frommathematicalphysicsto analysis:awalkinbarry simon ... · simon’s four-volume series, methods...

From Mathematical Physics toAnalysis: A Walk in BarrySimon’s Mathematical Garden

Editor’s Note: Fritz Gesztesy kindly accepted our invitation to put together this feature in honor of Barry Simon on theoccasion of Simon’s 2016 AMS Leroy P. Steele Prize for Lifetime Achievement and his 70th birthday conference thisAugust 28–September 1.

Fritz Gesztesy

This is a collection of contributions by collabo-rators, postdocs, and students of Barry Simonof the California Institute of Technology on theoccasion of Simon’s receiving the 2016 AMSLeroy P. Steele Prize for Lifetime Achievement.

The citation for the award mentions his tremendous im-pact on the education and research of a whole generationof mathematical scientists, and we will underscore thisby demonstrating his penetrating influence on topicsranging from quantum field theory, statistical mechanics,the general theory of Schrödinger operators, spectral andinverse spectral theory to orthogonal polynomials.

But we should start at the beginning: Barry was born toparents Minnie and Hy Simon in 1946, and together withhis older brother, Rick, grew up in Brooklyn, New York.There he attended James Madison High School, obtaininga perfect score on the MAA’s American High SchoolMathematics Examinaton in 1962 and thus becoming thesubject of an article in the New York Times at the tenderage of sixteen. Under the influence of Sam Marantz, aninspiring physics teacher in high school, he applied toHarvard and was admitted. While at Harvard he was a topfive Putnam Competition Winner in 1965 and received hisBA summa cum laude in physics in 1966. George Mackeyat Harvard recommended Barry pursue a doctorate withArthur Wightman at Princeton because Wightman waswell known for advocating the application of rigorousmathematics in physics.

Fritz Gesztesy is Mahala and Rose Houchins Professor of Mathe-matics at the University of Missouri, Columbia. His email addressis [email protected]. Since August 1 he is Jean andRalph Storm Chair of Mathematics at Baylor University. His emailaddress there is [email protected].

For permission to reprint this article, please contact:[email protected]: http://dx.doi.org/10.1090/noti1403

Barry Simon with his mother and brother, Rick (left),and with his father (right) (ca. 1950).

Barry Simon completed his doctorate in physics atPrinceton under Wightman’s supervision in 1970. Thebody of his work during the time of his doctoral re-search was of such importance that he was immediatelyappointed to assistant professor, jointly in mathematicsand physics, at Princeton. He rapidly rose to the rank offull professor by 1976. Several contributions below willattest to the electric atmosphere at Princeton in thosedays, making it a thriving center for quantum field theory,statistical mechanics, and nonrelativistic quantum me-chanics. Barry joined Caltech in 1981, holding the positionof IBM Professor of Mathematics and Theoretical Physicssince 1984. At Caltech, Barry’s interests further broad-ened into areas such as random and ergodic Schrödingeroperators, exotic spectra, inverse spectral theory, and theanalytic theory of orthogonal polynomials.

740 Notices of the AMS Volume 63, Number 7

Honorary doctorate, Ludwig-Maximilians-Universityof Munich, 2014.

… rigorousmathematicsin physics

Since many of Barry’s ma-jor research accomplishmentsare discussed in depth and putinto proper context in the vari-ous contributions to follow, weitemize only a few brief com-ments at this point, focusingon some key results he proved,fundamental concepts he ad-

vocated, and some of the important terms he and hiscollaborators first coined with lasting impact:

• a rigorous framework for resonances, complexand exterior scaling, Fermi’s golden rule, proofof the Oppenheimer formula for the Stark ef-fect, convergence of time-dependent perturbationtheory;

• constructive (Euclidean) quantum field theory intwo space-time dimensions, connections to sta-tistical mechanics, lattice approximations andcorrelation inequalities, 𝑃(𝜙)2 spatially cutofffield theories;

• hypercontractive and ultracontractive semi-groups;

• magnetic fields, diamagnetic inequality, Kato’sinequality;

• a proof of continuous symmetry breaking inclassical and quantum statistical models;

• Thomas–Fermi theory, semiclassical bounds, non-Weyl asymptotics;

• asymptotic perturbation theory of eigenvalues:Borel and Padé summability, Zeeman effect, anhar-monic oscillators, instanton tunneling, Birman–Schwinger principle, coupling constant thresh-olds;

• general theory of Schrödinger operators: essen-tial self-adjointness, pointwise bounds on eigen-functions, path integral techniques, absence ofsingular continuous spectrum in 𝑁-body sytems;

• Berry’s phase and holonomy, homotopic interpre-tation of the Thouless integers and topologicalstructure in the integer quantum Hall effect;

• random and almost periodic Schrödinger andJacobi operators, exotic spectral phenomena (Can-tor, singular continuous, and dense pure pointspectra) and their transition to becoming a cen-tral object in mathematical physics (the singu-lar continuous revolution), Wonderland theorem,Thouless formula, almost Mathieu equation;

• trace formulas for potential coefficients in termsof the Krein–Lifshitz spectral shift function,uniqueness theorems in inverse spectral theoryfor Schrödinger and Jacobi operators, oscillationtheory in gaps of the essential spectrum, inversespectral analysis with partial information on thepotential;

• a new approach (the analog of the continuedfraction method) to inverse spectral theory ofSchrödinger operators, his local Borg–Marchenkotheorem;

• a systematic application of operator theory tech-niques to orthogonal polynomials on the realline (OPRL) and on the unit circle (OPUC), CMVmatrices, Verblunsky coefficients;

• sum rules for Jacobi matrices and applications tospectral theory (Killip–Simon theorem), perturba-tions of OPRL and OPUC with periodic recursioncoefficients;

• Szegő asymptotics, a proof of Nevai’s conjectureand its finite gap extension, the finite gap analogof the Szegő-Shohat-Nevai theorem, the fine struc-ture of zeros of orthogonal polynomials (clockbehavior), higher-order Szegő theorems.

Barry Simon’s influence on our community by fartranscends his approximately four hundred papers, par-ticularly in viewof126coauthors, 50mentees, 31graduatestudents, and about 50 postdocs mentored. In this con-text, one must especially mention his twenty books, thefirst fifteen of which have educated scores of mathemati-cians and mathematical physicists, two generations bynow, and continuing into the foreseeable future.

One cannot overestimate the influence of Reed andSimon’s four-volume series, Methods of Modern Mathe-matical Physics, I–IV (1972–79). It took on the same levelof importance that Courant–Hilbert’s two volumes hadfor previous generations, and it continues to fill that role

August 2016 Notices of the AMS 741

all over the globe to this day. To gauge the importanceof Reed and Simon behind the Iron Curtain, we contactedAlbrecht Böttcher (TU Chemnitz, Germany). Like so manyin our generation, Albrecht has a very personal relation-ship with Volume I and underscored (his words) “the trulyingenious selection and presentation of the mathematicaltopics.” The latter sentiment, however, is by no meansunique to colleagues who read Reed-Simon in Russiantranslation; it is just as prevalent in the West.

One cannotoverestimate theinfluence of Reed

and Simon’sMethods ofModern

MathematicalPhysics, I–IV

Cumulative salesfigures for all fourvolumes to date top37,000 copies. Barry’sother books, most no-tably Functional Inte-gration and QuantumPhysics (1979), TraceIdeals and Their Appli-cations (1979, 2005),Orthogonal Polynomi-als on the Unit Circle,Parts 1, 2 (2005), andSzegő’s Theorem andIts Descendants (2011),profoundly influencedresearch in these ar-eas. Finally, we have

not even begun to understand the legacy which will becreated with his newest five-volume set, A ComprehensiveCourse in Analysis (2015), which offers a panorama fromreal to complex and harmonic analysis all the way tooperator theory.

In short, Barry has been a phenomenal force in math-ematical physics, encyclopedic in his knowledge and agrand master of mathematical structure and abstractanalysis. Above all, he truly enjoys collaboration and thehuman interactions that come with it. As a sign of histremendous influence on our community we note that todate MathSciNet lists 15,325 citations by 6,602 authors,and Google Scholar lists 61,680 citations and an h-indexof 104.

Barry has been recognized with honorary degrees fromthe Technion, Haifa; the University of Wales–Swansea,and LMU–Munich. He was awarded the Stampacchia Prizein 1982 with M. Aizenman, the Poincaré Prize of theInternational Association ofMathematical Physics in 2012,

The books Barry Simon has authored …thus far.

and the Bolyai Prize of theHungarianAcademyof Sciencesin 2015. He is a fellow of the American Physical Society(1981), theAmericanAcademyofArts and Sciences (2005),and the AMS (2013). He has served as vice president ofthe AMS and of IAMP.

Finally, on a personal note: Having been a frequentcollaborator of Barry’s, I have often been approachedwith the assumption of being one of Barry’s students orpostdocs, but this is not the case. On the other hand, likeso many of my generation, I learned about the tools of ourtrade through his writings and especially from Reed andSimon I–IV, so of course it feels like I was Barry’s student,even though the proper term appears to be that I’m one ofhis many mentees. Barry has been a constant inspirationtome for about forty years now; I feel incredibly fortunatethat he became my mentor and friend.

Evans M. HarrellSingular Perturbation Theory and ResonancesThe very first article in Barry Simon’s publication list,which appeared in Il Nuovo Cimento when he was a 22-year-old graduate student, was concerned with singularperturbation theory. This paper showed that a certainregularized, renormalized perturbation expansion for atwo-dimensional quantum field theory model convergeswith a positive radius of convergence. As Barry candidlyadmitted in that article, in itself the result was of limitedsignificance, but in a subject for which at that time “allthe mathematically suitable results…are of a negativenature,” it announced a new, more constructive era.

To the reader familiar with Barry Simon’s works onmathematical physics of the 1970s, it is striking howmany of the hallmarks of his technique are alreadyapparent in this first article. Before entering deeply intothe research, Barry first carried out a thorough andpenetrating review of the entire literature on the subject.This signature of his method was something those ofus who were students at Princeton in the 1970s wouldwitness every time Barry began a new research project:Seeing him emerge from the library shared by Jadwinand Fine Halls with a mountain of books and articles,it was humbling to realize that Barry was not merelybrave enough to collect all of the knowledge about thenext subject he wished to study, but seemingly overnighthe would absorb it in detail and carefully assess eachcontribution for its mathematical appropriateness. Trueto form, in that first article, Barry laid out which claims inthe literature were established with mathematical rigor,which were plausibly to be believed, perhaps with someextra attention to assumptions, and which were franklydubious. Finally, Barry’s own way of formulating theproblem was sparse and clear, and his reasoning incisive.

The perturbation theory that applies to nonrelativisticquantummechanics is a linear theory, allowingstraightfor-ward calculations of systematically corrected eigenvalues

Evans M. Harrell is professor emeritus at the School of Mathemat-ics at the Georgia Institute of Technology. His email address [email protected].


Lorenzo Sadun, Yosi Avron, Evans Harrell, BarrySimon (ca. 1988).

and eigenfunctions. Schrödinger’s adaptation of the pro-cedures of Rayleigh to calculate the shifts in hydrogenemission spectrum in the presence of an electric field(known as the Stark effect) went a long way in establish-ing the validity of his new quantum theory. This success isironic, given that the series for which Schrödinger and Ep-stein calculated the first terms has radius of convergencezero, and the spectrum of the Stark Hamiltonian for anynonzero value of the electric field is purely continuous,containing no discrete eigenvalues at all. The instant theinteraction is switched on, the nature of the spectrumchanges radically, and the perturbed eigenvalue becomesa resonance state! The Stark effect is known today tobelong to the realm of singular perturbation theory.

Somephysicists

were “usingmethods ofunknownvalidity”

In the two decadesafter Schrödinger’s work,mathematicians created a com-prehensive theory of linearoperators on Hilbert space.In the 1930s and 1940s Rel-lich and Kato produced amathematically rigorous the-ory of regular perturbations oflinear operators and some as-pects of singular perturbations.In their hands, perturbationtheory was concerned with ana-lytic operator-valued functions

of a complex variable, defined initially as convergentpower series with operator coefficients, which are mosttypically self-adjoint in applications to quantum mechan-ics. These functions exhibit the range of behavior ofordinary scalar analytic functions of a complex variable,with manageable complications when the spectrum con-sists of discrete eigenvalues of finite multiplicity, andsome new phenomena when there is an essential spec-trum. Physicists andchemists at this time solvedproblemsand developed perturbative techniques that sometimesfell into the domain of regular perturbation, but justas frequently produced series that could be calculatedterm by term while exhibiting singular features, as in a

The Stark effect: The spectrum of the hydrogen atomunder an increasingly intense electric field, from theCourtney-Spellmeyer-Jiao-Kleppner article appearingin Physical Review A, vol. 51 (1995).

zero radius of convergence or, worse, convergence but tothe wrong answer. (For instance, a resonance eigenvalueassociated with tunneling may have a nonzero imaginarypart that, typically, is represented in perturbation theoryby a series of the form ∑𝑛 𝑐𝑛𝛽𝑛 with 𝑐𝑛 = 0 for all 𝑛.)

At the time Barry Simon hit the scene, an industrywas thriving in attempts to get information from suchexpansions, whether by replacing the series by otherexpressions, especially Padé approximates𝑃[𝑚,𝑛], whichare ratios of polynomials of𝑚 and 𝑛 degrees, or by the useof analytic-function techniques like Borel summability tomake sense of divergent series. Many of Barry’s earlyworks addressed these topics.

One of the important models in quantum mechanicswith a singular perturbation is the quartic anharmonicoscillator. Its Hamiltonian is(1) 𝑝2 + 𝑥2 +𝛽𝑥4,and it was the subject of a landmark study by Benderand Wu in 1968-69 in which, “using methods of unknownvalidity”—in Barry’s memorable phrase—they painted afascinating and largely correct picture of the analyticstructure of the eigenvalues of the anharmonic oscilla-tor as functions of the coupling constant 𝛽, consideredas a complex variable. Among other things, Bender andWu conjectured that the power series expansion for theground-state eigenvalue of (1) had radius of convergence0. Then-new computational capabilities in symbolic alge-bra had allowed Bender and Wu and others to calculateperturbation series at high orders, and with the firstseventy-five coefficients 𝑎𝑛 for the ground-state eigen-value in hand, Bender and Wu specifically conjecturedthat

(2) 𝑎𝑛 ∼ 𝜋− 32√63𝑛Γ(𝑛+ 1

2) .

Barry’s response to the explorations of Bender and Wuwas to pen the definitive rigorous analysis of the analytic


Some of Barry’s coauthors, SimonFest 2006, on theoccasion of Barry Simon’s sixtieth birthday.

properties of the anharmonic oscillator in an extendedarticle in the Annals of Physics in 1970. This work is atimeless classic, a textbook model for how to do singularperturbation theory, and it remains one of Barry’s mosthighly cited works. Most of the claims of Bender and Wuwere put on a firm footing, and many further facts wereestablished. For example, it was shown that 𝛽 = 0 is athird-order branch point and an accumulation point ofsingularities. Moreover, Barry obtained sufficient controlon the growth rate of perturbation coefficients to showthat both the Padé and Borel methods were valid todetermine the eigenvalues for nonzero values of 𝛽. A fewyears later, with the aid of a dispersion relation derivedusing this understanding of the Riemann surfaces of theeigenvalues, Barry and coauthors proved the formula (2).

Later, several other models of singular perturbationtheory, including the Stark and Zeeman effects, receivedsimilar treatments in thehandsof Barry andhis associates,especially Yosi Avron and IraHerbst. These gemsprovideda foundation for developments in singular perturbationtheory and deepened the understanding of many of thetouchstones of quantum mechanics.

Barry has always been quick to recognize others’ goodideas when they appear and generous in promoting themin the community of mathematical physicists and beyond.A very pretty method introduced in that era by Aguilar,Balslev, and Combes in 1971 came to be called complexscaling. The original version made use of the dilatationsymmetry and complex analysis to move the essentialspectrum of a Schrödinger operator into the complexplane, while leaving discrete eigenvalues unaffected. The

unitary group of dilatations can be defined via(3) [𝑈(𝜃)𝑓] (x) ∶= 𝑒𝜈𝜃/2𝑓 (𝑒𝜃x)in terms of a real parameter 𝜃. If the scaled potentialenergy depends in an analytic way on 𝜃, then the pa-rameter can be complexified, and it is easy to see thatwith a compactness condition on the potential energythe essential spectrum of the complex-scaled Laplacianfor nonreal 𝜃 is simply rotated in the complex plane.Meanwhile, isolated eigenvalues are analytic as functionsof 𝜃, but since they are constant for real variations in𝜃, by unique continuation they are also constant for anyvariation of 𝜃, except that they can appear or disappearwhen they collide with the essential spectrum, at whichpoint analytic perturbation theory ceases to apply.

With this procedure the ad hoc tradition in physicsof treating resonances as nonreal eigenvalues somehowassociated with a self-adjoint Hamiltonian became math-ematically solid and canonical. I cannot do better indescribing complex scaling further than to point to Chap-ter XIII.10 of Reed-Simon, to which the reader is referredfor further details and context. Barry assiduously pro-moted this excellent tool for understanding resonances,evangelizing the technique to physicists and chemists, bywhom it was adopted and used in realistic problems.

I recall in particular when Barry took a delegation ofmathematical physicists to the 1978 Sanibel Workshopon Complex Scaling, organized by the noted quantumchemist Per-Olof Löwdin, at which the discussions be-tween the chemists and the believers in mathematicalmethods “of known validity” were quite fruitful and infor-mative on all sides. Of course, Barry not only recognized,clarified, and promoted the idea of complex scaling butmade his own fundamental advances in the subject, espe-cially by greatly extending the set of problems to whichit applied by his discovery that it suffices to performcomplex scaling on an exterior region.

A similar tale could be told of Barry’s recognition ofthe importance of the microlocal analysis of tunnelingphenomena by Helffer and Sjöstrand in the 1980s, whichBarry again promoted, clarified, and in certain waystransformed. But space here is limited, and besides, forthe reader interested in learning more about singularperturbation theory and resonances, there is an excellent1991 review article entitled “Fifty years of eigenvalueperturbation theory,” written by a master of the genre,Barry Simon himself.

Percy A. DeiftPrinceton in the 1970s; Exponential Decay ofEigenfunctions and Scattering TheoryThe 1970s were a very special time for mathematicalphysics at Princeton. One can read a lively account ofthose days, written by Barry himself, in the July 2012edition of the Bulletin of the International Association ofMathematical Physics. The main thrust of the activity

Percy A. Deift is Silver Professor of Mathematics at the CourantInstitute, NYU. His email address is [email protected].


Some of Barry Simon’s students, SimonFest 2006, onthe occasion of Barry Simon’s sixtieth birthday.

was in statistical mechanics, quantum field theory, andnonrelativistic quantum mechanics. The list of peoplewho participated in math-phys at Princeton Universityin those years as students, postdocs, junior faculty orsenior faculty, or just visitors for a day or two reads likea who’s who of mathematical physics. Leading the chargewere Arthur Wightman, Elliott Lieb, and Barry Simon. Butthere were also Eugene Wigner, Valentine Bargmann, andEd Nelson. And in applied mathematics, there was MartinKruskal, still flush with excitement from his seminal workon the Korteweg-de Vries equation, and across the wayat the Institute were Tulio Regge and Freeman Dyson,doing wonderful things. Barry was a dynamo, challengingus with open problems, understanding every lectureinstantaneously, writing paper after paper, often at theseminars themselves, all the while supervising seven oreight PhD students.

I was one of those students. I had an appointment tomeet with Barry once every two weeks. I would work veryhard preparing a list of questions that I did not know howto answer. Say there were ten questions; by the end of thefirst ten minutes in Barry’s office, the first six questionswere resolved. Regarding questions seven and eight, Barrywould think about them for about two or three minutesand then tell me how to do them. Regarding questionsnine and ten, Barry would think about them, also forabout two or three minutes, and say, “I don’t know how todo them. But if you look in such and such a book or paper,you will find the answer.” Invariably he was right. So inless than half an hour, all my questions were resolved,and as I walked out of the door there was the next studentwaiting his turn!

Barry’s first PhD student was Tony O’Connor.O’Connor’s thesis concerned exponential decay foreigenfunctions 𝜓 of 𝑁-body Schrödinger operators 𝐻,𝐻𝜓 = 𝜆𝜓, for 𝜆 below the essential spectrum of 𝐻. Here𝐻 = 𝐻0 +𝑉, where 𝐻0 is the kinetic energy and 𝑉 is theinteraction potential. For Schrödinger operators in onedimension, such results go back to the nineteenth century,but for 𝑁 > 1 particles moving in three dimensions,

completely different techniques were necessary. Overthe years many people have worked on the problem,including, to name a few, Stanislav Merkuriev in theformer Soviet Union, and John Morgan and Thomas andMaria Hoffmann-Ostenhof in the West. O’Connor had theidea of using the analyticity of the Fourier transform andobtained results in the 𝐿2 sense (i.e., 𝑒𝛼| ⋅ |𝜓 ∈ 𝐿2, 𝛼 > 0).Such bounds are optimal for isotropic decay.

O’Connor’s paper motivated Jean-Michel Combes andLarry Thomas to introduce an approach that has nowbecome standard under the general rubric of “boostanalyticity,” and in a set of three papers in the mid-1970s, Barry further developed these ideas to obtainpointwise exponential bounds on eigenfunctions undervarious assumptions on the asymptotic behavior of theinteraction potential 𝑉, proving eventually that if 𝑉(𝑥)was bounded below by |𝑥|2𝑚, say, then one obtainedsuperexponential decay for 𝜓(𝑥),

|𝜓(𝑥)| ≤ 𝑐 𝑒−𝛼|𝑥|𝑚+1 𝑐, 𝛼 > 0.Schrödinger operators typically involve interaction poten-tials 𝑉 which are sums of two-body interactions

𝑉(𝑥) = ∑1≤𝑖<𝑗≤𝑁

𝑉𝑖𝑗(𝑥𝑖 − 𝑥𝑗)

for particles 𝑥𝑖 in ℝ3, 𝑖 = 1,… ,𝑁. Although typically𝑉𝑖𝑗(𝑦) → 0 as |𝑦| → ∞,

𝑉(𝑥) clearly does not decay if |𝑥| → ∞ in such a waythat 𝑥𝑖 − 𝑥𝑗, say, remains bounded for some 𝑖 ≠ 𝑗. Suchnonisotropy in the potential 𝑉 suggests that isotropicbounds of the form

|𝜓(𝑥)| ≤ 𝑐𝑒−𝛼|𝑥|

are not optimal amongst all possible bounds.In a fourthpaperonexponential decay in1978, together

with Deift, Hunziker, and Vock, Barry constructed optimalnonisotropic bounds for eigenfunctions of the form

|𝜓(𝑥)| ≤ 𝑐𝑒(𝛼,𝑥)

for suitable 𝛼 = (𝛼1,… ,𝛼𝑁), 𝛼𝑖 ∈ ℝ3. The 𝛼’s reflect thegeometry of the channels where

𝑉(𝑥) ↛ 0 as |𝑥| → ∞.Eventually, in 1982, Agmon showed that for a very

general class of elliptic operators 𝐻 in ℝ3𝑁, there was anaturally associated Riemannian metric on ℝ3𝑁 such thateigenfunctions 𝜓,

𝐻𝜓 = 𝜆𝜓,with 𝜆 below the essential spectrum of 𝐻, satisfied thebound

|𝜓(𝑥)| ≤ 𝑐𝜖 𝑒−(1−𝜖) 𝜌(𝑥) ∀𝜖 > 0,where 𝜌(𝑥) is the geodesic distance from 𝑥 to the origin inℝ3𝑁 with respect to Agmon’s metric. The Agmon metriccan be used to derive, and so explain, the bounds in the1978 work of Deift, Hunziker, Simon, and Vock.

The scattering problem in quantum chemistry, goingback to the 1920s, can be stated informally as follows: In achemical reaction, do molecules go to molecules? In otherwords, suppose in the distant past that the particle systemis described by a collection of noninteracting molecules.


SimonFest in 2006 for Barry Simon’s sixtieth birthday.Percy Deift is in the front row, second from the right.

As time goes on, the atoms in the different moleculesbegin to interact with each other, and the molecules breakup. At large positive times, is the particle system againdescribed by molecules?

To be mathematically precise, consider a collection ofatoms

𝑥𝑖 ∈ ℝ3, 𝑖 = 1,… ,𝑁,with Hamiltonian

𝐻 = 𝐻0 +𝑉 = 𝐻0 + ∑1<𝑗≤𝑁

𝑉𝑖𝑗 (𝑥𝑖 − 𝑥𝑗) ,

in the center of mass frame. Let 𝐶(1),… ,𝐶(𝑚) be adecomposition of the atoms into 𝑚 clusters,

𝑚

∑𝑗=1

#(𝐶(𝑗)) = 𝑁.

Now suppose that for large negative times the clusters{𝐶(𝑗)} are far apart and in each cluster 𝐶(𝑘) the atomsare in an eigenstate 𝜓𝑘 of the cluster (read “molecular”)Hamiltonian, that is,

𝐻(𝑘)𝜓𝑘 = 𝐸𝑘𝜓𝑘, where 𝐻(𝑘) = 𝐻(𝑘)0 +𝑉(𝑘).

Here 𝐻(𝑘)0 is the kinetic energy for the cluster and

𝑉(𝑘) = ∑1≤𝑖≤𝑗≤# (𝐶(𝑘))

𝑥𝑖,𝑥𝑗∈𝐶(𝑘)

𝑉𝑖𝑗 (𝑥𝑖 − 𝑥𝑗)

is the interaction potential for the atoms in the cluster. Sothe scattering problem of quantum chemistry becomesthe following: At large positive times, is the particlesystem again described by molecules, that is, by somedecomposition of well-separated clusters ̃𝐶(1),… , ̃𝐶(�̃�)

with the atoms in each cluster ̃𝐶(𝑘) in an eigenstate ofthe cluster Hamiltonian �̃�(𝑘) (or more precisely, in somelinear combination of such molecular configurations)?

In the physical and mathematical literature, the scat-tering problem is known as the problem of “asymptoticcompleteness” or the “unitarity of the 𝑆-matrix.” Muchwork has been done on this problem by many peopleover the years, including the time-independent approachof Faddeev and his school, leading up to the eventualresolution of the problem in 1987 by Israel Michael Sigaland Avy Soffer using time-dependent methods pioneeredby Enss.

At the mathematical level, the first task in resolvingthe problem is to prove that such molecular states indeedexist. This boils down to proving that the so-calledwave operators 𝑊(𝐻,𝐻𝑐) exist, where 𝐻 is again theHamiltonian for the full system and𝐻𝑐 is the Hamiltonianfor the molecular system corresponding to the clusterdecomposition 𝐶 = {𝐶(1),… ,𝐶(𝑚)}. At the technical levelthis is a relatively easy thing to do. To prove asymptoticcompleteness, one must show that all states orthogonalto bound states of the full Hamiltonian 𝐻 are in the linearspan of these molecular wave operators 𝑊(𝐻,𝐻𝑐). Thisis a range question, and range questions in mathematicsare generically hard.

Barry is oneof the most

prolific math-ematicians of

hisgeneration

In 1978, together withDeift, Simon introduced anew class of wave operators,𝑊(𝐻𝑐, 𝐽𝑐,𝐻), and showed thatasymptotic completeness wasequivalent to proving the exis-tence of these wave operators.Here 𝐽𝑐 is an auxiliary func-tion reflecting the geometry ofthe cluster decomposition 𝐶.In this way the problem ofasymptotic completeness wastransformed froma rangeprob-lem to a potentially simpler

existence problem, and this is the path that Sigal andSoffer eventually followed in their resolution of asymp-totic completeness. As the ranges of the wave operators𝑊(𝐻,𝐻𝑐) lie in the absolutely continuous space of 𝐻, akey ingredient in proving asymptotic completeness wasto show that the singular continuous space for 𝐻 wastrivial. This key component was established by BarrySimon, together with Peter Perry and Israel Michael Sigal,in seminal work in 1980 using remarkable ideas of EricMourre [2].

On a Personal NoteBarry is one of the most prolific mathematicians ofhis generation. It was in the late 1970s, around thetime that we were working on nonisotropic bounds foreigenfunctions, that I got a glimpse of the speed withwhich Barry did things. Soon after Volker Enss introducedhis seminal time-dependent ideas on spectral theory andscattering theory, a few of us went to Barry’s house inEdison, New Jersey, to discuss a potential project inspiredby Enss’s work. We spent the afternoon laying out indetail a list of problems that needed to be addressed andleft in the late afternoon. The next morning Barry came


From about 1995 to 2000, Barry taught the first termof Caltech’s required freshman calculus, where hedeclared that “epsilon and delta are a calculusstudent’s finest weapons.” One year, on the last day,the students presented him with the boxing glovesshown.

into the office: Not only had he solved all the problemson our list, but he had in his hand the first draft of hissubsequent paper [3]! We were overwhelmed. For a youngperson like me, this was most discouraging. And I wasdoubly discouraged: Barry was younger than I was!

Barry has many fine qualities as a colleague and asa researcher, but I would like to focus on just one ofthem, viz., Barry’s keen sense of fairness and correctattribution of results. People in orthogonal polynomialsknow well Barry’s insistence on calling the recurrencecoefficients for orthogonal polynomials on the circleVerblunsky coefficients, in recognition of the almostforgotten seminal work of Samuel Verblunsky. But Iwould like to tell a different story. In the early 1980s Barrywas in Australia, where he met up with Michael Berry,who was also visiting. Berry began telling Barry aboutsome curious and puzzling calculations he had beenmaking in quantum adiabatic theory. Barry immediatelyunderstood that what was really going on was a matterof holonomy, and with characteristic speed he wrote andsent off a paper to Physical Review Letters, pointedly titled“Holonomy, the Quantum Adiabatic Theorem, and Berry’sphase.” In this way, a major discovery that could quiteeasily have become known as “Barry’s phase” was fixedin the literature as “Berry’s phase,” and justly so.

References[1] P. Deift and B. Simon, A time-dependent approach to the

completeness of multiparticle quantum systems, Commun.Pure Appl. Math. 30 (1977), 573–583. MR 0459397

[2] E. Mourre, Absence of singular continuous spectrum forcertain selfadjoint operators, Commun. Math. Phys. 78(1980/81), 391–408. MR 0603501

[3] B. Simon, Phase space analysis of simple scattering systems.Extensions of some work of Enss, Duke Math. J. 46 (1979),119–168. MR 0523604

Lon RosenBarry and I were both young pups when we collaboratedin the 1970s. It was an intense and exciting experiencefor me, one which I cherish and now take pleasure inrecalling for you. Warning: these personal recollectionshave no scientific content. If that’s what you’re lookingfor, please see my contribution to the Festschrift in honorof Barry Simon’s sixtieth birthday.

I must confess that my first meeting with Barry wasfar from auspicious. In 1967 I was in my first year ofdoctoral studies at the Courant Institute. Feeling isolated,I was reconsidering my decision not to have chosenPrinceton for graduate school. I asked a friend to arrangea lunch meeting for me with a typical student of ArthurWightman’s. I knew little about the “typical student” whowas chosen (Barry Simon), although I was familiar with hisname because Barry and I had both been Putnam Fellowsin the 1965 competition.

Some typical student! He practically tore my head off.Whatever I said about my interests or ideas, Barry wouldtrump it. I’d never met anyone else with such extensiveknowledge, amazing recall, and proofs at the ready. Istill haven’t. Thanks to Barry, I stayed put at Courant.Fortunately, James Glimm, who was to be my terrificthesis advisor, soon joined the faculty there. I learnedlater that Barry had been going through a rough patchin his personal and professional life around the time wemet and that the fire-breathing dragon who had me for

Lon Rosen is professsor emeritus at the Department of Mathemat-ics, University of British Columbia. His email address is [email protected].

Barry and Martha Simon, LMU Munich, 2014.


lunch was actually a gentle prince in disguise, if I may bepermitted a fairy tale metaphor.

Three years later I gave a seminar at Princeton on thesubject of higher-order estimates for the 𝑃(𝜙)2 model. Atthe conclusion of the seminar Barry showed me a cleverbootstrap trick that quickly established my most difficultestimate—or at least a weaker but perfectly acceptableversion of it. I was grateful and revised the publishedpaper accordingly.

Barry’simperative tounderstandeverything inthe simplestpossible way

This experience was notunique to me. As manyspeakers know, Barry’srapid-strike ability couldbe unnerving at seminars.He would sit front rowcentre, working on a pa-per, only to surface withastute observations, coun-terexamples, or shorterproofs. This penchant for“tricks” arises, it seemsto me, from Barry’s im-perative to understandeverything in the simplest possible way.

Barry and I both attended the Les Houches SummerSchool of 1970. It was there that I gained an appreciationfor Barry’s sense of humour. In particular, we had a lot offun putting on a skit which satirized the lecturing stylesand idiosyncrasies of the various celebrated speakers. Forexample, when Barry began an impersonation by firstbreaking a half dozen chalk sticks into small pieces,everyone roared, knowing that “Arthur Jaffe” was aboutto deliver his next lecture.

Sometimes the humour was (possibly) unintentional.Here’s a little story which Ed Nelson told me. Barry hadreturned from a trip to the former Soviet Union, where hehad great difficulty in arranging for kosher food. It wasapparently necessary for him to haul a suitcase filled withedibles. “Oh well,” sighed Barry, “I guess everyone has hiscross to bear.”

He would oftencome in with a

twinkle in his eye

In 1971 when Ijoined Barry in Prince-ton, we began ourjoint research by work-ing on what I wouldcall “incremental stuffusing available tech-niques,” things likecoupling constant an-

alyticity of the 𝑃(𝜙)2 Hamiltonian. One day ArthurWightman called us together for a presentation by aquiet visitor from Italy, Francesco Guerra, whom I barelyknew. Francesco proceeded to the chalkboard and madesome extraordinary claims about the vacuum energy den-sity in the 𝑃(𝜙)2 model. Barry and I gave each other asideways look as if to say, “he’s got to be kidding.” Hewasn’t! Francesco’s short proofs were stunning. The ironywas not lost on us that they were based on the Euclideanapproach of Ed Nelson of Princeton University. In anycase, that was the moment that the Euclidean Revolution

Barry Simon in Bangkok (ca. 2003).

began for Barry and me. The three of us (GRS) entered along and fruitful collaboration exploring and exploitingthe parallels between the 𝑃(𝜙)2 field theory model andclassical statistical mechanics. As usual, Barry snatchedup the new ideas like a dog with a bone.

During the GRS period, Barry had numerous otherprojects on the go, such as his research in mathematicalquantum mechanics and the Reed-Simon magnum opus.Thank goodness he was devoting only a fraction of histime to GRS, whereas I was on it full time. Otherwise,I would never have been able to keep up with him andcontribute my fair share. Barry’s joy in doing researchwas infectious. He would often come in with a twinkle inhis eye and say something like, “While I was standing inthe supermarket line, look what I discovered!” Barry wasalways appreciative of my efforts and extremely generousto others. I learned a tremendous amount from him bothdirectly during our collaboration and in subsequent yearsfrom his prodigious published output.

Jürg FröhlichBarry Simon and Statistical MechanicsWell, this is aboutmymentor and friendBarry Simon! I firstmet Barry at a summer school on constructive quantumfield theory and statistical mechanics at Les Houches,France, in 1970. We were only twenty-four years old atthe time, and I had just started my life as a PhD studentof Klaus Hepp, while Barry, a former PhD student of thelate Arthur S. Wightman, was already a “Herr Doktor”and—if my memory is correct—an assistant professorat Princeton University. As I wrote on the occasion ofhis sixtieth birthday celebrations [1], Barry would usuallybeat me in almost everything! To start with, he was borntwo and a half months before me.

At Les Houches, the late Oscar E. Lanford III lecturedon general functional analysis, including measure theory

Jürg Fröhlich is professor emeritus of the Department of Physicsat the ETH, Zürich. His email address is [email protected].


and the theory of operator algebras, tools rightly thoughtto be essential to understanding quantum theory andstatistical mechanics. Besides Barry andme, Alain Conneswas a student at that school. Barry and Alain were soonengaged in a competition to simplify Oscar’s proofs; Barrywould usually win—not only in mathematics but also inthe consumption of food.

Our common mentor and friend, the late EdwardNelson, wrote about Barry [1]: “In the late 1960s, Barrywas a graduate student in physics at Princeton andattended some courses I taught. I soon learned that Idid not need to prepare with great thoroughness; it wasenough to get things approximately right and Barry fromwhere he was sitting would tell us how to get themprecisely right. I miss Barry.” Well, Barry and I miss Ed!

My next encounter with Barry was in 1972 when hetaught a graduate course on quantum field theory inthe “3ième cycle” of French-speaking Switzerland. Thatcourse was the basis for his 𝑃(𝜙)2-book, which is stillremembered in the community. This is perhaps because itcontains those famous “Fröhlich bounds” or, more likely,because it is written in a very pedagogical manner—indeed, one of Barry’s outstanding strengths was and stillis to being able to write mathematical prose in a veryclear, pedagogical style.

In the fall of 1974 I accepted the offer of an assis-tant professorship at the mathematics department ofPrinceton University which had been prepared by ArthurWightman and Barry Simon. Major benefits of having hadBarry as a colleague were that I never had to submita grant proposal to the NSF; Barry’s proposal not onlycovered his own needs but also the ones of the lateValja Bargmann and me, and should one successfullycollaborate with him, he would always write the paper(except for appendices on tedious technicalities, such ascluster expansions, which he gracefully assigned to hiscollaborators).

Barry needed onlyroughly 5 percent

of the timeordinary mortalsneed to write a

paper

Barry needed onlyroughly 5 percent ofthe time ordinary mor-tals need to writea paper. He did hiswriting while listeningto seminar talks. Al-though he appeared tobe absorbed in his ac-tivity (carried out bymoving his left handalong rather peculiartrajectories), he wouldnevertheless be able to

point out errors to the lecturer or ask relevant questionsat the end. Barry was simply brilliant in “multitasking.”

Letme briefly describe two of our joint papers. The firstone is entitled “Pure states for general 𝑃(𝜙)2-theories:Construction, regularity and variational equality” andwas published in the Annals of Mathematics in 1977.In this paper, ideas and concepts from classical sta-tistical mechanics were transferred to Euclidean fieldtheory with the purpose of learning something new about

the latter. General concepts were illustrated on simpleexamples of Euclidean field theory in two dimensions,which, mathematically, may be defined as generalizedstochastic processes—more precisely, Markovian randomfields—over ℝ2 constructed as perturbations of Gaussianprocesses by local multiplicative functionals. From suchprocesses interacting scalar quantum field theories ontwo-dimensional Minkowski space can be reconstructed,furnishing examples of what has become known as con-structive quantum field theory. CQFT was first advocatedby Arthur Wightman in the early 1960s with the purposeof showing that quantum theory and the special theoryof relativity are compatible with each other, and wassubsequently pursued by Edward Nelson, James Glimm,Arthur Jaffe, and their followers.

The modern mathematically rigorous approach to sta-tistical mechanics was developed by, among many otherpeople, Roland L. Dobrushin, Oscar E. Lanford, and, mostimportantly, David Ruelle. Our paper is unthinkable with-out their work andwithout the discoveries of K. Symanzik,E. Nelson, F. Guerra, L. Rosen, and B. Simon in Euclideanfield theory, some of whose works are classics. They hadshown that in the Euclidean region (time purely imaginary)of complexified Minkowski space, a quantum field theoryof Bose fields looks like a model of classical statisticalmechanics. In my paper with Barry this fact is exploitedin an essential way.

We were to “win the jackpot” with the paper “Infraredbounds, phase transitions and continuous symmetrybreaking,” which was the result of joint work with ourfriend Tom Spencer and was published in Communica-tions in Mathematical Physics in 1976.1 Barry, Tom, andI decided to attempt to understand phase transitionsaccompanied by the spontaneous breaking of continuoussymmetries and long-range correlations (i.e., a divergentcorrelation length), in models of classical lattice spin sys-tems and lattice gases. We exploited ideas from quantumfield theory; in particular, we discovered an analogue ofthe so-called Källen-Lehmann spectral representation oftwo-point correlation functions in quantum field theory.For this representation to hold true it is necessary that themodel under scrutiny satisfy the Osterwalder-Schraderpositivity, also called reflection positivity, a property origi-nating in axiomatic quantum field theory (as described inwell-known books by Streater and Wightman and by Jost).

Here is an example: With each site 𝑥 of the latticeℤ𝑑 we associate a random variable 𝑆𝑥 ∈ ℝ𝑁, a classical“spin,” whose a priori distribution is given by a probabilitymeasure, 𝑑𝜇(⋅) on ℝ𝑁, invariant under rotations of ℝ𝑁;for instance,(4) 𝑑𝜇(𝑆) = const. 𝛿(|𝑆|2 − 1)𝑑𝑁𝑆.Let Λ be a finite cube in ℤ𝑑. The energy of a configuration,𝑆Λ ∶= {𝑆𝑥}𝑥∈Λ, of “spins” is given by a functional (called“Hamiltonian”),(5) 𝐻(𝑆Λ) ∶= − ∑

𝑥,𝑦∈Λ𝐽(𝑥 − 𝑦)𝑆𝑥 ⋅ 𝑆𝑦.

1At the beginning of my career I was privileged to have severalmentors, among whom Tom was undoubtedly the most importantone!


Birmingham, AL, Meeting on Differential Equations,1983. Back row: Fröhlich, Yajima, Simon, Temam, Enss,Kato, Schechter, Brezis, Carroll, Rabinowitz. Front row:Crandall, Ekeland, Agmon, Morawetz, Smoller, Lieb,Lax.

Here 𝐽(𝑥) is a function in ℓ1(ℤ𝑑) assumed to be reflection-positive and invariant under permutations of latticedirections. By a theorem of Bochner, these propertiesimply that it has an integral representation,

(6) 𝐽(𝑥1, 𝑥) = ∫1

−1𝜆|𝑥1|−1𝑒𝑖𝑘⋅𝑥𝑑𝜌(𝜆, 𝑘), 𝑥1 ≠ 0,

with 𝑥 = (𝑥2,… , 𝑥𝑑), where 𝑑𝜌(𝜆, 𝑘) is a positive measureon [−1, 1] × 𝕋𝑑−1, for example, 𝐽(𝑥) = 𝛿|𝑥|,1. In (5) we

impose periodic boundary conditions at the boundaryof Λ. The distribution of configurations 𝑆Λ of “spins” inthermal equilibrium at inverse temperature 𝛽 is given bythe Gibbs measure(7) 𝑑𝑃𝛽(𝑆Λ) ∶= 𝑍−1

𝛽,Λ𝑒𝑥𝑝[−𝛽𝐻(𝑆Λ)]∏𝑥∈Λ

𝑑𝜇(𝑆𝑥),

where 𝑍𝛽,Λ is a normalization factor (called “partitionfunction”). Let ⟨ ⋅ ⟩𝛽,Λ denote an expectation with respectto 𝑑𝑃𝛽. For “wave vectors” 𝑘 in the lattice dual to Λ,we define 𝜔𝛽,Λ(𝑘) to be the Fourier transform of thecorrelation function ⟨𝑆0 ⋅ 𝑆𝑥⟩𝛽,Λ, 0, 𝑥 ∈ Λ.

Simon, Spencer, and I proved that

(8) 0 ≤ 𝜔𝛽,Λ(𝑘) ≤𝑁

2𝛽( ̂𝐽(0) − ̂𝐽(𝑘))for 𝑘 ≠ 0,∀𝛽,

where ̂𝐽(𝑘) is the Fourier transformof 𝐽(𝑥), 𝑥 ∈ Λ. This so-called “infrared bound” is inspired by the Källen-Lehmannrepresentation of two-point functions in canonical rela-tivistic quantum field theory. The realization that (6)implies the upper bound in (8) is the basic result in ourwork. It is then an exercise to show that if the “couplingfunction” 𝐽(⋅) is such that(9) |Λ|−1 ∑

𝑘≠0[ ̂𝐽(0) − ̂𝐽(𝑘)]−1 ≤ const.,

uniformly in Λ, then in the thermodynamic limit Λ ↗ ℤ𝑑,phases with broken 𝑂(𝑁)-symmetry coexist and arepermuted among themselves under the action of the

symmetry group 𝑂(𝑁), provided 𝛽 is large enough. (By(4), ⟨|𝑆0|2⟩𝛽,Λ = 1; this and (8), (9) imply that the weightof the mode at 𝑘 = 0 in 𝜔𝛽,Λ(𝑘) is ∝ |Λ|.) For small 𝛽,however, the Gibbs state is well known to be unique.

It turns out that the bound (8) hasmany further applica-tions. It is an important ingredient in a beautiful analysisof critical behavior in the Ising model (by Aizenman,Duminil-Copin, and Sidoravicius) and in showing that thelarge-distance scaling limit of the nearest-neighbor Isingand the classical XY-model isGaussian in dimension 𝑑 > 4(“triviality of 𝜆𝜙4

𝑑-theory” in 𝑑 ≥ 4 dimensions); see [2],[3].

In 1980 Barry proved a correlation inequality, some-times referred to as the “Simon-Lieb inequality,” usefulto establish decay of correlations in models of classicalferromagnetic lattice spin systems, e.g., the one sketchedabove with 𝑁 = 1 or 2. Two years later, the basic idea ex-pressed in his inequality became a very useful ingredientin the analysis of multiscale problems, such as Andersonlocalization (as in work by Tom Spencer and me).

To conclude, letme draw the reader’s attention to BarrySimon’s book The Statistical Mechanics of Lattice Gases.

References[1] math.caltech.edu/SimonFest/stories.html[2] M. Aizenman, Geometric analysis of 𝜙4-fields and Ising

models, Commun. Math. Phys. 86 (1982), 1–48.[3] J. Fröhlich, On the triviality of 𝜆𝜙4

𝑑-theories and the approachto the critical point in 𝑑 ≥ 4 dimensions, Nucl. Phys. B 200(1982), 281–296.

Mike ReedOn Barry SimonWhen people ask, “How long did it take for you andBarry Simon to write those four volumes of Methods ofModern Mathematical Physics?” I usually say, “About tenyears,” since we started in the late 1960s when Barrywas a graduate student and I was a lecturer at Princeton,and we finished in the late 1970s. Writing those bookstook 50 percent of my research time for ten years butonly 10 percent of Barry’s research time, and that wasn’tbecause I contributed more—far from it. The reason isthat no one works faster than Barry. He instantly sees thethe significance of new ideas (whether in mathematics orphysics), understands the technical structures necessaryto bring the ideas to fruition, and immediately startswriting.

Barry’s legendaryspeedsometimesgothim into trouble.I remember going to seminars at Princeton with Barrycarrying new preprints frommore senior mathematicians.As the seminar proceeded, Barry would read the preprint,absorb the idea, understand the correct machinery toprove a stronger result, and begin writing. No one is moregenerous than Barry at giving credit to others; he alwaysdoes and did. Nevertheless, when Barry’s paper with astronger result and a better proof would appear before

Mike Reed is professor and Bass Fellow in Mathematics at DukeUniversity. His email address is [email protected].


http://math.caltech.edu/SimonFest/stories.html

Barry Simon and Michael Reed, Durham, NC, 2007.

the original result, the preprint’s author would sometimeshave hard feelings. These feelings would usually dissipatewhen he or she actually met Barry and discovered howopen and generous he is.

We wrote the books because we saw that the physicsof quantum mechanics and quantum field theory raiseddeep and interesting analysis questions. Of course ourancestors knew this too. But we saw how hard it wasfor mathematicians to understand the issues and fullyengage because it was so difficult to read the physicsliterature and translate the ideas and computations intomathematical questions. Originally, we were going towrite one small volume that would give the functionalanalysis background followed by short chapters introduc-ing mathematicians to problems in modern physics. Butwe were driven by Barry’s deep knowledge and intuitionabout physics and our shared enthusiasm to do and saymore, and the result is the four volumes that we wrote.

We had a terrific time! This was long ago, so ourhandwritten manuscripts were typed. Then we wouldtake the typed manuscript, usually 500–700 pages, andread it aloud. This was the only way to go slowly enough tocheck the English, the mathematics, and the physics. Onewould read, both would think, and the other would writedown corrections. Typically it took three weeks full timeto read a manuscript. For a couple of them, I lived with

He respects others,whatever theirtalent, whatevertheir station in life

Barry. We’d get upin the morning, getto work, and give upwhen we were tired inthe evening. We wereblessed by the toler-ance and good cookingof Barry’s wife, Martha.It was very rare thatwe’d be irritated or

angry at each other, because we both have strong person-alities that are not easily troubled and we were completelyfocused on the mathematics and the science. We did allthe problems in all the volumes, except the starred onesthat we sure were correct but couldn’t immediately seehow to do.

Of course, we were pleased and proud that so manycolleagues and students found our books useful. We bothstill teach out of them and field email questions aboutthe problems. Since we were so young when they werewritten, we got lots of funny remarks at conferences frommathematicians who didn’t know us, such as, “You can’tbe the Simon who wrote those books; you’re too young,”and, “Hah! I always thought that Reed was Simon’s firstname.”

There are lots of things to celebrate about Barry Simon:his stupendous achievements, his many students, hissense of humor, his generosity to colleagues. I celebratehis deep sense of common humanity with other humanbeings. He respects others, whatever their talent, whatevertheir station in life, and this sense of common humanitymakes him very special.

This article will continue next month with contribu-tions by S. Jitomirskaya, Y. Avron, D. Damanik, J. Breuer,Y. Last, and A. Martinez-Finkelshtein.

The Leroy P. Steele Lifetime Achievement2016 Barry Simon2015 Victor Kac2014 Phillip A. Griffiths2013 Yakov G. Sinai2012 Ivo M. Babuška2011 John W. Milnor2010 William Fulton2009 Luis Caffarelli2008 George Lusztig2007 Henry P. McKean2006 Frederick W. Gehring, Dennis P. Sullivan2005 Israel M. Gelfand2004 Cathleen Synge Morawetz2003 Ronald Graham, Victor Guillemin2002 Michael Artin, Elias Stein2001 Harry Kesten2000 Isadore M. Singer1999 Richard V. Kadison1998 Nathan Jacobson1997 Ralph S. Phillips1996 Goro Shimura1995 John T. Tate1994 Louis Nirenberg1993 Eugene B. Dynkin


Barry Simon’s Students at PrincetonAnthony O’Connor, 1972Jay Rosen, 1974Robert Israel, 1975Percy Deift, 1976Evans Harrell II, 1976George Hagedorn, 1978Mark Ashbaugh, 1980Antti Kupiainen, 1980Steven Levin, 1980Peter Perry, 1981Keith Miller, 1982

Barry Simon’s Students at the California Instituteof Technology

Byron Siu, 1984Nestor Caticha Alfonso, 1985Barton Huxtable, 1987Kristiana Odencrantz, 1987Clemens Glaffig, 1988Askell Hardarson, 1988John Lindner, 1989Vojkan Jaksic, 1992Yunfeng Zhu, 1996Alexander Kiselev, 1997Andrei Khodakovsky, 1999Rowan Killip, 2000Andrej Zlatos, 2003Irina Nenciu, 2005Mihai Stoiciu, 2005Manwah Wong, 2009Rostyslav Kozhan, 2010Anna Maltsev, 2010Milivoje Lukic, 2011Brian Zachary Simanek, 2012

Creditsp. 740 Childhood photo of Simon with family, courtesyof Barry Simon.p. 741 Photo of Simon receiving his honorary doctorate,photographer Heinrich Steinlein. Used with permission.p. 742 Photo of Simon’s books, courtesy of Barry Simon.p. 743 Photo of Sadun, Avron, Harrell, and Simon, courtesyof Evans Harrell.p. 743. The Stark Effect, used with permission of TheAmerican Physical Society.p. 744 Photo of Simon’s coauthors, courtesy of The Cali-fornia Institute of Technology.p. 745 Photo of Simon’s students, courtesy of The Califor-nia Institute of Technology.p. 746 Photo of 2006 SimonFest, courtesy of The Califor-nia Institute of Technology.p. 747 Photo of Simon in boxing gloves, courtesy of BarrySimon.p. 747 Photo of Barry and Martha Simon, photographerHeinrich Steinlein. Used with permission.p. 748 Photo of Simon in Bangkok, courtesy of BarrySimon.p. 750 Photo of Birmingham Meeting on Differential Equa-tions, courtesy of Barry Simon.p. 751 Photo of Simon and Reed, courtesy of Barry Simon.


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