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 university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...
who specializes in
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
,
condensed matter theory
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
, 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 space#Definition, inner product, Norm (mathematics)#Defini ...
.
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
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
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 particles ...
,
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 el ...
.
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 of ...
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
, mottoeng = Through efforts to heights
, established = 1825 – Birmingham School of Medicine and Surgery1836 – Birmingham Royal School of Medicine and Surgery1843 – Queen's College1875 – Mason Science College1898 – Mason Univers ...
in England in 1956.
Lieb was a
Fulbright Fellow at
Kyoto University
, mottoeng = Freedom of academic culture
, established =
, type = National university, Public (National)
, endowment = ¥ 316 billion (2.4 1000000000 (number), billion USD)
, faculty = 3,480 (Teaching Staff)
, administrative_staff ...
, Japan (1956–1957),
and worked as the Staff
Theoretical Physicist for
IBM from 1960 to 1963.
In 1961–1962, Lieb was on leave as professor of applied mathematics at
Fourah Bay College, 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 int ...
.
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
The American Physical Society (APS) is a not-for-profit membership organization of professionals in physics and related disciplines, comprising nearly fifty divisions, sections, and other units. Its mission is the advancement and diffusion of k ...
and the
American Institute of Physics (1978), the
Max Planck Medal 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 (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
The American Physical Society (APS) is a not-for-profit membership organization of professionals in physics and related disciplines, comprising nearly fifty divisions, sections, and other units. Its mission is the advancement and diffusion of k ...
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 significan ...
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 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 mor ...
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 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.
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
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
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 and
ferroelectricity, 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 el ...
.
Together with Daniel Mattis and Theodore Schultz he solved in 1964 the two-dimensional
Ising model (with a new derivation of the exact solution by
Lars Onsager via the
Jordan-Wigner transformation of the transfer matrices) and in 1961 the
XY model
The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's ''n''-vector model for .
Definition
G ...
, 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
In the mathematical field of topology, knot theory is the study of knot (mathematics), 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 ...
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 algebras) ...
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 alge ...
.
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
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. Though ...
. 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
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than 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. 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 transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
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
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 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 mor ...
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.
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
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, who c ...
, 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
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
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 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 space#Definition, inner product, Norm (mathematics)#Defini ...
, 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 lemma ...
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
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 Re ...
. 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 (geometry), plane (the ''projection plane'') perpendicular to ...
, 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
The Schrödinger–Newton equation, sometimes referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational potential, where the gravitational p ...
,
which can describe a self gravitating object or an electron moving in a polarizable medium (
polaron). 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
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, who c ...
and
Jean-Michel Coron
Jean-Michel Coron (born August 8, 1956) is a French people, 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 differ ...
. 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
A university press is an academic publishing hou ...
, 2010
* Lieb, Elliott H.;
Loss, Michael. ''Analysis''.
Graduate Studies in Mathematics Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats.
List of books
*1 ''The General To ...
, 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 Heise ...
*
Araki–Lieb–Thirring inequality
*
Borell–Brascamp–Lieb inequality
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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 ...
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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' ...
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Carlen-Lieb extension
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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 ...
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Ice-type model In statistical mechanics, the ice-type models or six-vertex models are a family of vertex models for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice. ...
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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 In quantum information theory, the Wehrl entropy, named after Alfred Wehrl, is a classical entropy of a quantum-mechanical density matrix. It is a type of quasi-entropy defined for the Husimi Q representation of the phase-space quasiprobability di ...
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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 ...
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Lieb–Oxford inequality
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Lieb–Robinson bounds
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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 el ...
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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 ...
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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 ...
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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 ...
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Strong subadditivity of quantum entropy
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Temperley–Lieb algebra In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable models, knot theory and the braid group, quant ...
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Von Neumann entropy
References
External links
Faculty pageat 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