Elliott H. Lieb
   HOME

TheInfoList



OR:

Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist and professor of mathematics and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
at
Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...
who specializes in statistical mechanics, condensed matter theory, and
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
. Lieb is a prolific author, with over 400 publications both in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
and mathematics. In particular, his scientific works pertain to quantum and classical many-body problem,
atomic structure Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, an ...
, the
stability of matter Stability of matter refers to the problem of showing rigorously that a large number of charged quantum particles can coexist and form macroscopic objects, like ordinary matter. The first proof was provided by Freeman Dyson and Andrew Lenard in 196 ...
, functional inequalities, the theory of
magnetism Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particle ...
, and the
Hubbard model The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard. The Hubbard model states that each ...
.


Biography

He received his B.S. in physics from the
Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private Land-grant university, land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern t ...
in 1953 and his PhD in mathematical physics from the
University of Birmingham The University of Birmingham (informally Birmingham University) is a Public university, public research university located in Edgbaston, Birmingham, United Kingdom. It received its royal charter in 1900 as a successor to Queen's College, Birmingha ...
in England in 1956. Lieb was a
Fulbright Fellow The Fulbright Program, including the Fulbright–Hays Program, is one of several United States Cultural Exchange Programs with the goal of improving intercultural relations, cultural diplomacy, and intercultural competence between the people ...
at
Kyoto University , mottoeng = Freedom of academic culture , established = , type = Public (National) , endowment = ¥ 316 billion (2.4 billion USD) , faculty = 3,480 (Teaching Staff) , administrative_staff = 3,978 (Total Staff) , students = 22 ...
, Japan (1956–1957), and worked as the Staff
Theoretical Physicist Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
for IBM from 1960 to 1963. In 1961–1962, Lieb was on leave as professor of applied mathematics at
Fourah Bay College Fourah Bay College is a public university in the neighbourhood of Mount Aureol in Freetown, Sierra Leone. Founded on 18 February 1827, it is the first western-style university built in Sub-Saharan Africa and, furthermore, the first university- ...
, the
University of Sierra Leone The University of Sierra Leone is the name of the former unitary public university system in Sierra Leone. Established in February 1827, it is the oldest university in Africa. As of May 2005, the University of Sierra Leone was reconstituted in ...
. He has been a professor at Princeton since 1975, following a leave from his professorship at MIT. He is married to fellow Princeton professor
Christiane Fellbaum Christiane D. Fellbaum is a Lecturer with Rank of Professor in the Program in Linguistics and the Computer Science Department at Princeton University. The co-developer of the WordNet project, she is also its current director. Biography Fellbaum r ...
. For years, he has rejected the standard practice of transferring copyright of his research articles to
academic publishers Academic publishing is the subfield of publishing which distributes academic research and scholarship. Most academic work is published in academic journal articles, books or theses. The part of academic written output that is not formally publ ...
. Instead, he would only give publishers his consent to publish.


Awards

Lieb has been awarded several prizes in mathematics and physics, including the Heineman Prize for Mathematical Physics of the American Physical Society and the
American Institute of Physics The American Institute of Physics (AIP) promotes science and the profession of physics, publishes physics journals, and produces publications for scientific and engineering societies. The AIP is made up of various member societies. Its corpora ...
(1978), the
Max Planck Medal The Max Planck medal is the highest award of the German Physical Society , the world's largest organization of physicists, for extraordinary achievements in theoretical physics. The prize has been awarded annually since 1929, with few exceptions, ...
of the
German Physical Society The German Physical Society (German: , DPG) is the oldest organisation of physicists. The DPG's worldwide membership is cited as 60,547, as of 2019, making it the largest physics society in the world. It holds an annual conference () and multiple ...
(1992), the
Boltzmann medal The Boltzmann Medal (or Boltzmann Award) is a prize awarded to physicists that obtain new results concerning statistical mechanics; it is named after the celebrated physicist Ludwig Boltzmann. The Boltzmann Medal is awarded once every three years ...
of the
International Union of Pure and Applied Physics The International Union of Pure and Applied Physics (IUPAP ) is an international non-governmental organization whose mission is to assist in the worldwide development of physics, to foster international cooperation in physics, and to help in the ...
(1998), the Schock Prize (2001), the
Henri Poincaré Prize The Henri Poincaré Prize is awarded every three years since 1997 for exceptional achievements in mathematical physics and foundational contributions leading to new developments in the field. The prize is sponsored by the Daniel Iagolnitzer Foundat ...
of the
International Association of Mathematical Physics The International Association of Mathematical Physics (IAMP) was founded in 1976 to promote research in mathematical physics. It brings together research mathematicians and theoretical physicists, including students. The association's ordinary memb ...
(2003), and th
Medal of the Erwin Schrödinger Institute for Mathematics and Physics
(2021). In 2022 he was awarded the Medal for Exceptional Achievement in Research from the American Physical Society for ″major contributions to theoretical physics through obtaining exact solutions to important physical problems, which have impacted condensed matter physics, quantum information, statistical mechanics, and atomic physics″ and the Carl Friedrich Gauss Prize at the
International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...
″for deep mathematical contributions of exceptional breadth which have shaped the fields of quantum mechanics, statistical mechanics, computational chemistry, and quantum information theory.″ Also in 2022 he received the
Dirac Medal The Dirac Medal is the name of four awards in the field of theoretical physics, computational chemistry, and mathematics, awarded by different organizations, named in honour of Professor Paul Dirac, one of the great theoretical physicists of the 2 ...
of the ICTP jointly with Joel Lebowitz and David Ruelle. Lieb is a member of the
U.S. National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Natio ...
and has twice served (1982–1984 and 1997–1999) as the President of the
International Association of Mathematical Physics The International Association of Mathematical Physics (IAMP) was founded in 1976 to promote research in mathematical physics. It brings together research mathematicians and theoretical physicists, including students. The association's ordinary memb ...
. Lieb was awarded the
Austrian Decoration for Science and Art The Austrian Decoration for Science and Art (german: Österreichisches Ehrenzeichen für Wissenschaft und Kunst) is a state decoration of the Republic of Austria and forms part of the Austrian national honours system. History The "Austrian ...
in 2002. In 2012 he became a fellow of the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
and in 2013 a
Foreign Member of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural knowledge, including mathemati ...
.


Works

Lieb has made fundamental contributions to both theoretical physics and mathematics. Only some of them are outlined here. His main research papers are gathered in four Selecta volumes. More details can also be found in two books published by
EMS Press The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current ...
in 2022 on the occasion of his 90th birthday. His research is reviewed there in more than 50 chapters.


Statistical mechanics, soluble systems

Lieb is famous for many groundbreaking results in statistical Mechanics concerning, in particular, soluble systems. His numerous works have been collected in the Selecta ''″Statistical mechanics″'' and ''″Condensed matter physics and exactly soluble models″'', as well as in a book with Daniel Mattis. They treat (among other things) Ising-type models, models for
ferromagnetism Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials a ...
and
ferroelectricity Ferroelectricity is a characteristic of certain materials that have a spontaneous electric polarization that can be reversed by the application of an external electric field. All ferroelectrics are also piezoelectric and pyroelectric, with the a ...
, the exact solution of the 6-vertex models for two-dimensional `ice model', the one-dimensional delta Bose gas (now called the Lieb-Liniger model) and the
Hubbard model The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard. The Hubbard model states that each ...
. Together with Daniel Mattis and Theodore Schultz he solved in 1964 the two-dimensional
Ising model The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
(with a new derivation of the exact solution by
Lars Onsager Lars Onsager (November 27, 1903 – October 5, 1976) was a Norwegian-born American physical chemist and theoretical physicist. He held the Gibbs Professorship of Theoretical Chemistry at Yale University. He was awarded the Nobel Prize in C ...
via the Jordan-Wigner transformation of the transfer matrices) and in 1961 the XY model, an explicitly solvable one-dimensional spin-1/2 model. In 1968, together with
Fa-Yueh Wu Fa-Yueh Wu (January 5, 1932 – January 21, 2020) was a Chinese-born theoretical physicist, mathematical physicist, and mathematician who studied and contributed to solid-state physics and statistical mechanics. Life Early stage Born on Jan ...
, he gave the exact solution of the one-dimensional Hubbard model. In 1971 he and Neville Temperley introduced the Temperley-Lieb algebra in order to build certain transfer matrices. This algebra also has links with
knot theory In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...
and the
braid group A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...
, quantum groups and subfactors of
von Neumann algebras In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algeb ...
. Together with
Derek W. Robinson Derek William Robinson (25 June 1935 – 31 August 2021) was a British-Australian theoretical mathematician and physicist. He was a researcher at the Australian National University. Early life Derek W. Robinson was born in southern England. ...
in 1972 he derived bounds on the propagation speed of information in non relativistic spin systems with local interactions. They have become known as Lieb-Robinson bounds and play an important role, for instance, in error bounds in the thermodynamic limit or in
quantum computing Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thou ...
. They can be used to prove the exponential decay of correlations in spin systems or to make assertions about the gap above the ground state in higher-dimensional spin systems (generalized Lieb-Schultz-Mattis theorems). In 1972 he and
Mary Beth Ruskai Mary Beth Ruskai (born 1944) is an American mathematical physicist and Professor Emerita of Mathematics at the University of Massachusetts, with interest in mathematical problems in quantum theory. She is a Fellow of the AAAS, AMS, APS, and AWM ...
proved the strong subadditivity of quantum entropy, a theorem that is fundamental for
quantum information theory Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both ...
. This is closely related to what is known as the data processing inequality in quantum information theory. The Lieb-Ruskai proof of strong subadditivity is based on an earlier paper where Lieb solved several important conjectures about operator inequalities, including the Wigner-Yanase-Dyson conjecture. In the years 1997–99, Lieb provided a very original rigorous treatment of the increase of entropy in the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...
and
adiabatic accessibility Adiabatic accessibility denotes a certain relation between two equilibrium states of a thermodynamic system (or of different such systems). The concept was coined by Constantin Carathéodory in 1909 ("adiabatische Erreichbarkeit") and taken up 90 ...
with
Jakob Yngvason Jakob Yngvason (born 23 November 1945) is an Icelandic/Austrian physicist and emeritus professor of mathematical physics at the University of Vienna. He has made important contributions to local quantum field theory, thermodynamics, and the quantu ...
.


Many-body quantum systems and the stability of matter

In 1975, Lieb and Walter Thirring found a proof of the
stability of matter Stability of matter refers to the problem of showing rigorously that a large number of charged quantum particles can coexist and form macroscopic objects, like ordinary matter. The first proof was provided by Freeman Dyson and Andrew Lenard in 196 ...
that was shorter and more conceptual than that of
Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum m ...
and Andrew Lenard in 1967. Their argument is based on a new inequality in spectral theory, which became known as the Lieb-Thirring inequality. The latter has become a standard tool in the study of large fermionic systems, e.g. for (pseudo-)relativistic fermions in interaction with classical or quantized electromagnetic fields. On the mathematical side, the Lieb-Thirring inequality has also generated a huge interest in the spectral theory of Schrödinger operators. This fruitful research program has led to many important results that can be read in his Selecta ''″The stability of matter : from atoms to stars″'' as well as in his book ''″The stability of matter in quantum mechanics″'' with Robert Seiringer. Based on the original Dyson-Lenard theorem of stability of matter, Lieb together with Joel Lebowitz had already provided in 1973 the first proof of the existence of thermodynamic functions for quantum matter. With Heide Narnhofer he did the same for Jellium, also called the
homogeneous electron gas Jellium, also known as the uniform electron gas (UEG) or homogeneous electron gas (HEG), is a quantum mechanical model of interacting electrons in a solid where the positive charges (i.e. atomic nuclei) are assumed to be uniformly distributed i ...
, which is at the basis of most functionals in
Density Functional Theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-bo ...
. In the 1970s, Lieb together with Barry Simon studied several nonlinear approximations of the many-body Schrödinger equation, in particular Hartree-Fock theory and the Thomas-Fermi model of atoms. They provided the first rigorous proof that the latter furnishes the leading order of the energy for large non-relativistic atoms. With Rafael Benguria and Haïm Brezis, he studied several variations of the Thomas-Fermi model. The ionization problem in mathematical physics asks for a rigorous upper bound on the number of electrons that an atom with a given nuclear charge can bind. Experimental and numerical evidence seems to suggest that there can be at most one, or possibly two extra electrons. To prove this rigorously is an open problem. A similar question can be asked concerning molecules. Lieb proved a famous upper bound on the number of electrons a nucleus can bind. Moreover, together with
Israel Michael Sigal Israel Michael Sigal (born 31 August 1945 in Kiev, Ukrainian SSR) is a Canadian mathematician specializing in mathematical physics. He is a professor at the University of Toronto Department of Mathematics. He was an invited speaker at Internat ...
, Barry Simon and Walter Thirring, he proved, for the first time, that the excess charge is asymptotically small compared to the nuclear charge. Together with
Jakob Yngvason Jakob Yngvason (born 23 November 1945) is an Icelandic/Austrian physicist and emeritus professor of mathematical physics at the University of Vienna. He has made important contributions to local quantum field theory, thermodynamics, and the quantu ...
, he gave a rigorous proof of a formula for the ground state energy of dilute Bose gases. Subsequently, together with Robert Seiringer and
Jakob Yngvason Jakob Yngvason (born 23 November 1945) is an Icelandic/Austrian physicist and emeritus professor of mathematical physics at the University of Vienna. He has made important contributions to local quantum field theory, thermodynamics, and the quantu ...
he studied the Gross-Pitaevskii equation for the ground state energy of dilute bosons in a trap, starting with many-body quantum mechanics. Lieb's works with Joseph Conlon and Horng-Tzer Yau and with Jan Philip Solovej on what is known as the N^ law for bosons provide the first rigorous justification of Bogoliubov's pairing theory. In quantum chemistry Lieb is famous for having provided in 1983 the first rigorous formulation of Density Functional Theory using tools of convex analysis. The universal Lieb functional gives the lowest energy of a Coulomb system with a given density profile, for mixed states. In 1980, he proved with Stephen Oxford the Lieb-Oxford inequality which provides an estimate on the lowest possible classical Coulomb energy at fixed density and was later used for calibration of some functionals such as PBE and SCAN. More recently, together with Mathieu Lewin and Robert Seiringer he gave the first rigorous justification of the Local-density approximation for slowly varying densities.


Analysis

In the 70s Lieb entered the mathematical fields of
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
and
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
, where he made fundamental contributions. An important theme was the quest of best constants in several inequalities of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, which he then used to rigorously study nonlinear quantum systems. His results in this direction are collected in the Selecta ''″Inequalities″''. Among the inequalities where he determined the sharp constants are Young's inequality and the Hardy-Littlewood-Sobolev inequality, to be further discussed below. He also developed tools now considered standard in analysis, such as
rearrangement inequalities Rearrangement may refer to: Chemistry * Rearrangement reaction Mathematics * Rearrangement inequality * The Riemann rearrangement theorem, also called the Riemann series theorem ** see also Lévy–Steinitz theorem * A permutation of the term ...
or the Brezis-Lieb lemma which provides the missing term in
Fatou's lemma In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's l ...
for sequences of functions converging almost everywhere. With Herm Brascamp and
Joaquin Luttinger Joaquin (Quin) Mazdak Luttinger (December 2, 1923 – April 6, 1997) was an American physicist well known for his contributions to the theory of interacting electrons in one-dimensional metals (the electrons in these metals are said to be in ...
, he proved in 1974 a generalization of the
Riesz rearrangement inequality In mathematics, the Riesz rearrangement inequality, sometimes called Riesz–Sobolev inequality, states that any three non-negative functions f : \mathbb^n \to \mathbb^+, g : \mathbb^n \to \mathbb^+ and h : \mathbb^n \to \mathbb^+ satisfy the inequ ...
, stating that certain multilinear integrals increase when all the functions are replaced by their symmetric decreasing rearrangement. With Frederick Almgren, he clarified the continuity properties of rearrangement. Rearrangement is often used to prove the existence of solutions for some nonlinear models. In two papers (one in 1976 with Herm Brascamp and another one alone in 1990), Lieb determined the validity and the best constants of a whole family of inequalities that generalizes, for instance, the
Hölder's inequality In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of spaces. :Theorem (Hölder's inequality). Let be a measure space and let with . ...
, Young's inequality for convolutions, and the Loomis-Whitney inequality. This is now known as the Brascamp-Lieb inequality. The spirit is that the best constant is determined by the case where all functions are Gaussians. The Brascamp-Lieb inequality has found applications and extensions, for instance, in harmonic analysis. Using rearrangement inequalities and compactness methods, Lieb proved in 1983 the existence of optimizers for the Hardy-Littlewood-Sobolev inequality and of the Sobolev inequality. He also determined the best constant in some cases, discovering and exploiting the conformal invariance of the problem and relating it, via
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter th ...
, to a conformally equivalent, but more tractable problem on the sphere. A new rearrangement-free proof was provided later with Rupert Frank, allowing to treat the case of the Heisenberg group. In a 1977 work he also proved the uniqueness (up to symmetries) of the ground state for the Choquard-Pekar equation, also called Schrödinger–Newton equation, which can describe a self gravitating object or an electron moving in a polarizable medium (
polaron A polaron is a quasiparticle used in condensed matter physics to understand the interactions between electrons and atoms in a solid material. The polaron concept was proposed by Lev Landau in 1933 and Solomon Pekar in 1946 to describe an electr ...
). With Lawrence Thomas he provided in 1997 a variational derivation of the Choquard-Pekar equation from a model in quantum field theory (the Fröhlich Hamiltonian). This had been solved earlier by
Monroe Donsker Monroe David Donsker (October 17, 1924 – June 8, 1991) was an American mathematician and a professor of mathematics at New York University (NYU). His research interest was probability theory.. Education and career Donsker was born in Burl ...
and
Srinivasa Varadhan Sathamangalam Ranga Iyengar Srinivasa Varadhan FRS (born 2 January 1940) is an Indian American mathematician, widely recognised as one of the most influential mathematicians of the 20th century. He is known for his fundamental contributions to ...
using a probabilistic path integral method. In another work with Herm Brascamp in 1976, Lieb extended the Prékopa-Leindler inequality to other types of convex combinations of two positive functions. He strengthened the inequality and the Brunn-Minkowski inequality by introducing the notion of essential addition. Lieb also wrote influential papers on harmonic maps, among others with Frederick Almgren, Haïm Brezis and
Jean-Michel Coron Jean-Michel Coron (born August 8, 1956) is a French mathematician. He first studied at École Polytechnique, where he worked on his PhD thesis advised by Haïm Brezis. Since 1992, he has studied the control theory of partial differential equatio ...
. In particular, Algrem and Lieb proved a bound on the number of singularities of energy minimizing harmonic maps. Finally, his textbook ''″Analysis″'' with
Michael Loss Michael Loss (born 1954) is a mathematician and mathematical physics, mathematical physicist who works as a professor of mathematics at the Georgia Institute of Technology. Loss obtained his Ph.D. in 1982 from ETH Zurich, with a dissertation on the ...
should be mentioned. It has become a standard for graduate courses in mathematical analysis. It develops all the traditional tools of analysis in a concise, intuitive and eloquent form, with a view towards applications.


Publication list (partial)


Books

* Lieb, Elliott H.; Seiringer, Robert. ''The stability of matter in quantum mechanics''.
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...
, 2010 * Lieb, Elliott H.; Loss, Michael. ''Analysis''. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997. xviii+278 pp. * Lieb, Elliott H.; Seiringer, Robert; Solovej, Jan Philip; Yngvason, Jakob. ''The mathematics of the Bose gas and its condensation''. Oberwolfach Seminars, 34. Birkhäuser Verlag, Basel, 2005. viii+203 pp. ; 3-7643-7336-9


Selecta of research articles

* ''Statistical mechanics. Selecta of Elliott H. Lieb''. Edited, with a preface and commentaries, by B. Nachtergaele, J. P. Solovej and J. Yngvason. Springer-Verlag, Berlin, 2004. x+505 pp. * ''Condensed matter physics and exactly soluble models. Selecta of Elliott H. Lieb''. Edited by B. Nachtergaele, J. P. Solovej and J. Yngvason. Springer-Verlag, Berlin, 2004. x+675 pp. * ''The Stability of Matter: From Atoms to Stars. Selecta of Elliott H. Lieb'' (4th edition). Edited by W. Thirring, with a preface by F. Dyson. Springer-Verlag, Berlin, 2005. xv+932 pp. * ''Inequalities. Selecta of Elliott H. Lieb''. Edited, with a preface and commentaries, by M. Loss and M. B. Ruskai. Springer-Verlag, Berlin, 2002. xi+711 pp.


As editor

* Lieb, Elliott H. and Mattis, Daniel C., editors. ''Mathematical Physics in One Dimension: Exactly Soluble Models of Interacting Particles'', Academic Press, New York, 1966.


Other


''The Physics and Mathematics of Elliott Lieb''
Edited by R. L. Frank, A. Laptev, M. Lewin and R. Seiringer. EMS Press, July 2022, 1372 pp. These are two books published by
EMS Press The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current ...
on the occasion of Lieb's 90th birthday, which contain around 50 chapters about his impact on a very broad range of topics and the resulting subsequent developments. Many contributions are of an expository character and thus accessible to non-experts.


See also

*
Adiabatic accessibility Adiabatic accessibility denotes a certain relation between two equilibrium states of a thermodynamic system (or of different such systems). The concept was coined by Constantin Carathéodory in 1909 ("adiabatische Erreichbarkeit") and taken up 90 ...
*
AKLT model The AKLT model is an extension of the one-dimensional quantum Heisenberg spin model. The proposal and exact solution of this model by Ian Affleck, Elliott H. Lieb, Tom Kennedy and provided crucial insight into the physics of the spin-1 Heisenb ...
* Araki–Lieb–Thirring inequality * Borell–Brascamp–Lieb inequality * Brascamp–Lieb inequality *
Brezis–Lieb lemma In the mathematical field of analysis, the Brezis–Lieb lemma is a basic result in measure theory. It is named for Haïm Brézis and Elliott Lieb, who discovered it in 1983. The lemma can be viewed as an improvement, in certain settings, of Fatou' ...
* Carlen-Lieb extension *
Entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
* Ice-type model *
Lieb conjecture In quantum information theory, the Lieb conjecture is a theorem concerning the Wehrl entropy of quantum systems for which the classical phase space is a sphere. It states that no state of such a system has a lower Wehrl entropy than the SU(2) coh ...
on the Wehrl entropy * Lieb–Liniger Model *
Lieb–Oxford inequality In quantum chemistry and physics, the Lieb–Oxford inequality provides a lower bound for the indirect part of the Coulomb energy of a quantum mechanical system. It is named after Elliott H. Lieb and Stephen Oxford. The inequality is of impor ...
* Lieb–Robinson bounds *
Lieb–Thirring inequality In mathematics and physics, Lieb–Thirring inequalities provide an upper bound on the sums of powers of the negative eigenvalues of a Schrödinger operator in terms of integrals of the potential. They are named after Elliott H. Lieb, E. H. Lieb a ...
*Lieb-Wu equation for the
Hubbard model The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard. The Hubbard model states that each ...
*
Lieb's square ice constant Lieb's square ice constant is a mathematical constant used in the field of combinatorics to quantify the number of Eulerian orientations of grid graphs. It was introduced by Elliott H. Lieb in 1967. Definition An ''n'' × ''n'' grid ...
*
Lieb's concavity theorem In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices.E. Carlen, Trace Inequalities and Quantum Entr ...
*
Stability of matter Stability of matter refers to the problem of showing rigorously that a large number of charged quantum particles can coexist and form macroscopic objects, like ordinary matter. The first proof was provided by Freeman Dyson and Andrew Lenard in 196 ...
* Strong subadditivity of quantum entropy * Temperley–Lieb algebra *
Von Neumann entropy In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density mat ...


References


External links


Faculty page
at Princeton. {{DEFAULTSORT:Lieb, Elliott 1932 births Living people 20th-century American mathematicians 21st-century American mathematicians Members of the United States National Academy of Sciences Alumni of the University of Birmingham 20th-century American physicists 21st-century American physicists MIT Department of Physics alumni Massachusetts Institute of Technology faculty Princeton University faculty Recipients of the Austrian Decoration for Science and Art Fellows of the American Mathematical Society Rolf Schock Prize laureates Foreign Members of the Royal Society Scientists from Massachusetts People from Boston Mathematical physicists Winners of the Max Planck Medal Presidents of the International Association of Mathematical Physics