# Barry Simon

Barry Simon
Born16 April 1946 (age 76)
Alma materA.B., Harvard University
Ph.D., Princeton University
TitleProfessor
SpouseMartha Simon
AwardsHenri Poincaré Prize (2012)
Bolyai Prize (2015)
Steele Prize (2016)
James Madison Wall of Distinction (2016)[1]
Scientific career
InstitutionsPrinceton University
Caltech
Doctoral studentsPercy Deift
Alexander Kiselev
Antti Kupiainen

Barry Martin Simon (born 16 April 1946) is an American mathematical physicist and was the IBM Professor of Mathematics and Theoretical Physics at Caltech,[2] known for his prolific contributions in spectral theory, functional analysis, and nonrelativistic quantum mechanics (particularly Schrödinger operators), including the connections to atomic and molecular physics. He has authored more than 400 publications on mathematics and physics.

His work has focused on broad areas of mathematical physics and analysis covering: quantum field theory, statistical mechanics, Brownian motion, random matrix theory, general nonrelativistic quantum mechanics (including N-body systems and resonances), nonrelativistic quantum mechanics in electric and magnetic fields, the semi-classical limit, the singular continuous spectrum, random and ergodic Schrödinger operators, orthogonal polynomials, and non-selfadjoint spectral theory.[3]

## Early life

Barry Simon's mother was a school teacher, his father was an accountant. Simon attended James Madison High School in Brooklyn.[4][5]

## Career

During his high school years, Simon started attending college courses for highly gifted pupils at Columbia University. In 1962, Simon won a MAA mathematics competition. The New York Times reported that in order to receive full credits for a faultless test result he had to make a submission with MAA. In this submission he proved that one of the problems posed in the test was ambiguous.[5][3]

In 1962, Simon entered Harvard with a stipend. He became a Putnam Fellow in 1965 at 19 years old.[6] He received his A.B. in 1966 from Harvard College and his Ph.D. in Physics at Princeton University in 1970, supervised by Arthur Strong Wightman. His dissertation dealt with Quantum mechanics for Hamiltonians defined as quadratic forms.[3]

Following his doctoral studies, Simon took a professorship at Princeton for several years, often working with colleague Elliott H. Lieb on the Thomas-Fermi Theory and Hartree-Fock Theory of atoms in addition to phase transitions and mentoring many of the same students as Lieb. He eventually was persuaded to take a post at Caltech, from which he retired in the summer of 2016.[3][7]

His status is legendary in mathematical physics and he is renowned for his ability to write scientific manuscripts "in five percent of the time ordinary mortals need to write such papers."[8]

A former graduate student of Simon's, in a tale revealing of his brilliance, once stated:

Barry has always been remarkable for his vast knowledge of mathematics, so it was many years before I can recall ever telling him a published theorem he didn't already know. One day I saw Barry in Princeton shortly after a meeting and told him about an old inequality for PDEs, which, as I could tell from his intent look, was new to him. I said, "It seems to be useful. Do you want to see the proof?" His response "No, that's OK." Then he went to the board and wrote down a flawless proof on the spot.[9]

## Selected publications

### Articles

• Resonances in n-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory, Annals of Mathematics 97 (1973), 247–274 doi:10.2307/1970847 (over 700 citations)
• (with F. Guerra and L. Rosen) The P(φ)2 quantum theory as classical statistical mechanics, Annals of Mathematics 101 (1975), 111–259 doi:10.2307/1970988
• (with E. Lieb) The Thomas-Fermi theory of atoms, molecules and solids, Advances in Mathematics 23 (1977), 22–116 doi:10.1016/0001-8708(77)90108-6 (over 700 citations)
• (with J. Fröhlich and T. Spencer) Infrared bounds, phase transitions and continuous symmetry breaking, Commun. Math. Phys. 50 (1976), 79–85 doi:10.1007/BF01608557
• (with P. Perry and I. M. Sigal) Spectral analysis of multiparticle Schrödinger operators, Annals of Mathematics 114 (1981), 519–567 doi:10.2307/1971301
• Schrödinger semigroups, Bulletin of the American Mathematical Society 7 (1982), 447–526 doi:10.1090/S0273-0979-1982-15041-8 (over 1500 citations)
• (with M. Aizenman) Brownian motion and Harnack's inequality for Schrödinger operators, Commun. Pure Appl. Math. 35 (1982), 209–273 (over 600 citations)
• Holonomy, the quantum adiabatic theorem and Berry's phase, Phys. Rev. Lett. 51 (1983), 2167–2170 doi:10.1103/PhysRevLett.51.2167 (over 2050 citations)
• (with Joseph E. Avron and Ruedi Seiler) Homotopy and quantization in condensed matter physics, Phys. Rev. Lett. 51 (1983) 51–53 doi:10.1103/PhysRevLett.51.51 (over 600 citations)
• Semiclassical analysis of low lying eigenvalues, II. Tunneling, Annals of Mathematics 120 (1984), 89–118 doi:10.2307/2007072
• (with T. Wolff) Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Commun. Pure Appl. Math. 39 (1986), 75–90 doi:10.1002/cpa.3160390105
• Operators with singular continuous spectrum: I. General operators, Annals of Mathematics 141 (1995), 131–145 doi:10.2307/2118629

### Books

• Quantum mechanics for hamiltonians defined as quadratic forms. Princeton University Press, Princeton NJ 1971, ISBN 0-691-08090-9.
• with Michael C. Reed: Methods of Modern Mathematical Physics. 4 vols. Academic Press, New York, NY etc. 1972–1978;
• The ${\displaystyle \Phi ^{2}}$ Euclidean (Quantum) Field Theory. Princeton University Press, Princeton NJ 1974, ISBN 0-691-08144-1.
• as editor with Elliott H. Lieb and Arthur S. Wightman: Studies in mathematical physics. Essays in Honor of Valentine Bargmann. Princeton University Press, Princeton NJ 1976, ISBN 0-691-08180-8, contributions by Barry Simon:
• pp. 305–326: On the number of bound states of two body Schrödinger operators – a review. online PDF; 377 kB.
• pp. 327–349: Quantum dynamics: from automorphism to hamiltonian. online PDF; 573 kB.
• Functional integration and quantum physics (= Pure and Applied Mathematics. 86). Academic Press, New York NY etc. 1979, ISBN 0-12-644250-9 (2nd edition: American Mathematical Society, Providence RI 2005, ISBN 0-8218-3582-3).
• Trace Ideals and their applications (= London Mathematical Society. Lecture Note Series. 35). Cambridge University Press, Cambridge etc. 1979, ISBN 0-521-22286-9 (2nd edition: (= Mathematical Surveys and Monographs. 120). American Mathematical Society, Providence RI 2005, ISBN 0-8218-3581-5).
• with Hans L. Cycon, Richard G. Froese, and Werner Kirsch: Schrödinger Operators. Springer, Berlin etc. 1987, ISBN 3-540-16758-7 (corrected and extended 2nd printing: Springer 2008, ISBN 978-3-540-16758-7).
• The Statistical mechanics of lattice gases. vol. 1. Princeton University Press, Princeton NJ 1993, ISBN 0-691-08779-2.
• Orthogonal polynomials on the unit circle (= American Mathematical Society Colloquium Publications. 54, 1–2). 2 vols. American Mathematical Society, Providence RI 2005;
• Convexity. An analytic viewpoint (= Cambridge Tracts in Mathematics. 187). Cambridge University Press, Cambridge etc. 2011, ISBN 1-107-00731-3.[16]
• Szegő´s theorem and its descendants. Spectral theory for ${\displaystyle L^{2}}$ perturbations of orthogonal polynomials. Princeton University Press, Princeton NJ 2011, ISBN 978-0-691-14704-8.
• A Comprehensive Course in Analysis. 4 vols. with vol. 2 published in 2 parts, American Mathematical Society, Providence RI 2015, ISBN 978-1-4704-1098-8.
• vol. 1: Real Analysis.[17]
• vol. 2A: Basic Complex Analysis.
• vol. 2B: Advanced Complex Analysis.
• vol. 3: Harmonic Analysis.
• vol. 4: Operator Theory.
• Loewner's theorem on monotone matrix functions Springer, 2019, ISBN 978-3-030-22421-9[18]

## References

1. ^ "Dr. Barry Simon Speaking at the James Madison High School May 15th 2016 Wall of Distinction Event". YouTube. Steve Mekler. May 31, 2016.
2. ^ Caltech Math Faculty page
3. Gesztesy, Fritz; Harrell, Evans M.; Deift, Percy A.; Rosen, Lon; Fröhlich, Jürg; Reed, Mike (1 August 2016). "From Mathematical Physics to Analysis: A Walk in Barry Simon's Mathematical Garden". Notices of the American Mathematical Society. 63 (7): 740–752. doi:10.1090/noti1403.
4. ^ Vogel, Mike (June 10, 2015). "What is it about Brooklyn's James Madison High?". am New York. The school has selected honorees for its 2016 Wall of Distinction, former Madison alumni association president Richard Kossoff told me. They include food critic Arthur Schwartz, renowned physicist and mathematician Barry Simon, screenwriter Roger Schulman, who co-wrote "Shrek," and the late David "Sonny" Werblin, the NY Jets owner who brought Joe Namath to the team.
5. ^ a b Terte, Robert H. "One Student Plus one Challenge Equals One Perfect Math Score". The New York Times. p. 39.
6. ^ "Putnam Competition Individual and Team Winners". Mathematical Association of America. Retrieved December 9, 2021.
7. ^ Clavin, Whitney (August 10, 2016). "Lifetime of Numbers: Q&A with Barry Simon". Caltech News. California Institute of Technology.
8. ^ "Simonfest Barry Stories: Jürg Fröhlich". math.caltech.edu.
9. ^ "Simonfest Barry Stories: Evans Harrell". math.caltech.edu.
10. ^ "Approximation of Feynman integrals and Markov fields by spin systems by Barry Simon". Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974). Vol. 2. Montreal: Canad. Math. Congress. 1975. pp. 399–402. MR 0441161.
11. ^ "APS Fellow Archive". American Physical Soceity. (Search on year 1981 & institution Princeton University.)
12. ^ List of Fellows of the American Mathematical Society. Retrieved 2013-07-20.
13. ^ Bolyai Prize goes to Barry Simon Archived 2015-12-22 at the Wayback Machine Communication by Hungarian Academy of Sciences (mta.hu). Retrieved 12 April 2015
14. ^ "Dannie Heineman Prize for Mathematical Physics". American Physical Society. Retrieved 7 September 2018.
15. ^ Galvin, Molly (April 30, 2019). "National Academy of Sciences Elects Members and Foreign Associates; Historic Number of Women Elected to Its Membership". News from the National Academy of Sciences. National Academy of Sciences.
16. ^ Poplicher, Mihaela (April 19, 2021). "Review of Convexity: An Analytic Viewpoint by Barry Simon". MAA Reviews, Mathematical Association of America.
17. ^ Berg, Michael (January 28, 2016). "Review of A Comprehensive Course in Analysis. Part I by Barry Simon". MAA Reviews, Mathematical Association of America.
18. ^ McCarthy, John E. (2019-12-16). "Book Review: Loewner's theorem on monotone matrix functions". Bulletin of the American Mathematical Society. 57 (4): 679–684. doi:10.1090/bull/1688. ISSN 0273-0979. S2CID 213549854.