/R008272/1)
This work was supported by Engineering and Physical Sciences Research Council Grants No. EP/L016516/1, No. EP/R008272/1, No. EP/N014588/1, and No. EP/ L003457/1, as well as a Royal Society University Research Fellowship.