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Not to be confused with Simon's problem in computational complexity.
In mathematics, the Simon problems (or Simon's problems) are a series of fifteen questions posed in the year 2000 by Barry Simon, an American mathematical physicist.[1][2] Inspired by other collections of mathematical problems and open conjectures, such as the famous list by David Hilbert, the Simon problems concern quantum operators.[3] Eight of the problems pertain to anomalous spectral behavior of Schrödinger operators, and five concern operators that incorporate the Coulomb potential.[1][4]
In 2014, Artur Avila won a Fields Medal for work including the solution of three Simon problems.[5][6] Among these was the problem of proving that the set of energy levels of one particular abstract quantum system was in fact the Cantor set, a challenge known as the "Ten Martini Problem" after the reward that Mark Kac offered for solving it.[6][7]
The 2000 list was a refinement of a similar set of problems that Simon had posed in 1984.[8][9]
^ abSimon, Barry (2000). "Schrödinger Operators in the Twenty-First Century". Mathematical Physics 2000. Imperial College London. pp. 283–288. doi:10.1142/9781848160224_0014. ISBN 978-1-86094-230-3.
^Marx, C. A.; Jitomirskaya, S. (2017). "Dynamics and Spectral Theory of Quasi-Periodic Schrödinger-type Operators". Ergodic Theory and Dynamical Systems. 37 (8): 2353–2393. arXiv:1503.05740. doi:10.1017/etds.2016.16. S2CID 119317111.
^Damanik, David. "Dynamics of SL(2,R)-Cocycles and Applications to Spectral Theory; Lecture 1: Barry Simon's 21st Century Problems" (PDF). Beijing International Center for Mathematical Research, Peking University. Retrieved 2018-07-07.
^"Simon's Problem" (PDF). University of Colorado Boulder.
^"Fields Medal awarded to Artur Avila". Centre national de la recherche scientifique. 2014-08-13. Retrieved 2018-07-07.
^ abBellos, Alex (2014-08-13). "Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained". The Guardian. Retrieved 2018-07-07.
^Simon, Barry (1984). "Fifteen problems in mathematical physics". Perspectives in Mathematics: Anniversary of Oberwolfach 1984(PDF). Birkhäuser. pp. 423–454. Retrieved 24 June 2021.
^Coley, Alan A. (2017). "Open problems in mathematical physics". Physica Scripta. 92 (9): 093003. arXiv:1710.02105. Bibcode:2017PhyS...92i3003C. doi:10.1088/1402-4896/aa83c1. S2CID 3892374.
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