Lieb's works pertain to quantum and classical many-body problem,[1][2][3] atomic structure,[3] the stability of matter,[3] functional inequalities,[4] the theory of magnetism,[2] and the Hubbard model.
Together with Derek W. Robinson in 1972, Lieb derived bounds on the propagation speed of information in non-relativistic spin systems with local interactions.
[26] In the years 1997–99, Lieb provided a rigorous treatment of the increase of entropy in the second law of thermodynamics and adiabatic accessibility with Jakob Yngvason.
[27] In 1975, Lieb and Walter Thirring found a proof of the stability of matter that was shorter and more conceptual than that of Freeman Dyson and Andrew Lenard in 1967.
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.
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.
Together with Jakob Yngvason, Lieb gave a rigorous proof of a formula for the ground state energy of dilute Bose gases.
Subsequently, together with Robert Seiringer and Jakob Yngvason he studied the Gross-Pitaevskii equation for the ground state energy of dilute bosons in a trap, starting with many-body quantum mechanics.
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[31] 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.
[32] In the 70s Lieb entered the mathematical fields of calculus of variations and partial differential equations, where he made fundamental contributions.
An important theme was the quest of best constants in several inequalities of functional analysis, which he then used to rigorously study nonlinear quantum systems.
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).
In another work with Herm Brascamp in 1976, Lieb extended the Prékopa-Leindler inequality to other types of convex combinations of two positive functions.
Lieb also wrote influential papers on harmonic maps, among others with Frederick Almgren, Haïm Brezis and Jean-Michel Coron.
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.