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SUMMARY:How to compute spectral properties of operators on Hilbert spaces 
 with error control - Matthew Colbrook -- DAMTP
DTSTART:20200603T103000Z
DTEND:20200603T113000Z
UID:TALK142717@talks.cam.ac.uk
CONTACT:Angela Harper
DESCRIPTION:Computing spectra of operators is fundamental in the sciences\
 , with wide-ranging applications in condensed-matter physics\, quantum mec
 hanics and chemistry\, statistical mechanics\, etc. While there are algori
 thms that in certain cases converge to the spectrum (e.g. Bloch's theorem 
 for periodic operators)\, no general procedure is known that (a) always co
 nverges\, (b) provides bounds on the errors of approximation\, and (c) pro
 vides 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 alg
 orithms provided are optimal\, realizing the boundary of what digital comp
 uters can achieve. The algorithms work for discrete operators and operator
 s over the continuum such as PDEs. Moreover\, they are easy to implement a
 nd parallelize\, offer fundamental speed-ups\, and allow problems that bef
 ore\, regardless of computing power\, were out of reach. Results are demon
 strated 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 w
 ill also discuss extensions to compute other spectral properties such as m
 easures. The main paper for this talk can be found here https://journals.a
 ps.org/prl/abstract/10.1103/PhysRevLett.122.250201 and more details on thi
 s programme can be found here http://www.damtp.cam.ac.uk/user/mjc249/home.
 html
LOCATION:Zoom
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