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Elliott Hershel Lieb (born July 31, 1932) is an American
mathematical physicist Mathematical physics refers to the development of mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developmen ...
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 r ...
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 ...
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 o ...
. 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 r ...
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, and ...
, the stability of matter, functional inequalities, the theory of magnetism, and the Hubbard model.


Biography

He received his
B.S. A Bachelor of Science (BS, BSc, SB, or ScB; from the Latin ') is a bachelor's degree awarded for programs that generally last three to five years. The first university to admit a student to the degree of Bachelor of Science was the University ...
in physics from the
Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...
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 of ...
at Kyoto University, 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 experime ...
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-l ...
, the University of Sierra Leone. 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. For years, he has rejected the standard practice of transferring copyright of his research articles to academic publishers. 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 Dannie Heineman Prize for Mathematical Physics is an award given each year since 1959 jointly by the American Physical Society and American Institute of Physics. It is established by the Heineman Foundation in honour of Dannie Heineman. As of 201 ...
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 (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 The Rolf Schock Prizes were established and endowed by bequest of philosopher and artist Rolf Schock (1933–1986). The prizes were first awarded in Stockholm Stockholm () is the capital and largest city of Sweden as well as the largest ...
(2001), the Henri Poincaré Prize 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 The Carl Friedrich Gauss Prize for Applications of Mathematics is a mathematics award, granted jointly by the International Mathematical Union and the German Mathematical Society for "outstanding mathematical contributions that have found significa ...
at the International Congress of Mathematicians ″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 20 ...
of the ICTP jointly with
Joel Lebowitz Joel Louis Lebowitz (born May 10, 1930) is a mathematical physicist widely acknowledged for his outstanding contributions to statistical physics, statistical mechanics and many other fields of Mathematics and Physics. Lebowitz has published m ...
and
David Ruelle David Pierre Ruelle (; born 20 August 1935) is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term '' strange attractor'', and developed a ...
. Lieb is a member of the U.S. National Academy of Sciences 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 mathematic ...
.


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 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 ...
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. 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, 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 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 In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...
and subfactors of von Neumann algebras. Together with Derek W. Robinson 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 In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...
or in quantum computing. 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 proved the
strong subadditivity of quantum entropy In quantum information theory, strong subadditivity of quantum entropy (SSA) is the relation among the von Neumann entropies of various quantum subsystems of a larger quantum system consisting of three subsystems (or of one quantum system with thre ...
, a theorem that is fundamental for
quantum information theory Quantum information is the information of the quantum state, 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 re ...
. This is closely related to what is known as the
data processing inequality The data processing inequality is an information theoretic concept which states that the information content of a signal cannot be increased via a local physical operation. This can be expressed concisely as 'post-processing cannot increase inform ...
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 quant ...
.


Many-body quantum systems and the stability of matter

In 1975, Lieb and
Walter Thirring Walter Thirring (29 April 1927 – 19 August 2014) was an Austrian physicist after whom the Thirring model in quantum field theory is named. He was the son of the physicist Hans Thirring.Thirring, H. Über die Wirkung rotierender ferner Massen ...
found a proof of the stability of matter 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 Robert Seiringer (1 September 1976, Vöcklabruck) is an Austrian mathematical physicist. Life and work Seiringer studied physics at the University of Vienna, where in 1999 he acquired his diploma and in 2000 with Jakob Yngvason as thesis advi ...
. Based on the original Dyson-Lenard theorem of stability of matter, Lieb together with
Joel Lebowitz Joel Louis Lebowitz (born May 10, 1930) is a mathematical physicist widely acknowledged for his outstanding contributions to statistical physics, statistical mechanics and many other fields of Mathematics and Physics. Lebowitz has published m ...
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 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 in ...
, also called the homogeneous electron gas, 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-body ...
. In the 1970s, Lieb together with
Barry Simon Barry Martin Simon (born 16 April 1946) is an American mathematical physicist and was the IBM professor of Mathematics and Theoretical Physics at Caltech, known for his prolific contributions in spectral theory, functional analysis, and no ...
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 Haïm Brezis (born 1 June 1944) is a French mathematician, who mainly works in functional analysis and partial differential equations. Biography Born in Riom-ès-Montagnes, Cantal, France. Brezis is the son of a Romanian immigrant father, wh ...
, 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 Barry Martin Simon (born 16 April 1946) is an American mathematical physicist and was the IBM professor of Mathematics and Theoretical Physics at Caltech, known for his prolific contributions in spectral theory, functional analysis, and no ...
and
Walter Thirring Walter Thirring (29 April 1927 – 19 August 2014) was an Austrian physicist after whom the Thirring model in quantum field theory is named. He was the son of the physicist Hans Thirring.Thirring, H. Über die Wirkung rotierender ferner Massen ...
, 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 quant ...
, he gave a rigorous proof of a formula for the ground state energy of dilute Bose gases. Subsequently, together with
Robert Seiringer Robert Seiringer (1 September 1976, Vöcklabruck) is an Austrian mathematical physicist. Life and work Seiringer studied physics at the University of Vienna, where in 1999 he acquired his diploma and in 2000 with Jakob Yngvason as thesis advi ...
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 quant ...
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 Horng-Tzer Yau (; born 1959 in Taiwan) is a Taiwanese-American mathematician. He received his B.Sc. in 1981 from National Taiwan University and his Ph.D. in 1987 from Princeton University. Yau joined the faculty of NYU in 1988, and became a full ...
and with
Jan Philip Solovej Jan Philip Solovej (born 14 June 1961) is a Danish mathematician and mathematical physicist working on the mathematical theory of quantum mechanics. He is a professor at University of Copenhagen. Biography Solovej obtained his Ph.D. in 198 ...
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 Mathieu Lewin (born 14 November 1977 in Senlis, Oise, France) is a French mathematician and mathematical physicist who deals with partial differential equations, mathematical quantum field theory and mathematics of quantum mechanical many-body sy ...
and
Robert Seiringer Robert Seiringer (1 September 1976, Vöcklabruck) is an Austrian mathematical physicist. Life and work Seiringer studied physics at the University of Vienna, where in 1999 he acquired his diploma and in 2000 with Jakob Yngvason as thesis advi ...
he gave the first rigorous justification of the
Local-density approximation Local-density approximations (LDA) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space (and no ...
for slowly varying densities.


Analysis

In the 70s Lieb entered the mathematical fields of calculus of variations 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 o ...
, 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 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 lemm ...
for sequences of functions converging almost everywhere. With Herm Brascamp and Joaquin Luttinger, he proved in 1974 a generalization of the Riesz rearrangement inequality, stating that certain multilinear integrals increase when all the functions are replaced by their
symmetric decreasing rearrangement In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function. Definition for sets Given a measurable set, A, in ...
. With
Frederick Almgren Frederick Justin Almgren Jr. (July 3, 1933 – February 5, 1997) was an American mathematician working in geometric measure theory. He was born in Birmingham, Alabama. Almgren received a Guggenheim Fellowship in 1974. Between 1963 and 1992 he wa ...
, 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, 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 In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the ...
. He also determined the best constant in some cases, discovering and exploiting the conformal invariance of the problem and relating it, via stereographic projection, 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 electro ...
). 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 and Srinivasa Varadhan 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 Frederick Justin Almgren Jr. (July 3, 1933 – February 5, 1997) was an American mathematician working in geometric measure theory. He was born in Birmingham, Alabama. Almgren received a Guggenheim Fellowship in 1974. Between 1963 and 1992 he wa ...
,
Haïm Brezis Haïm Brezis (born 1 June 1944) is a French mathematician, who mainly works in functional analysis and partial differential equations. Biography Born in Riom-ès-Montagnes, Cantal, France. Brezis is the son of a Romanian immigrant father, wh ...
and Jean-Michel Coron. 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 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 King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pre ...
, 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 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 * Araki–Lieb–Thirring inequality *
Borell–Brascamp–Lieb inequality In mathematics, the Borell–Brascamp–Lieb inequality is an integral inequality due to many different mathematicians but named after Christer Borell, Herm Jan Brascamp and Elliott Lieb. The result was proved for ''p'' > 0 by Henst ...
*
Brascamp–Lieb inequality In mathematics, the Brascamp–Lieb inequality is either of two inequalities. The first is a result in geometry concerning integrable functions on ''n''-dimensional Euclidean space \mathbb^. It generalizes the Loomis–Whitney inequality and Höl ...
* Brezis–Lieb lemma * 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 thermodynam ...
* Ice-type model * Lieb conjecture on the Wehrl entropy *
Lieb–Liniger Model The Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics. Introduction A model of a gas of particles moving in one dimension and satisfying Bose–Einstein statistics was introduced in ...
* Lieb–Oxford inequality * Lieb–Robinson bounds * Lieb–Thirring inequality *Lieb-Wu equation for the Hubbard model * Lieb's square ice constant * Lieb's concavity theorem * Stability of matter *
Strong subadditivity of quantum entropy In quantum information theory, strong subadditivity of quantum entropy (SSA) is the relation among the von Neumann entropies of various quantum subsystems of a larger quantum system consisting of three subsystems (or of one quantum system with thre ...
*
Temperley–Lieb algebra In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrix, transfer matrices, invented by Harold Neville Vazeille Temperley, Neville Temperley and Elliott H. Lieb, Elliott Lieb. It is also rela ...
*
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 matrix ...


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