How to Compute Spectra with Error Control.
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Publication Date
2019-06-28Journal Title
Physical Review Letters
ISSN
1079-7114
Publisher
American Physical Society
Volume
122
Number
250201
Language
eng
Type
Article
This Version
VoR
Metadata
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Colbrook, M., Roman, B., & Hansen, A. (2019). How to Compute Spectra with Error Control.. Physical Review Letters, 122 (250201)https://doi.org/10.1103/PhysRevLett.122.250201
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.
Sponsorship
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.
Funder references
EPSRC (1804238)
EPSRC (EP/N014588/1)
EPSRC (EP/L003457/1)
EPSRC (EP/L016516/1)
EPSRC (EP/R008272/1)
Identifiers
External DOI: https://doi.org/10.1103/PhysRevLett.122.250201
This record's URL: https://www.repository.cam.ac.uk/handle/1810/295873
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