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Barry Martin Simon (born 16 April 1946) is an American mathematical physicist and was the IBM professor of Mathematics and Theoretical Physics at Caltech,[1] 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.

Key Information

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.[2]

Early life

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Barry Simon's mother was a school teacher, his father was an accountant. His ancestors were from Odesa and Grodno. His grandfather got the new surname, Simon, at Ellis Island; his original surname was Slopak.[3] Simon attended James Madison High School in Brooklyn.[4][5] Simon is an observant Jew.[3]

Career

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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][2]

In 1962, Simon entered Harvard with a stipend. He became a Putnam Fellow in 1965 at 19 years old.[6] He received his AB in 1966 from Harvard College and his PhD in Physics at Princeton University in 1970, supervised by Arthur Strong Wightman. His dissertation dealt with Quantum mechanics for Hamiltonians defined as quadratic forms.[2]

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.[2][7]

Honors and awards

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Selected publications

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Articles

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  • Simon, Barry (1973). "Resonances in n-Body Quantum Systems With Dilatation Analytic Potentials and the Foundations of Time-Dependent Perturbation Theory". The Annals of Mathematics. 97 (2): 247–274. doi:10.2307/1970847. JSTOR 1970847. (over 700 citations)
  • Guerra, F.; Rosen, L.; Simon, B. (1975). "The P(φ)2 Euclidean Quantum Field Theory as Classical Statistical Mechanics". The Annals of Mathematics. 101 (1): 111-259. doi:10.2307/1970988. JSTOR 1970988.
  • Lieb, Elliott H; Simon, Barry (1977). "The Thomas-Fermi theory of atoms, molecules and solids". Advances in Mathematics. 23 (1): 22–116. doi:10.1016/0001-8708(77)90108-6. (over 700 citations)
  • Fröhlich, J.; Simon, B.; Spencer, T. (1976). "Infrared bounds, phase transitions and continuous symmetry breaking". Communications in Mathematical Physics. 50 (1): 79–95. Bibcode:1976CMaPh..50...79F. doi:10.1007/BF01608557. ISSN 0010-3616.
  • Perry, P.; Sigal, I. M.; Simon, B. (1981). "Spectral Analysis of N-Body Schrodinger Operators". The Annals of Mathematics. 114 (3): 519-567. doi:10.2307/1971301. JSTOR 1971301.
  • Simon, Barry (1982). "Schrödinger semigroups". Bulletin of the American Mathematical Society. 7 (3): 447–526. doi:10.1090/S0273-0979-1982-15041-8. ISSN 0273-0979. (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)
  • Simon, Barry (1983-12-12). "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase". Physical Review Letters. 51 (24): 2167–2170. Bibcode:1983PhRvL..51.2167S. doi:10.1103/PhysRevLett.51.2167. ISSN 0031-9007. (over 2050 citations)
  • Avron, J. E.; Seiler, R.; Simon, B. (1983-07-04). "Homotopy and Quantization in Condensed Matter Physics". Physical Review Letters. 51 (1): 51–53. Bibcode:1983PhRvL..51...51A. doi:10.1103/PhysRevLett.51.51. ISSN 0031-9007. (over 600 citations)
  • Simon, Barry (1984). "Semiclassical Analysis of Low Lying Eigenvalues, II. Tunneling". The Annals of Mathematics. 120 (1): 89–118. doi:10.2307/2007072. JSTOR 2007072.
  • Simon, Barry; Wolff, Tom (1986). "Singular continuous spectrum under rank one perturbations and localization for random hamiltonians". Communications on Pure and Applied Mathematics. 39 (1): 75–90. doi:10.1002/cpa.3160390105. ISSN 0010-3640.
  • Simon, Barry (1995). "Operators with Singular Continuous Spectrum: I. General Operators". The Annals of Mathematics. 141 (1): 131–145. doi:10.2307/2118629. JSTOR 2118629.

Books

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  • 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 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 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]

See also

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References

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Further reading

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[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Barry Simon

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Barry Simon (born April 16, 1946) is an American mathematical physicist and analyst renowned for his pioneering work in , Schrödinger operators, , , and related areas of . Simon earned an A.B. summa cum laude from in 1966 and a Ph.D. in physics from in 1970 under the supervision of Arthur Wightman, with a thesis on for Hamiltonians defined as quadratic forms. He began his academic career as an instructor at Princeton in 1969, advancing to assistant in 1970, associate in 1972, and full in 1976, remaining on the faculty until 1981. In 1981, he joined the (Caltech) as a , becoming the IBM Professor of Mathematics and in 1984—a position he held until 2016, after which he became . During his tenure at Caltech, he also served as (chair) of the Department of Mathematics from 1997 to 2003 and again from 2009 to 2013. Simon's research has profoundly influenced modern , with over 400 published papers and more than 20 books that have become standard references in the field, including the four-volume Methods of Modern Mathematical Physics co-authored with Michael Reed. His contributions include foundational advances in the of Schrödinger operators, eigenvalue , orthogonal polynomials, Berry's phase, and non-relativistic , often bridging rigorous analysis with physical applications. He has supervised 30 Ph.D. theses, shaping generations of researchers in these domains. Among his numerous honors, Simon received the 2012 Poincaré Prize for his work in and , the 2015 International Bolyai Prize, the 2016 Leroy P. Steele Prize for Lifetime Achievement from the , and the 2018 Dannie Heineman Prize for Mathematical Physics from the . He is a winner of the Putnam Mathematical Competition, holds three honorary doctorates from the Technion-Israel Institute of Technology, the University of Wales Swansea, and Ludwig Maximilians-Universität München, and was elected to the in 2019, as well as the American Academy of Arts and Sciences and the .

Early life and education

Early life

Barry Simon was born on April 16, 1946, in , , to a Jewish family. His parents were Hyman Simon (1910–1994), a post office clerk who held an accounting degree from , and Minnie Landa (1912–2002), an assistant buyer at Macy’s who had studied at . He had an older brother, Richard C. Simon, born in 1942. Simon attended a Reform Hebrew elementary school and a junior high school in Brooklyn before enrolling at James Madison High School, located near his family's apartment. At James Madison, he demonstrated exceptional talent in mathematics and physics, skipping grades to take advanced courses such as calculus during his junior year and earning a mathematics medal upon graduation in 1963. His early interest in science was nurtured through the Science Honors Program, which he joined in 1958 and where he studied topics including , vector analysis, and under the guidance of teacher Sam Marantz; he also achieved a perfect score of 150 on the 1962 American High School Mathematics Examination. Following high school, Simon transitioned to undergraduate studies at .

Education

Simon earned his AB degree from in 1966, with a focus on coursework in both and physics. His undergraduate studies included advanced topics in these fields, bolstered by his participation as an Individual Putnam Fellow in the 1965 , where the Harvard team ranked first. He pursued graduate studies at , obtaining his PhD in physics in 1970 under the supervision of Arthur Wightman. Simon's doctoral thesis, titled Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, addressed foundational aspects of through . Wightman's mentorship profoundly influenced Simon's early work in , particularly through exposure to Wightman's seminars on axiomatic and related rigorous methods.

Professional career

Early positions

Following his PhD in physics from in 1970 under the supervision of Arthur Wightman, which equipped him with a strong foundation in , Barry Simon immediately transitioned into faculty roles at the same institution. He began as an instructor in the Department of Mathematics during the 1969–1970 academic year, overlapping with the completion of his doctorate. From 1970 to 1972, Simon served as an with a joint appointment in the Departments of and Physics at Princeton, where he received tenure by December 1971. He advanced to in 1972 and held that position until 1976, after which he was promoted to full professor, maintaining the joint appointment until 1981. During this period, Simon negotiated a reduced load to one course per semester, supported by external grants that supplemented his salary. His classes typically enrolled 25–30 advanced students. Simon also took on administrative responsibilities in the Physics Department, serving as director of graduate studies from 1973 to 1974 and as chair of the graduate admissions committee from 1979 to 1980. In these roles, he managed , including admissions processes and appeals for draft deferments during the era, and handled departmental challenges such as security incidents involving graduate students. A key early collaboration during his Princeton years was with Michael Reed, resulting in the co-authored multi-volume series Methods of Modern Mathematical Physics, with the first volume on published in 1972. This work marked a significant portion of his initial research output, establishing foundational texts in the field. In 1976–1977, Simon took a one-year leave from Princeton to serve as visiting professor in the Department of Physics at in , commuting from Princeton at the invitation of Joel Lebowitz.

Caltech tenure

In 1981, Barry Simon joined the as a professor of mathematics and theoretical physics, marking the beginning of his long tenure at the institution. His prior administrative experience at , including directing graduate studies in physics, positioned him to contribute significantly to Caltech's academic leadership from the outset. Simon's roles at Caltech expanded in 1984 when he was appointed the Professor of Mathematics and , a position he held until 2016. He also served as chair of the Caltech Theoretical Physics Staffing Committee in 1983, aiding in faculty recruitment efforts. From 1997 to 2003 and again from 2009 to 2013, Simon acted as (department chair) of the Mathematics Department, overseeing operations, hiring initiatives, and responses to challenges such as faculty shortages in the early . During his second term, he implemented a math camp for incoming students with weaker preparation and adapted the curriculum to accommodate new exam formats. Additionally, as a member of the Stevenson Committee from 1994 to 1996, he helped redesign Caltech's core undergraduate curriculum, reducing the required math and physics quarters from six to five while incorporating and components. Simon retired in 2016, becoming the Professor of Mathematics and , . Despite retirement, he remained actively involved at Caltech, continuing research collaborations and major book projects, and supervising Ph.D. students, with a total of 30 under his guidance across his career.

Research areas

Spectral theory and Schrödinger operators

Barry Simon made foundational contributions to the rigorous of Schrödinger operators, which are central to nonrelativistic and model the Hamiltonian H=−Δ+VH = -\Delta + V for a potential VV. His early work established essential self-adjointness for these operators under broad conditions on VV, ensuring unique self-adjoint extensions and well-defined via the Stone's . This framework extended classical results by Kato and Rellich, incorporating singular potentials and magnetic fields to handle realistic . In scattering theory, Simon co-authored a comprehensive that provided the modern axiomatic approach to wave operators and asymptotic completeness for . This work resolved key and completeness problems for short-range potentials, enabling the study of particle interactions over long times and distances. His results on the absence of ballistic motion for operators with point spectrum further clarified propagation properties in disordered systems. Simon's eigenvalue estimates advanced the understanding of bound states in Schrödinger operators. For instance, he proved bounds on the growth of the number of negative eigenvalues as the potential strength increases, providing quantitative control essential for stability analyses in . A landmark result is the Simon-Wolff theorem (1986), which demonstrates that for rank-one perturbations of an operator with absolutely continuous spectrum, embedded eigenvalues are absent for a dense GδG_\delta set of coupling constants, implying purely singular continuous spectrum in certain intervals. This theorem has profound implications for embedded eigenvalues in random Hamiltonians and . Simon contributed to inequalities pivotal in , including extensions and applications of the Hardy inequality to ensure positivity and domain properties of operators. In particular, his work utilized Hardy-type estimates to analyze perturbations and self-adjointness for potentials with Coulomb singularities, facilitating proofs of estimates. Applications of Simon's extend to atomic and molecular Hamiltonians, where he examined potential perturbations to model interactions. For example, rank-one perturbations of free Laplacians yield singular continuous spectra densely in parameter space, with proofs relying on Borel-Stieltjes transforms and resolvent estimates to exclude point spectrum. These results apply to perturbed atomic systems, showing how small changes in boundary conditions or potentials can induce exotic spectral measures without embedded eigenvalues. In collaboration with Elliott Lieb, Simon addressed the stability of matter, proving that the total of fermionic systems is bounded below proportionally to the number of particles, preventing collapse under forces. Their 1977 work on Thomas-Fermi theory rigorously justified the model for large atoms and molecules, showing it becomes exact in the limit of high nuclear charge and deriving asymptotic eigenvalue distributions for molecular Hamiltonians via potential perturbations. This collaboration integrated spectral estimates with variational methods to confirm the extensivity of matter's .

Orthogonal polynomials and random matrices

Barry Simon made significant contributions to the theory of orthogonal polynomials, particularly orthogonal polynomials on the real line (OPRL) and on the unit circle (OPUC), and their deep connections to random matrix theory. His two-volume work, Orthogonal Polynomials on the Unit Circle, provides a comprehensive treatment of the classical and spectral theory of OPUC, emphasizing the role of Verblunsky coefficients in relating recurrence relations to measures on the unit circle. In this framework, Simon established key asymptotic behaviors for the norms of orthogonal polynomials, such as Szegő's theorem, which states that for a measure dμ=w(θ)dθ2πd\mu = w(\theta) \frac{d\theta}{2\pi} on the unit circle, lim⁡n→∞∥ϕn∥2=exp⁡(12π∫02πlog⁡w(θ) dθ)\lim_{n \to \infty} \|\phi_n\|^2 = \exp\left( \frac{1}{2\pi} \int_0^{2\pi} \log w(\theta) \, d\theta \right), where ϕn\phi_n are the orthonormal polynomials. These results underpin the spectral analysis of associated operators and extend to OPRL via Jacobi matrices. Simon's proofs of asymptotic behaviors for orthogonal polynomials on the real line include extensions relevant to random matrix ensembles, such as refinements to the Marchenko-Pastur law. In the context of the Pastur-Ishii theorem, he demonstrated that "bad sets" for the limiting spectral measures—regions where the support does not align with the equilibrium measure—have zero logarithmic capacity using potential theory, ensuring robust convergence properties for Wishart-type random matrices. This work connects the eigenvalue distributions of large random matrices to the zeros of orthogonal polynomials, where the Marchenko-Pastur distribution emerges as the limiting density for sample covariance matrices. Furthermore, Simon's analysis of the fine structure of zeros in orthogonal polynomials reveals universal spacing patterns, linking discrete spectral measures to continuous limits in random matrix models. A central theme in Simon's research is the role of determinantal point processes in random matrix theory, realized through the Christoffel-Darboux (CD) kernel. The CD kernel for OPRL is given by Kn(z,ζ)=∑j=0npj(z)pj(ζ)=an+1[pn+1(z)pn(ζ)−pn(z)pn+1(ζ)]z−ζ,K_n(z, \zeta) = \sum_{j=0}^n p_j(z) p_j(\zeta) = \frac{a_{n+1} [p_{n+1}(z) p_n(\zeta) - p_n(z) p_{n+1}(\zeta)]}{z - \zeta}, where pjp_j are orthonormal polynomials and ana_n are recursion coefficients. Simon's review highlights its use in establishing universality of local eigenvalue statistics: for measures with absolutely continuous spectrum, 1nKn(x+a/n,x+b/n)/Kn(x,x)→sin⁡(πρe(x)(b−a))πρe(x)(b−a)\frac{1}{n} K_n(x + a/n, x + b/n) / K_n(x,x) \to \frac{\sin(\pi \rho_e(x) (b - a))}{\pi \rho_e(x) (b - a)} in the bulk, where ρe\rho_e is the equilibrium density. This sine-kernel limit demonstrates that eigenvalue correlations in diverse random matrix ensembles, including Gaussian and Jacobi types, follow universal laws independent of specific distributions. In collaboration with others, Simon proved bulk universality and clock spacing for zeros of orthogonal polynomials associated with ergodic Jacobi matrices possessing absolutely continuous spectrum. Specifically, for almost every xx in the spectral support Σac\Sigma_{ac}, the normalized CD kernel satisfies lim⁡n→∞1nKn(x,x)=ρ∞(x)w(x)\lim_{n \to \infty} \frac{1}{n} K_n(x,x) = \frac{\rho_\infty(x)}{w(x)}, where ρ∞\rho_\infty is the limiting density of zeros and ww is the a.c. spectral weight, leading to quasi-clock spacing where consecutive zeros Ej+1(n)(x)−Ej(n)(x)≈1/(nρ∞(x))E_{j+1}^{(n)}(x) - E_j^{(n)}(x) \approx 1/(n \rho_\infty(x)). These results extend classical universality from fixed measures to random, ergodic settings, mirroring behaviors in random matrix eigenvalue distributions. The CD kernel's properties also facilitate applications to integrable systems in statistical mechanics, such as the Calogero-Moser model, where orthogonal polynomials describe joint eigenfunctions and correlation structures in many-body interactions.

Other contributions

Simon's early career included significant contributions to constructive , where he advanced axiomatic approaches to establishing rigorous foundations for quantum fields. In his early career, following his PhD under Wightman at Princeton, he focused on Wightman functions and their role in verifying the for theories like P(φ)_2, providing mathematical rigor to previously heuristic models in relativistic . These efforts helped bridge abstract mathematical structures with physical predictions, influencing subsequent developments in Euclidean . In , Simon made key advances in understanding phase transitions, particularly for s and related lattice systems. He developed correlation inequalities that illuminated the behavior of spin systems at criticality, such as those for plane rotors and higher-dimensional vector models, which provided bounds on correlation functions essential for proving the existence of phase transitions. His work on reflection positivity further served as a powerful tool for analyzing the , enabling proofs of long-range order and in ferromagnetic systems. These contributions extended applications to statistical models, offering insights into thermodynamic limits. Beyond these areas, Simon explored orthogonal polynomial ensembles in contexts outside random matrices, such as analytic theory on the unit circle, which connected to broader problems in and approximation theory. He also made brief forays into and pure , including estimates for non-normal matrices and connections to zero distributions. Throughout his career, Simon's expository writings and interdisciplinary approach solidified his role as a bridge between mathematical physics subfields, fostering deeper conceptual understanding across , field theory, and statistical systems.

Recognition and honors

Major awards

In 1965, Barry Simon was named a Fellow in the , recognizing his outstanding performance as an undergraduate at . Simon received three honorary doctorates for his contributions to : a D.Sc. honoris causa from the Technion-Israel Institute of Technology in 1999, an Honorary Fellowship from the University of Wales in 2006, and a D.h.c. from Ludwig Maximilians-Universität München in 2010. In 2012, Barry Simon received the Henri Poincaré Prize from the International Association of Mathematical Physics (IAMP) for his outstanding contributions to . This triennial award recognizes exceptional achievements in the field, and Simon was one of four recipients that year, honored for his profound influence on areas such as and . The International Prize, awarded by the in 2015, was given to Simon for his seminal work in , particularly through his comprehensive monographs on orthogonal polynomials. Presented every five years to the author of the most outstanding mathematical treatise from the preceding period, the prize underscores Simon's rigorous advancements in analyzing operator spectra and their applications. In 2016, the (AMS) bestowed upon Simon the Leroy P. Steele Prize for Lifetime Achievement, acknowledging his enduring impact across . This prestigious honor, awarded annually for sustained excellence in research, highlights Simon's broad contributions that have shaped education and scholarship in the discipline over decades. Simon was awarded the 2018 Dannie Heineman Prize for Mathematical Physics by the (AIP) and the (APS), recognizing his foundational advancements in . The prize, which celebrates pioneering work bridging mathematics and physics, specifically cited Simon's rigorous developments in quantum many-body systems and related topics.

Professional memberships

Barry Simon was elected to the in 2019, recognizing his contributions to . He has been a fellow of the American Academy of Arts and Sciences since his election in 2005. Simon was among the inaugural class of fellows elected by the in 2013. Additionally, he serves as a corresponding member of the , a position he has held since 1990. Simon's influence within the mathematical community is further evidenced by his service on the Editorial Boards Committee of the from 1989 to 1993.

Publications

Books

Barry Simon has authored or co-authored over 20 books and monographs, many of which have become standard references in , analysis, and related fields, influencing graduate education through their rigorous treatments and extensive use in advanced courses. These works often integrate his research themes, such as , providing pedagogical bridges between abstract mathematics and physical applications. A landmark collaboration is the four-volume series Methods of Modern Mathematical Physics, co-authored with Michael Reed and published by Academic Press between 1972 and 1979 (Volume I: Functional Analysis, 1972, revised 1980; Volume II: Fourier Analysis, Self-Adjointness, 1975; Volume III: Scattering Theory, 1979; Volume IV: Analysis of Operators, 1978). This series establishes foundational principles of , , and , serving as a core text for graduate students and researchers in . Its comprehensive approach to operators and has made it a highly influential reference, with individual volumes cited thousands of times in the literature. In 2015, Simon published the five-volume A Comprehensive Course in Analysis with the (Part 1: Real Analysis; Part 2A: Basic Complex Analysis; Part 2B: Advanced Complex Analysis; Part 3: Harmonic Analysis; Part 4: Operator Theory). This set offers a modern, unified treatment of real and complex analysis, emphasizing insightful proofs, historical context, and advanced topics like the spectral theorem and , while including over 3,700 references and extensive exercises. It functions as both a graduate textbook and research reference, beta-tested in courses and praised for its depth and selectivity in covering the contemporary analysis corpus. Among his other monographs, The P(φ)₂ Euclidean ( (Princeton University Press, 1974) provided a complete of existence for two-dimensional scalar fields, establishing benchmarks for Euclidean quantum field models. The two-part Orthogonal Polynomials on the Unit Circle (, 2005; Part 1: Classical Theory; Part 2: ) revolutionized the field by linking orthogonal polynomials to operator and via innovations like the CMV matrix and multiple proofs of key , such as the strong Szegő theorem. This work, drawing over 1,100 references, has had profound impact on probability measures on the unit circle and related areas in . (, 1993) provides a rigorous survey of classical and quantum lattice models, advancing understanding of phase transitions and correlation inequalities. Additional notable titles include Trace Ideals and Their Applications (, 1979; second edition, , 2005), which details applications in spectral asymptotics, and Szegő's Theorem and Its Descendants (, 2009), extending spectral analysis of orthogonal polynomials. These books, along with others like Convexity: An Analytic Viewpoint (, 2011) and Loewner's Theorem on Monotone Matrix Functions (Springer, 2019), underscore Simon's prolific output and enduring pedagogical influence in graduate programs worldwide.

Selected papers

Barry Simon has authored more than 450 research papers over five decades, accumulating over 95,000 citations and an of 121 on (as of November 2025). His publications evolved from foundational contributions in constructive during the 1970s to rigorous analyses of Schrödinger operators in the , and later to orthogonal polynomials, random matrix theory, and spectral problems from the through the and beyond. Post-2016 works continued this trajectory, emphasizing advanced and analytic extensions of earlier themes, including papers through 2024 on topics such as sum rules and large deviations in random systems. In the 1970s, Simon's papers advanced through rigorous constructions and analytic techniques. A key example is his 1970 work on "Coupling constant analyticity for the anharmonic oscillator," co-authored with A. Dicke, which proved analyticity in for quantum mechanical models, enabling perturbative expansions and cited over 800 times. The 1977 collaboration with E. H. Lieb on "The Thomas-Fermi theory of atoms, molecules and solids" derived variational principles for density functionals, foundational for and cited nearly 900 times. Shifting to spectral theory in the 1980s, Simon's 1982 paper "Schrödinger semigroups," published in the Bulletin of the , offered a comprehensive review of semigroup methods for time-dependent Schrödinger equations, influencing and cited over 1,600 times. His 1983 solo paper "Holonomy, the quantum , and Berry's phase" in formalized Berry's phase in , linking adiabatic evolution to geometric phases and becoming one of his most cited works with over 2,400 citations. On Schrödinger operators, the 1982 collaboration with M. Aizenman, "Brownian motion and Harnack inequality for Schrödinger operators," introduced and applied the IMS localization formula—a partition-of-unity technique bounding eigenvalues via potential decompositions—proving Harnack inequalities for positive solutions and cited over 700 times. The IMS formula, further developed in Simon's 1983 semiclassical analysis paper, enabled precise eigenvalue asymptotics for low-lying states in multi-dimensional potentials. Simon's 1980s and 1990s contributions extended to random systems and orthogonal polynomials. The 1985 paper "Localization in general one dimensional random systems, I. Jacobi matrices" established dynamical localization for disordered Schrödinger operators, a in theory for understanding spectral gaps and , with broad impact in . In orthogonal polynomials, his work on CMV matrices—unitary representations of measures on the unit circle—culminated in the 2006 review "CMV matrices: Five years after," which synthesized their as analogs to Jacobi matrices, facilitating computations of zeros and Verblunsky coefficients, and cited extensively in theory. This built on earlier 1980s papers like "Kotani theory for one dimensional Jacobi matrices" (1983), linking matrices to Lyapunov exponents in ensembles. From the 2000s onward, Simon's papers emphasized high-impact intersections of these themes. The 1994 collaboration "Singular continuous spectrum is generic" proved the ubiquity of singular continuous spectra in perturbed operators, resolving open questions in random matrix theory and cited over 500 times. Post-2016 publications, such as those on sum rules for meromorphic Herglotz functions (2017), extended spectral estimates to large deviation principles in random systems, maintaining his focus on precise analytic tools. Many of these articles form the basis for his monographs, providing rigorous extensions of core ideas.

References

  1. https://www.nasonline.org/directory-entry/barry-simon-muzpov/
  2. https://mathshistory.st-andrews.ac.uk/Biographies/Simon_Barry/
  3. https://www.pma.caltech.edu/people/barry-m-simon
  4. https://heritageproject.caltech.edu/interviews-updates/barry-simon
  5. https://www.mathgenealogy.org/id.php?id=11905
  6. https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-21/issue-3/Hamiltonians-defined-as-quadratic-forms/cmp/1103857333.pdf
  7. http://www.math.caltech.edu/SimonPapers/barry_simon=ams_notices=2016_07and08.pdf
  8. http://math.caltech.edu/simon/bsimon_cv.pdf
  9. https://heritageproject.caltech.edu/interviews/barry-simon
  10. http://math.caltech.edu/SIMON/bsimon_biblio.pdf
  11. https://www.jstor.org/stable/24714042
  12. https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-165/issue-1/Operators-with-singular-continuous-spectrum-II-Rank-one-operators/cmp/1104271034.pdf
  13. http://math.caltech.edu/SimonPapers/R54.pdf
  14. https://arxiv.org/pdf/0806.1528
  15. http://math.caltech.edu/papers/bsimon/p324.pdf
  16. https://www.jstor.org/stable/1970990
  17. https://inspirehep.net/literature/1698988
  18. http://math.caltech.edu/SimonPapers/LatticeGasesSamplerSimon.pdf
  19. https://www.cambridge.org/us/universitypress/subjects/mathematics/mathematical-physics/phase-transitions-theory-lattice-gases
  20. https://arxiv.org/pdf/math-ph/0607025
  21. http://www.math.caltech.edu/papers/bsimon/l.pdf
  22. https://www.caltech.edu/about/news/caltech-professor-barry-simon-wins-henri-poincare-prize-23607
  23. https://www.iamp.org/poincare/bs12-laud.pdf
  24. https://www.caltech.edu/about/news/simon-wins-international-mathematics-prize-46655
  25. http://math.caltech.edu/SimonPapers/TotikBolyai.pdf
  26. https://www.eurekalert.org/news-releases/812792
  27. https://www.eurekalert.org/news-releases/870739
  28. https://www.amacad.org/person/barry-martin-simon
  29. https://www.ams.org/cgi-bin/fellows/fellows.cgi
  30. http://www.oeaw.ac.at/en/m/simon-barry
  31. https://mathshistory.st-andrews.ac.uk/Extras/Simon_Barry_books/
  32. https://scholar.google.com/citations?user=OM0D_3wAAAAJ&hl=en
  33. https://www.ams.org/bookpages/simon
  34. https://www.ams.org/publications/journals/notices/201901/rnoti-p74.pdf
  35. https://www.ams.org/journals/bull/2007-44-03/S0273-0979-07-01135-4/S0273-0979-07-01135-4.pdf
  36. http://www.math.caltech.edu/simon/biblio.html
  37. http://www.math.caltech.edu/simon/simon.html
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