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CATEGORIES:Electronic Structure Discussion Group
SUMMARY:How to compute spectral properties of operators on
Hilbert spaces with error control - Matthew Colbr
ook -- DAMTP
DTSTART;TZID=Europe/London:20200603T113000
DTEND;TZID=Europe/London:20200603T123000
UID:TALK142717AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/142717
DESCRIPTION:Computing spectra of operators is fundamental in t
he sciences\, with wide-ranging applications in co
ndensed-matter physics\, quantum mechanics and che
mistry\, statistical mechanics\, etc. While there
are algorithms that in certain cases converge to t
he spectrum (e.g. Bloch's theorem for periodic ope
rators)\, no general procedure is known that (a) a
lways converges\, (b) provides bounds on the error
s of approximation\, and (c) provides approximate
eigenvectors. This may lead to incorrect simulatio
ns. 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 b
oundary of what digital computers can achieve. The
algorithms work for discrete operators and operat
ors over the continuum such as PDEs. Moreover\, th
ey are easy to implement and parallelize\, offer f
undamental speed-ups\, and allow problems that bef
ore\, regardless of computing power\, were out of
reach. Results are demonstrated on difficult probl
ems such as the spectra of quasicrystals and non-H
ermitian phase transitions in optics. This algorit
hm is part of a wider programme on determining wha
t is computationally possible and optimal for spec
tral properties in infinite-dimensional spaces. If
time permits\, we will also discuss extensions to
compute other spectral properties such as measure
s. The main paper for this talk can be found here
https://journals.aps.org/prl/abstract/10.1103/Phys
RevLett.122.250201 and more details on this progra
mme can be found here http://www.damtp.cam.ac.uk/u
ser/mjc249/home.html
LOCATION:Zoom
CONTACT:Angela Harper
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