Selecta of Elliott H. Lieb

The Stability of Matter: From Atoms to Stars Selecta of Elliott H. Lieb Edited by W. Thirring With a Preface by F. Dyson Springer-Verlag Berlin Heidelberg New York 2004 Fourth Edition

Inequalities Selecta of Elliott H. Lieb Edited by M. Loss and M. B. Ruskai Springer-Verlag Berlin Heidelberg New York 2002

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 Heidelberg New York 2004

Statistical Mechanics Selecta of Elliott H. Lieb Edited by B. Nachtergaele, J.P. Solovej and J. Yngvason Springer-Verlag Berlin Heidelberg New York 2004

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Publications of Elliott H. Lieb

1. Second Order Radiative Corrections to the Magnetic Moment of a Bound Electron, Phil. Mag. Vol. 46,311-316(1955).

2. A Non-Perturbation Method for Non-Linear Field Theories, Proc. Roy. Soc. 241A, 339-363 (1957).

3. (with K. Yamazaki) Ground State Energy and Effective Mass of the Polaron, Phys. Rev. 111, 728-733 (1958).

4. (with H. Koppe) Mathematical Analysis of a Simple Model Related to the Stripping Reaction, Phys. Rev. 116, 367-371 (1959).

5. Hard Sphere Bose Gas - An Exact Momentum Space Formulation, Proc. U.S. Nat. Acad. Sci. 46, 1000-1002 (1960).

6. Operator Formalism in Statistical Mechanics, J. Math. Phys. 2, 341-343 (1961).

7. (with D.C. Mattis) Exact Wave Functions in Superconductivity, J. Math. Phys. 2, 602-609 (1961). (with T.D. Schultz and D.C. Mattis) Two Soluble Models of an Antiferromag-netic Chain, Annals of Phys. (N.Y.) 16, 407–466 (1961). (with D.C. Mattis) Theory of Ferromagnetism and the Ordering of Electronic Energy Levels, Phys. Rev. 125,164-172 (1962). (with D.C. Mattis) Ordering Energy Levels of Interacting Spin Systems, J. Math. Phys. 3, 749-751 (1962).

11. New Method in the Theory of Imperfect Gases and Liquids, J. Math. Phys. 4, 671-678 (1963). (with W. Liniger) Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State, Phys. Rev. 130,1605-1616 (1963). Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum, Phys. Rev. 130,1616-1624(1963). Simplified Approach to the Ground State Energy of an Imperfect Bose Gas, Phys. Rev. 130,2518-2528(1963).

* means the paper appears in: Stability of Matter, ed by W. Thirring, 4th Edition (Springer Berlin Heidelberg 2004) $ means the paper appears in: Inequalities, ed by M. Loss and M.B. Ruskai (Springer Berlin Heidelberg 2002) It means the paper appears in: Condensed Matter Physics and Exactly Soluble Models, ed by B. Nachtergaele, J.P. Solovej and J. Yngvason (Springer Berlin Heidelberg 2004) t means the paper appears in: Statistical Mechanics, ed by B. Nachtergaele, J.P. Solovej and J. Yngvason (Springer Berlin Heidelberg 2004)

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15. (with A. Sakakura) Simplified Approach to the Ground State Energy of an Imperfect Bose Gase. II. The Charged Bose Gas at High Density, Phys. Rev. 133, A899–A906 (1964).

16. (with W. Liniger) Simplified Approach to the Ground State Energy of an Imperfect Bose Gas. III. Application to the One-Dimensional Model, Phys. Rev. 134, A312–A315 (1964). (with T.D. Schultz and D.C. Mattis) Two-Dimensional Ising Model as a Sol­uble Problem of Many Fermions, Rev. Mod. Phys. 36, 856–871 (1964).

18. The Bose Fluid, Lectures in Theoretical Physics, Vol. VIIC, (Boulder summer school), University of Colorado Press, 175-224 (1965). (with D.C. Mattis) Exact Solution of a Many-Fermion System and Its Asso­ciated Boson Field, J. Math. Phys. 6, 304-312 (1965). (with S.Y. Larsen, J.E. Kilpatrick and H.F. Jordan) Suppression at High Tem­perature of Effects Due to Statistics in the Second Virial Coefficient of a Real Gas, Phys. Rev. 140, A129-A130 (1965).

21. (with D.C. Mattis) Book Mathematical Physics in One Dimension, Academic Press, New York (1966).

$22. Proofs of Some Conjectures on Permanents, J. of Math. and Mech. 16, 127–139 (1966).

23. Quantum Mechanical Extension of the Lebowitz-Penrose Theorem on the van der Waals Theory, J. Math. Phys. 7, 1016–1024 (1966).

24. (with D.C. Mattis) Theory of Paramagnetic Impurities in Semiconductors, J. Math. Phys. 7, 2045–2052 (1966).

25. (with T. Burke and J.L. Lebowitz) Phase Transition in a Model Quantum System: Quantum Corrections to the Location of the Critical Point, Phys. Rev. 149, 118–122 (1966).

26. Some Comments on the One-Dimensional Many-Body Problem, unpublished Proceedings of Eastern Theoretical Physics Conference, New York (1966). Calculation of Exchange Second Virial Coefficient of a Hard Sphere Gas by Path Integrals, J. Math. Phys. 8, 43–52 (1967). (with Z. Rieder and J.L. Lebowitz) Properties of a Harmonic Crystal in a Stationary Nonequilibrium State, J. Math. Phys. 8, 1073–1078 (1967). Exact Solution of the Problem of the Entropy of Two-Dimensional Ice, Phys. Rev. Lett. 18, 692–694 (1967). Exact Solution of the F Model of an Antiferroelectric, Phys. Rev. Lett. 18, 1046–1048 (1967). Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric, Phys. Rev. Lett. 19, 108–110 (1967).

†† 32. Residual Entropy of Square Ice, Phys. Rev. 162, 162–172 (1967). 33. Ice, Ferro- and Antiferroelectrics, in Methods and Problems in Theoretical

Physics, in honour of R.E. Peierls, Proceedings of the 1967 Birmingham conference, North-Holland, 21–28 (1970).

34. Exactly Soluble Models, in Mathematical Methods in Solid State and Super-fluid Theory, Proceedings of the 1967 Scottish Universities’ Summer School of Physics, Oliver and Boyd, Edinburgh 286–306 (1969).

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Solution of the Dimer Problem by the Transfer Matrix Method, J. Math. Phys. 8, 2339–2341 (1967). (with M. Flicker) Delta-Function Fermi Gas with Two-Spin Deviates, Phys. Rev. 161, 179–188 (1967).

$37. Concavity Properties and a Generating Function for Stirling Numbers,J. Com­binatorial Theory 5, 203–206 (1968).

38. A Theorem on Pfaffians, J. Combinatorial Theory 5, 313–319 (1968). (with F.Y. Wu) Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension, Phys. Rev. Lett. 20, 1445–1448 (1968). Two-Dimensional Ferroelectric Models, J. Phys. Soc. (Japan) 26 (supple­ment), 94–95(1969).

41. (with W.A. Beyer) Clusters on a Thin Quadratic Lattice, Studies in Appl. Math. 48, 77–90 (1969).

42. (with C.J. Thompson) Phase Transition in Zero Dimensions: A Remark on the Spherical Model, J. Math. Phys. 10, 1403–1406 (1969). (with J.L. Lebowitz) Existence of Thermodynamics for Real Matter with Coulomb Forces, Phys. Rev. Lett. 22, 631–634 (1969).

44. Two Dimensional Ice and Ferroelectric Models, in Lectures in Theoretical Physics, XI D, (Boulder summer school) Gordon and Breach, 329–354 (1969).

45. Survey of the One Dimensional Many Body Problem and Two Dimensional Ferroelectric Models, in Contemporary Physics: Trieste Symposium 1968, International Atomic Energy Agency, Vienna, vol. 1, 163–176 (1969).

46. Models, in Phase Transitions, Proceedings of the 14th Solvay Chemistry Conference, May 1969, Interscience, 45–56 (1971).

$47. (with H. Araki) Entropy Inequalities, Commun. Math. Phys. 18, 160–170 (1970). (with O.J. Heilmann) Violation of the Noncrossing Rule: The Hubbard Hamil-tonian for Benzene, Trans. N.Y. Acad. Sci. 33, 116–149 (1970). Also in Annals N.Y. Acad. Sci. 172, 583–617 (1971). (Awarded the 1970 Boris Pregel award for research in chemical physics.)

†49. (with O.J. Heilmann) Monomers and Dimers,Phys. Rev. Lett. 24, 1412–1414 (1970).

50. Book Review of “Statistical Mechanics” by David Ruelle, Bull. Amer. Math. Soc. 76, 683–687 (1970).

51. (with J.L. Lebowitz) Thermodynamic Limit for Coulomb Systems,in Syste`mes a un Nombre Infini de Degre´s de Liberte´, Colloques Internationaux de Centre National de la Recherche Scientifique 181, 155–162 (1970).

52. (with D.B. Abraham,T. Oguchi and T. Yamamoto) On the Anomolous Specific Heat of Sodium Trihydrogen Selenite, Progr. Theor. Phys. (Kyoto) 44, 1114– 1115 (1970).

53. (with D.B. Abraham)Anomalous Specific Heatof Sodium Trihydrogen Selen-ite – An Associated Combinatorial Problem, J. Chem. Phys. 54, 1446–1450 (1971).

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54. (with O.J. Heilmann, D. Kleitman and S. Sherman) Some Positive Definite Functions on Sets and Their Application to the Ising Model, Discrete Math. 1, 19–27 (1971).

55. (with Th. Niemeijer and G. Vertogen) Models in Statistical Mechanics, in Statistical Mechanics and Quantum Field Theory, Proceedings of 1970 Ecole d’Ete de Physique The´orique (Les Houches), Gordon and Breach, 281–326 (1971). (with H.N.V. Temperley) Relations between the ‘Percolation’ and ‘Colouring’ Problem and Other Graph-Theoretical Problems Associated with Regular Planar Lattices: Some Exact Results for the ‘Percolation’ Problem, Proc. Roy. Soc. A322, 251–280 (1971).

57. (with M. de Llano) Some Exact Results in the Hartree-Fock Theory of a Many-Fermion System at High Densities, Phys. Letts. 37B, 47–49 (1971). (with J.L. Lebowitz) The Constitution of Matter: Existence of Thermody­namics for Systems Composed of Electrons and Nuclei, Adv. in Math. 9, 316–398 (1972).

59. (with F.Y. Wu) Two Dimensional Ferroelectric Models, in Phase Transitions and Critical Phenomena, C. Domb and M. Green eds., vol. 1, Academic Press 331–490 (1972). (with D. Ruelle) A Property of Zeros of the Partition Function for Ising Spin Systems, J. Math. Phys. 13, 781–784 (1972). (with O.J. Heilmann) Theory of Monomer-Dimer Systems, Commun. Math. Phys. 25, 190–232 (1972). Errata 27, 166 (1972). (with M.L. Glasser and D.B. Abraham) Analytic Properties of the Free Energy for the “Ice” Models, J. Math. Phys. 13, 887–900 (1972). (with D.W. Robinson) The Finite Group Velocity of Quantum Spin Systems, Commun. Math. Phys. 28, 251–257 (1972).

64. (with J.L. Lebowitz) Phase Transition in a Continuum Classical System with Finite Interactions, Phys. Lett. 39A, 98–100 (1972).

65. (with J.L. Lebowitz) Lectures on the Thermodynamic Limit for Coulomb Systems, in Statistical Mechanics and Mathematical Problems, Battelle 1971 Recontres, Springer Lecture Notes in Physics 20, 136–161 (1973).

66. (with J.L. Lebowitz) Lectures on the Thermodynamic Limit for Coulomb Systems, in Lectures in Theoretical Physics XIV B, (Boulder summer school), Colorado Associated University Press, 423–460 (1973).

$67. Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture, Adv. in Math. 11, 267–288 (1973).

$68. (with M.B. Ruskai) A Fundamental Property of Quantum Mechanical Entropy, Phys. Rev. Lett. 30, 434–436 (1973).

$69. (with M.B. Ruskai) Proof of the Strong Subadditivity of Quantum-Mechanical Entropy, J. Math. Phys. 14, 1938–1941 (1973).

70. (with K. Hepp) On the Superradiant Phase Transition for Molecules in a Quantized Radiation Field: The Dicke Maser Model, Annals of Phys. (N.Y.) 76, 360–404 (1973).

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158.

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(with K. Hepp) Phase Transitions in Reservoir-Driven Open Systems with Applications to Lasers and Superconductors, Helv. Phys. Acta 46, 573–602 (1973). (with K. Hepp) Equilibrium Statistical Mechanics of Matter Interacting with the Quantized Radiation Field, Phys. Rev. A8, 2517–2525 (1973). (with K. Hepp) Constructive Macroscopic Quantum Electrodynamics, in Con­structive Quantum Field Theory, Proceedings of the 1973 Erice Summer School, G. Velo and A. Wightman, eds., Springer Lecture Notes in Physics 25, 298–316 (1973). The Classical Limit of Quantum Spin Systems, Commun. Math. Phys. 31, 327–340 (1973).

75. (with B. Simon) Thomas-Fermi Theory Revisited, Phys. Rev. Lett. 31, 681–683 (1973).

$76. (with M.B. Ruskai) Some Operator Inequalities of the Schwarz Type, Adv. in Math. 12, 269–273 (1974).

77. Exactly Soluble Models in Statistical Mechanics, lecture given at the 1973 I.U.P.A.P. van der Waals Centennial Conference on Statistical Mechanics, Physica 73, 226–236 (1974).

78. (with B. Simon) On Solutions to the Hartree-Fock Problem for Atoms and Molecules, J. Chem. Physics 61, 735–736 (1974).

79. Thomas-Fermi and Hartree-Fock Theory, lecture at 1974 International Con­gress of Mathematicians, Vancouver. Proceedings, 2, 383–386 (1975).

$80. Some Convexity and Subadditivity Properties of Entropy, Bull. Amer. Math. Soc. 81, 1–13 (1975).

$81. (with H.J. Brascamp and J.M. Luttinger) A General Rearrangement Inequality for Multiple Integrals, Jour. Funct. Anal. 17, 227–237 (1975).

$82. (with H.J. Brascamp) Some Inequalities for Gaussian Measures and the Long-Range Order of the One-Dimensional Plasma, lecture at Conference on Func­tional Integration, Cumberland Lodge, England. Functional Integration and its Applications, A.M. Arthurs ed., Clarendon Press, 1–14 (1975). (with K. Hepp) The Laser: A Reversible Quantum Dynamical System with Irreversible Classical Macroscopic Motion, in Dynamical Systems, Battelle 1974 Rencontres, Springer Lecture Notes in Physics 38, 178–208 (1975). Also appears in Melting, Localization and Chaos, Proc. 9th Midwest Solid State Theory Symposium, 1981, R. Kalia and P. Vashishta eds., North-Holland, 153–177 (1982).

*84. (with P. Hertel and W. Thirring) Lower Bound to the Energy of Complex Atoms, J. Chem. Phys. 62, 3355–3356 (1975).

*85. (with W. Thirring) Bound for the Kinetic Energy of Fermions which Proves the Stability of Matter, Phys. Rev. Lett. 35, 687–689 (1975). Errata 35, 1116 (1975). (with H.J. Brascamp and J.L. Lebowitz) The Statistical Mechanics of Anhar-monic Lattices, in the proceedings of the 40th session of the International Statistics Institute, Warsaw, 9, 393–404 (1975).

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186.

$87. (with H.J. Brascamp) Best Constants in Young’s Inequality, Its Converse and Its Generalization to More Than Three Functions, Adv. in Math. 20, 151–172 (1976).

$88. (with H.J. Brascamp) On Extensions of the Brunn-Minkowski and Pre´kopa-Leindler Theorems, Including Inequalities for Log Concave Functions and with an Application to the Diffusion Equation, J. Funct. Anal. 22, 366–389 (1976).

89. (with J.F. Barnes and H.J. Brascamp) Lower Bounds for the Ground State Energy of the Schroedinger Equation Using the Sharp Form of Young’s In­equality, in Studies in Mathematical Physics, Lieb, Simon, Wightman eds., Princeton Press, 83–90 (1976).

$90. Inequalities for Some Operator and Matrix Functions, Adv. in Math. 20, 174–178 (1976).

*91. (with H. Narnhofer) The Thermodynamic Limit for Jellium, J. Stat. Phys. 12, 291–310 (1975). Errata J. Stat. Phys. 14, 465 (1976).

*92. The Stability of Matter, Rev. Mod. Phys. 48, 553–569 (1976). 93. Bounds on the Eigenvalues of the Laplace and Schroedinger Operators, Bull.

Amer. Math. Soc. 82, 751–753 (1976). 94. (with F.J. Dyson and B. Simon) Phase Transitions in the Quantum Heisenberg

Model, Phys. Rev. Lett. 37, 120–123 (1976). (See no. 104.) $*95. (with W. Thirring) Inequalities for the Moments of the Eigenvalues of the

Schro¨dinger Hamiltonian and Their Relation to Sobolev Inequalities, in Stud­ies in Mathematical Physics, E. Lieb, B. Simon, A. Wightman eds., Princeton University Press, 269–303 (1976).

96. (with B. Simon and A. Wightman) Book Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann, Princeton University Press (1976).

97. (with B. Simon) Thomas-Fermi Theory of Atoms, Molecules and Solids, Adv. in Math. 23, 22–116 (1977).

†98. (with O.E. Lanford and J.L. Lebowitz) Time Evolution of Infinite Anharmonic Systems, J. Stat. Phys. 16, 453–461 (1977).

99. The Stability of Matter, Proceedings of the Conference on the Fiftieth An­niversary of the Schroedinger equation, Acta Physica Austriaca Suppl. XVII, 181–207 (1977).

$100. Existence and Uniqueness of the Minimizing Solution of Choquard’s Non­Linear Equation, Studies in Appl. Math. 57, 93–105 (1977). (with J. Fro¨hlich) Existence of Phase Transitions for Anisotropic Heisenberg Models, Phys. Rev. Lett. 38, 440–442 (1977).

*102. (with B. Simon) The Hartree-Fock Theory for Coulomb Systems, Commun. Math. Phys. 53, 185–194 (1977).

103. (with W. Thirring) A Lower Bound for Level Spacings, Annals of Phys. (N.Y.) 103, 88–96 (1977). (with F.J. Dyson and B. Simon) Phase Transitions in Quantum Spin Systems with Isotropic and Non-Isotropic Interactions, J. Stat. Phys. 18, 335–383 (1978).

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105. Many Particle Coulomb Systems, lectures given at the 1976 session on statis­tical mechanics of the International Mathematics Summer Center (C.I.M.E.). In Statistical Mechanics, C.I.M.E. 1 Ciclo 1976, G. Gallavotti, ed., Liguore Editore, Naples, 101–166 (1978).

*106. (with R. Benguria) Many-Body Atomic Potentials in Thomas-Fermi Theory, Annals of Phys. (N.Y.) 110, 34–45 (1978).

*107. (with R. Benguria) The Positivity of the Pressure in Thomas-Fermi Theory, Commun. Math. Phys. 63, 193–218 (1978). Errata 71, 94 (1980).

108. (with M. de Llano) Solitons and the Delta Function Fermion Gas in Hartree-Fock Theory, J. Math. Phys. 19, 860–868 (1978). (with J. Fro¨hlich) Phase Transitions in Anisotropic Lattice Spin Systems, Commun. Math. Phys. 60, 233–267 (1978). (with J. Fro¨hlich, R. Israel and B. Simon) Phase Transitions and Reflection Positivity. I. General Theory and Long Range Lattice Models, Commun. Math. Phys. 62, 1–34 (1978). (See no. 124.)

$111. (with M. Aizenman and E.B. Davies) Positive Linear Maps Which are Order Bounded on C* Subalgebras, Adv. in Math. 28, 84–86 (1978).

$*112. (with M. Aizenman) On Semi-Classical Bounds for Eigenvalues of Schro¨dinger Operators, Phys. Lett. 66A, 427–429 (1978).

113. New Proofs of Long Range Order, in Proceedings of the International Confer­ence on Mathematical Problems in Theoretical Physics (June 1977), Springer Lecture Notes in Physics, 80, 59–67 (1978).

$114. Proof of an Entropy Conjecture of Wehrl, Commun. Math. Phys. 62, 35–41 (1978).

115. (with B. Simon) Monotonicity of the Electronic Contribution to the Born-Oppenheimer Energy, J. Phys. B. 11, L537-L542 (1978). (with O.J. Heilmann) Lattice Models for Liquid Crystals, J. Stat. Phys. 20, 679–693 (1979).

117. (with H. Brezis) Long Range Atomic Potentials in Thomas-Fermi Theory, Commun. Math. Phys. 65, 231–246 (1979).

*118. The N5/3 Law for Bosons, Phys. Lett. 70A, 71–73 (1979). 119. A Lower Bound for Coulomb Energies, Phys. Lett. 70A, 444–446 (1979). 120. Why Matter is Stable, Kagaku 49, 301–307 and 385–388 (1979). (In Japanese). 121. The Number of Bound States of One-Body Schro¨dinger Operators and the

Weyl Problem, Symposium of the Research Inst. of Math. Sci., Kyoto University, (1979).

122. Some Open Problems About Coulomb Systems, in Proceedings of the Lau­sanne 1979 Conference of the International Association of Mathematical Physics, Springer Lecture Notes in Physics, 116, 91–102 (1980).

$*123. The Number of Bound States of One-Body Schro¨dinger Operators and the Weyl Problem, Proceedings of the Amer. Math. Soc. Symposia in Pure Math., 36, 241–252 (1980). (with J. Fro¨hlich, R.B. Israel and B. Simon) Phase Transitions and Reflection Positivity. II. Lattice Systems with Short-Range and Coulomb Interactions. J. Stat. Phys. 22, 297–347 (1980). (See no. 110.)

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125. Why Matter is Stable, Chinese Jour. Phys. 17, 49–62 (1980). (English version of no. 120). A Refinement of Simon’s Correlation Inequality, Commun. Math. Phys. 77, 127–135 (1980).

127. (with B. Simon) Pointwise Bounds on Eigenfunctions and Wave Packets in N-Body Quantum Systems. VI. Asymptotics in the Two-Cluster Region, Adv. in Appl. Math. 1, 324–343 (1980).

128. The Uncertainty Principle, article in Encyclopedia of Physics, R. Lerner and G. Trigg eds., Addison Wesley, 1078–1079 (1981).

$*129. (with S. Oxford) An Improved Lower Bound on the Indirect Coulomb Energy, Int. J. Quant. Chem. 19, 427–439 (1981).

*130. (with R. Benguria and H. Brezis) The Thomas-Fermi-vonWeizsaecker Theory of Atoms and Molecules, Commun. Math. Phys. 79, 167–180 (1981). (with M. Aizenman) The Third Law of Thermodynamics and the Degeneracy of the Ground State for Lattice Systems, J. Stat. Phys. 24, 279–297 (1981). (with J.-R. Bricmont, J. Fontaine, J.L. Lebowitz and T. Spencer) Lattice Systems with a Continuous Symmetry III. Low Temperature Asymptotic Expansion for the Plane Rotator Model, Commun. Math. Phys. 78, 545–566 (1981). (with A.D. Sokal) A General Lee-Yang Theorem for One-Component and Multicomponent Ferromagnets, Commun. Math. Phys. 80, 153–179 (1981).

*134. Variational Principle for Many-Fermion Systems, Phys. Rev. Lett. 46, 457–459 (1981). Errata 47, 69 (1981).

135. Thomas-Fermi and Related Theories of Atoms and Molecules, in Rigorous Atomic and Molecular Physics, G. Velo and A. Wightman, eds., Plenum Press 213–308 (1981).

*136. Thomas-Fermi and Related Theories of Atoms and Molecules, Rev. Mod. Phys. 53, 603–641 (1981). Errata 54, 311 (1982). (Revised version of no. 135.)

137. Statistical Theories of Large Atoms and Molecules, in Proceedings of the 1981 Oaxtepec conference on Recent Progress in Many-Body Theories, Springer Lecture Notes in Physics, 142, 336–343 (1982).

138. Statistical Theories of Large Atoms and Molecules, Comments Atomic and Mol. Phys. 11, 147–155 (1982).

*139. Analysis of the Thomas-Fermi-von Weizsa¨cker Equation for an Infinite Atom without Electron Repulsion, Commun. Math. Phys. 85, 15–25 (1982).

140. (with D.A. Liberman) Numerical Calculation of the Thomas-Fermi-von Weiz-sa cker Function for an Infinite Atom without Electron Repulsion, Los Alamos National Laboratory Report, LA-9186-MS (1982).

*141. Monotonicity of the Molecular Electronic Energy in the Nuclear Coordinates, J. Phys. B.: At. Mol. Phys. 15, L63-L66 (1982).

142. Comment on “Approach to Equilibrium of a Boltzmann Equation Solution”, Phys. Rev. Lett. 48, 1057 (1982).

143. Density Functionals for Coulomb Systems, in Physics as Natural Philosophy: Essays in honor of Laszlo Tisza on his 75th Birthday, A. Shimony and H. Feshbach eds., M.I.T. Press, 111–149 (1982).

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$144. An Lp Bound for the Riesz and Bessel Potentials of Orthonormal Functions, J. Funct. Anal. 51, 159–165 (1983).

$145. (with H. Brezis) A Relation Between Pointwise Convergence of Functions and Convergence of Functionals, Proc. Amer. Math. Soc. 88, 486–490 (1983).

*146. (with R. Benguria) A Proof of the Stability of Highly Negative Ions in the Absence of the Pauli Principle, Phys. Rev. Lett. 50, 1771–1774 (1983).

$147. Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities, Annals of Math. 118, 349–374 (1983).

$148. Density Functionals for Coulomb Systems (a revised version of no. 143), Int. Jour. Quant. Chem. 24, 243–277 (1983). An expanded version appears in Density Functional Methods in Physics, R. Dreizler and J. da Providencia eds., Plenum Nato ASI Series 123, 31–80 (1985).

149. The Significance of the Schro¨dinger Equation for Atoms,Molecules and Stars, lecture given at the Schro¨dinger Symposium, Dublin Institute of Advanced Studies, October 1983, unpublished Proceedings.

*150. (with I. Daubechies) One Electron Relativistic Molecules with Coulomb In­teraction, Commun. Math. Phys. 90, 497–510 (1983).

151. (with I. Daubechies) Relativistic Molecules with Coulomb Interaction, in Differential Equations, Proc. of the Conference held at the University of Alabama in Birmingham, 1983, I. Knowles and R. Lewis eds., Math. Studies Series, 92, 143–148 North-Holland (1984).

152. Some Vector Field Equations, in Differential Equations, Proc. of the Confer­ence held at the University of Alabama in Birmingham, 1983, I. Knowles and R. Lewis eds., Math. Studies Series 92, 403–412 North-Holland (1984).

$153. On the Lowest Eigenvalue of the Laplacian for the Intersection of Two Do­mains, Inventiones Math. 74, 441–448 (1983).

154. (with J. Chayes and L. Chayes) The Inverse Problem in Classical Statistical Mechanics, Commun. Math. Phys. 93, 57–121 (1984).

$155. On Characteristic Exponents in Turbulence, Commun. Math. Phys. 92, 473–480 (1984).

*156. Atomic and Molecular Negative Ions, Phys. Rev. Lett. 52, 315–317 (1984). *157. Bound on the Maximum Negative Ionization of Atoms and Molecules, Phys.

Rev. 29A, 3018–3028 (1984). 158. (with W. Thirring) Gravitational Collapse in Quantum Mechanics with Rela-

tivistic Kinetic Energy, Annals of Phys. (N.Y.) 155, 494–512 (1984). 159. (with I.M. Sigal, B. Simon and W. Thirring) Asymptotic Neutrality of Large-Z

Ions, Phys. Rev. Lett. 52, 994–996 (1984). (See no. 185.) *160. (with R. Benguria) The Most Negative Ion in the Thomas-Fermi-von Weiz-

sa cker Theory of Atoms and Molecules, J. Phys. B: At. Mol. Phys. 18, 1045–1059 (1985).

$161. (with H. Brezis) Minimum Action Solutions of Some Vector Field Equations, Commun. Math. Phys. 96, 97–113 (1984).

$162. (with H. Brezis) Sobolev Inequalities with Remainder Terms, J. Funct. Anal. 62, 73–86 (1985).

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$163. Baryon Mass Inequalities in Quark Models, Phys. Rev. Lett. 54, 1987–1990 (1985).

*164. (with J. Fro¨hlich and M. Loss) Stability of Coulomb Systems with Magnetic Fields I. The One-Electron Atom, Commun. Math. Phys. 104, 251–270 (1986).

*165. (with M. Loss) Stability of Coulomb Systems with Magnetic Fields II. The Many-Electron Atom and the One-Electron Molecule, Commun. Math. Phys. 104, 271–282 (1986).

*166. (with W. Thirring) Universal Nature of van der Waals Forces for Coulomb Systems, Phys. Rev. A 34, 40–46 (1986).

167. Some Ginzburg-Landau Type Vector-Field Equations, in Nonlinear systems of Partial Differential Equations in Applied Mathematics, B. Nicolaenko, D. Holm and J. Hyman eds., Amer. Math. Soc. Lectures in Appl. Math. 23, Part 2, 105–107 (1986).

†† 168. (with I. Affleck) A Proof of Part of Haldane’s Conjecture on Spin Chains, Lett. Math. Phys. 12, 57–69 (1986).

$169. (with H. Brezis and J-M. Coron) Estimations d’Energie pour des Applications de R3 a Valeurs dans S2, C.R. Acad. Sci. Paris 303 Ser. 1, 207–210 (1986).

170. (with H. Brezis and J-M. Coron) Harmonic Maps with Defects, Commun. Math. Phys. 107, 649–705 (1986).

171. Some Fundamental Properties of the Ground States of Atoms and Molecules, in Fundamental Aspects of Quantum Theory, V. Gorini and A. Frigerio eds., Nato ASI Series B, 144, 209–214, Plenum Press (1986).

172. (with T. Kennedy) A Model for Crystallization: A Variation on the Hubbard Model, in Statistical Mechanics and Field Theory: Mathematical Aspects, Springer Lecture Notes in Physics 257, 1–9 (1986).

173. (with T. Kennedy) An Itinerant Electron Model with Crystalline or Magnetic Long Range Order, Physica 138A, 320–358 (1986).

†† 174. A Model for Crystallization: A Variation on the Hubbard Model, Physica 140A, 240–250 (1986) (Proceedings of IUPAP Statphys 16, Boston).

†† 175. (with T. Kennedy) Proof of the Peierls Instability in One Dimension, Phys. Rev. Lett. 59, 1309–1312 (1987).

†† 176. (with I. Affleck, T. Kennedy and H. Tasaki) Rigorous Results on Valence-Bond Ground States in Antiferromagnets, Phys. Rev. Lett. 59, 799–802 (1987).

*177. (with H.-T. Yau) The Chandrasekhar Theory of Stellar Collapse as the Limit of Quantum Mechanics, Commun. Math. Phys. 112, 147–174 (1987).

178. (with H.-T. Yau) A Rigorous Examination of the Chandrasekhar Theory of Stellar Collapse, Astrophys. Jour. 323, 140–144 (1987).

$179. (with F. Almgren) Singularities of Energy Minimizing Maps from the Ball to the Sphere, Bull. Amer. Math. Soc. 17, 304–306 (1987). (See no. 190.)

180. Bounds on Schro¨dinger Operators and Generalized Sobolev Type Inequalities, Proceedings of the International Conference on Inequalities, University of Birmingham, England, 1987, Marcel Dekker Lecture Notes in Pure and Appl. Math., W.N. Everitt ed., volume 129, pages 123–133 (1991).

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(with I. Affleck, T. Kennedy and H. Tasaki) Valence Bond Ground States in Isotropic Quantum Antiferromagnets, Commun. Math. Phys. 115, 477–528 (1988).

182. (with T. Kennedy and H. Tasaki) A Two Dimensional Isotropic Quantum Antiferromagnet with Unique Disordered Ground State, J. Stat. Phys. 53, 383–416 (1988). (with T. Kennedy and B.S. Shastry) Existence of Ne el Order in Some Spin 1/2 Heisenberg Antiferromagnets, J. Stat. Phys. 53, 1019–1030 (1988). (with T. Kennedy and B.S. Shastry) The XY Model has Long-Range Order for all Spins and all Dimensions Greater than One, Phys. Rev. Lett. 61, 2582–2584 (1988).

*185. (with I.M. Sigal, B. Simon and W. Thirring) Approximate Neutrality of Large-Z Ions, Commun. Math. Phys. 116, 635–644 (1988). (See no. 159.)

*186. (with H.-T. Yau) The Stability and Instability of Relativistic Matter, Commun. Math. Phys. 118, 177–213 (1988).

187. (with H.-T. Yau) Many-Body Stability Implies a Bound on the Fine Structure Constant, Phys. Rev. Lett. 61, 1695–1697 (1988).

*188. (with J. Conlon and H.-T. Yau) The N7/5 Law for Charged Bosons, Commun. Math. Phys. 116, 417–448 (1988).

$189. (with F. Almgren and W. Browder) Co-area, Liquid Crystals, and Minimal Surfaces, in Partial Differential Equations, S.S. Chern ed., Springer Lecture Notes in Math. 1306, 1–22 (1988).

190. (with F. Almgren) Singularities of Energy Minimizing Maps from the Ball to the Sphere: Examples, Counterexamples and Bounds, Ann. of Math. 128, 483–530 (1988).

$191. (with F. Almgren) Counting Singularities in Liquid Crystals, in IXth Interna­tional Congress on Mathematical Physics, B. Simon, A. Truman, I.M. Davies eds., Hilger, 396–409 (1989). This also appears in: Symposia Mathemat-ica, vol. XXX, Ist. Naz. Alta Matem. Francesco Severi Roma, 103–118, Academic Press (1989); Variational Methods, H. Berestycki, J-M. Coron, I. Ekeland eds., Birkha¨user, 17–36 (1990); How many singularities can there be in an energy minimizing map from the ball to the sphere?, in Ideas and Meth­ods in Mathematical Analysis, Stochastics, and Applications, S. Albeverio, J.E. Fenstad, H. Holden, T. Lindstrom eds., Cambridge Univ. Press, vol. 1, 394–408 (1992).

$192. (with F. Almgren) Symmetric Decreasing Rearrangement can be Discontinu­ous, Bull. Amer. Math. Soc. 20, 177–180 (1989).

$193. (with F. Almgren) Symmetric Decreasing Rearrangement is Sometimes Con­tinuous, Jour. Amer. Math. Soc. 2, 683–773 (1989). A summary of this work (using ‘rectifiable currents’) appears as The (Non)continuity of Symmet­ric Decreasing Rearrangement in Symposia Mathematica, vol. XXX, Ist. Naz. Alta Matem. Francesco Severi Roma, 89–102, Academic Press (1989) and in Variational Methods, H. Berestycki, J-M. Coron, I. Ekeland eds., Birkha¨user, 3–16 (1990).

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†† $194. Two Theorems on the Hubbard Model, Phys. Rev. Lett. 62, 1201–1204 (1989). Errata 62, 1927 (1989).

195. (with J. Conlon and H.-T. Yau) The Coulomb gas at Low Temperature and Low Density, Commun. Math. Phys. 125, 153–180 (1989).

$196. Gaussian Kernels have only Gaussian Maximizers, Invent. Math. 102, 179– 208 (1990).

$*197. Kinetic Energy Bounds and their Application to the Stability of Matter, in Schro¨dinger Operators, Proceedings Sønderborg Denmark 1988, H. Holden and A. Jensen eds., Springer Lecture Notes in Physics 345, 371–382 (1989). Expanded version of no. 180.

*198. The Stability of Matter: From Atoms to Stars, 1989 Gibbs Lecture, Bull. Amer. Math. Soc. 22, 1–49 (1990).

$199. Integral Bounds for Radar Ambiguity Functions and Wigner Distributions, J. Math. Phys. 31, 594–599 (1990).

200. On the Spectral Radius of the Product of Matrix Exponentials, Linear Alg. and Appl.141, 271–273 (1990).

†† $201. (with M. Aizenman) Magnetic Properties of Some Itinerant-Electron Systems at T >0, Phys. Rev. Lett. 65, 1470–1473 (1990).

202. (with H. Siedentop) Convexity and Concavity of Eigenvalue Sums, J. Stat. Phys. 63, 811–816 (1991).

$203. (with J.P. Solovej) Quantum Coherent Operators: A Generalization of Coher­ent States, Lett. Math. Phys. 22, 145–154 (1991).

204. The Flux-Phase Problem on Planar Lattices, Helv. Phys. Acta 65, 247– 255 (1992). Proceedings of the conference “Physics in Two Dimensions”, Neuchâtel, August 1991.

205. Atome in starken Magnetfeldern, Physikalische Bla tter 48, 549–552 (1992). Translation by H. Siedentop of the Max-Planck medal lecture (1 April 1992) “Atoms in strong magnetic fields”.

206. Absence of Ferromagnetism for One-Dimensional Itinerant Electrons, in Probabilistic Methods in Mathematical Physics, Proceedings of the Interna­tional Workshop Siena, May 1991, F. Guerra, M. Loffredo and C. Marchioro eds., World Scientific pp. 290–294 (1992). A shorter version appears in Rigorous Results in Quantum Dynamics, J. Dittrich and P. Exner eds., World Scientific, pp. 243–245 (1991).

207. (with J.P. Solovej and J. Yngvason) Heavy Atoms in the Magnetic Field of a Neutron Star, Phys. Rev. Lett. 69, 749–752 (1992).

*208. (with J.P. Solovej) Atoms in the Magnetic Field of a Neutron Star, in Dif­ferential Equations with Applications to Mathematical Physics, W.F. Ames, J.V. Herod and E.M. Harrell II eds., Academic Press, pages 221–237 (1993). Also in Spectral Theory and Scattering Theory and Applications, K. Yajima, ed., Advanced Studies in Pure Math. 23, 259–274, Math. Soc. of Japan, Kinokuniya (1994). This is a summary of nos. 215, 216. Earlier summaries also appear in: (a) M e´thodes Semi-Classiques, Colloque internatinal (Nantes 1991), Asterisque 210, 237–246 (1991); (b) Some New Trends on Fluid Dy­namics and Theoretical Physics, C.C. Lin and N. Hu eds., 149–157, Peking

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University Press (1993); (c) Proceedings of the International Symposium on Advanced Topics of Quantum Physics, Shanxi, J.Q. Lang, M.L. Wang, S.N. Qiao and D.C. Su eds., 5–13, Science Press, Beijing (1993).

209. (with M. Loss and R. McCann) Uniform Density Theorem for the Hubbard Model, J. Math. Phys. 34, 891–898 (1993).

210. Remarks on the Skyrme Model, in Proceedings of the Amer. Math. Soc. Symposia in Pure Math. 54, part 2, 379–384 (1993). (Proceedings of Summer Research Institute on Differential Geometry at UCLA, July 8–28, 1990.)

$211. (with E. Carlen) Optimal Hypercontractivity for Fermi Fields and Related Noncommutative Integration Inequalities, Commun. Math. Phys. 155, 27– 46 (1993).

212. (with E. Carlen) Optimal Two-Uniform Convexity and Fermion Hypercon-tractivity, in Quantum and Non-Commutative Analysis, Proceedings of June, 1992 Kyoto Conference, H. Araki et.al. eds., Kluwer (1993), pp. 93–111. (Condensed version of no. 211.) (with M. Loss) Fluxes, Laplacians, and Kasteleyn’s Theorem, Duke Math. Journal 71, 337–363 (1993).

214. (with V. Bach, R. Lewis and H. Siedentop) On the Number of Bound States of a Bosonic N-Particle Coulomb System, Zeits. f. Math. 214, 441–460 (1993).

215. (with J.P. Solovej and J. Yngvason) Asymptotics of Heavy Atoms in High Magnetic Fields: I. Lowest Landau Band Region, Commun. Pure Appl. Math. 47, 513–591 (1994).

216. (with J.P. Solovej and J. Yngvason) Asymptotics of Heavy Atoms in High Magnetic Fields: II. Semiclassical Regions, Commun. Math. Phys. 161, 77–124 (1994).

*217. (with V. Bach, M. Loss and J.P. Solovej) There are No Unfilled Shells in Unrestricted Hartree-Fock Theory, Phys. Rev. Lett. 72, 2981–2983 (1994).

$218. (with K. Ball and E. Carlen) Sharp Uniform Convexity and Smoothness Inequalities for Trace Norms, Invent. Math. 115, 463–482 (1994).

$219. Coherent States as a Tool for Obtaining Rigorous Bounds, Proceedings of the Symposium on Coherent States, past, present and future, Oak Ridge, D.H. Feng, J. Klauder and M.R. Strayer eds., World Scientific (1994), pages 267–278.

†† 220. The Hubbard model: Some Rigorous Results and Open Problems, in Pro­ceedings of 1993 conference in honor of G.F. Dell’Antonio, Advances in Dynamical Systems and Quantum Physics, S. Albeverio et.al. eds., pp. 173– 193, World Scientific (1995). A revised version appears in Proceedings of 1993 NATO ASI The Hubbard Model, D. Baeriswyl et.al. eds., pp. 1–19, Plenum Press (1995). A further revision appears in Proceedings of the XIth International Congress of Mathematical Physics, Paris, 1994, D. Iagolnitzer ed., pp. 392–412, International Press (1995).

221. (with V. Bach and J.P. Solovej) Generalized Hartree-Fock Theory of the Hubbard Model, J. Stat. Phys. 76, 3–90 (1994).

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†† 222. Flux Phase of the Half-Filled Band, Phys. Rev. Lett. 73, 2158–2161 (1994). $223. (with M. Loss) Symmetry of the Ginzburg-Landau Minimizer in a Disc, Math.

Res. Lett. 1, 701–715 (1994). 224. (with J.P. Solovej and J. Yngvason) Quantum Dots, in Proceedings of the

Conference on Partial Differential Equations and Mathematical Physics, Uni­versity of Alabama, Birmingham, 1994, I. Knowles, ed., International Press (1995), pages 157–172.

*225. (with J.P. Solovej and J. Yngvason) Ground States of Large Quantum Dots in Magnetic Fields, Phys. Rev. B 51, 10646–10665 (1995).

226. (with J. Freericks) The Ground State of a General Electron-Phonon Hamilto-nian is a Spin Singlet, Phys. Rev. B 51, 2812–2821 (1995).

†† 227. (with B. Nachtergaele) Stability of the Peierls Instability for Ring-Shaped Molecules, Phys. Rev. B 51, 4777–4791 (1995).

228. (with B. Nachtergaele) Dimerization in Ring-Shaped Molecules: The Stabil­ity of the Peierls Instability in Proceedings of the XIth International Congress of Mathematical Physics, Paris, 1994, D. Iagolnitzer ed., pp. 423–431, Inter­national Press (1995).

229. (with B. Nachtergaele) Bond Alternation in Ring-Shaped Molecules: The Stability of the Peierls Instability. In Proceedings of the conference The Chemical Bond, Copenhagen 1994, Int. J. Quant. Chem. 58, 699–706 (1996).

230. Fluxes and Dimers in the Hubbard Model, in Proceedings of the Interna­tional Congress of Mathematicians, Zu rich, 1994, S.D. Chatterji ed., vol. 2, pp. 1279–1280, Birkha¨user (1995).

*231. (with M. Loss and J. P. Solovej) Stability of Matter in Magnetic Fields, Phys. Rev. Lett. 75, 985–989 (1995).

*232. (with O. J. Heilmann) Electron Density near the Nucleus of a large Atom, Phys. Rev A 52, 3628–3643 (1995).

*233. (with A. Iantchenko and H. Siedentop) Proof of a Conjecture about Atomic and Molecular Cores Related to Scott’s Correction, J. reine u. ang. Math. 472, 177–195 (1996).

†† 234. (with L.E. Thomas) Exact Ground State Energy of the Strong-Coupling Po-laron, Commun. Math. Phys. 183, 511–519 (1997). Errata 188, 499–500 (1997).

$235. (with L. Cafarelli and D. Jerison) On the Case of Equality in the Brunn-Minkowski Inequality for Capacity, Adv. in Math. 117, 193–207 (1996).

*236. (with M. Loss and H. Siedentop) Stability of Relativistic Matter via Thomas-Fermi Theory, Helv. Phys. Acta 69, 974–984 (1996).

†† 237. Some of the Early History of Exactly Soluble Models, in Proceedings of the 1996 Northeastern University conference on Exactly Soluble Models, Int. Jour. Mod. Phys. B 11, 3–10 (1997).

*238. (with H. Siedentop and J. P. Solovej) Stability and Instability of Relativistic Electrons in Magnetic Fields, J. Stat. Phys. 89, 37–59 (1997).

239. (with H. Siedentop and J-P. Solovej) Stability of Relativistic Matter with Magnetic Fields, Phys. Rev. Lett. 79, 1785–1788 (1997).

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240. Stability of Matter in Magnetic Fields, in Proceedings of the Conference on Unconventional Quantum Liquids, Evora, Portugal, 1996 Zeits. f. Phys. B 933, 271-274 (1997).

241. Birmingham in the Good Old Days, in Proceedings of the Conference on Unconventional Quantum Liquids, Evora, Portugal, 1996 Zeits. f. Phys. B 933,125-126 (1997).

242. (with M. Loss) Book Analysis, American Mathematical Society (1997). 243. Doing Math with Fred, in In Memoriam Frederick J. Almgren Jr., 1937-1997,

Experimental Math. 6, 2-3 (1997). *244. (with J.P. Solovej and J. Yngvason) Asymptotics of Natural and Artificial

Atoms in Strong Magnetic Fields, in The Stability of Matter: From Atoms to Stars, Selecta ofE. H. Lieb, W. Thirring ed., second edition, Springer Verlag, pp. 145-167(1997). This is a summary of nos. 207,208,215,216,224,225.

245. Stability and Instability of Relativistic Electrons in Classical Electromagnetic Fields, in Proceedings of Conference on Partial Differential Eqations and Mathematical Physics, Georgia Inst. of Tech., March, 1997, Amer. Math. Soc. Contemporary Math. series, E. Carlen, E. Harrell, M. Loss eds., 217, 99-108 (1998).

*246. (with J. Yngvason) Ground State Energy of the Low Density Bose Gas, Phys. Rev. Lett. 80,2504-2507(1998). arXiv math-ph/9712138,mp_arc 97-631. (with J. Yngvason) A Guide to Entropy and the Second Law of Thermody­namics, Notices of the Amer. Math. Soc. 45, 571-581 (1998). arXiv math-ph/9805005, mp_arc 98-339. http://www.ams.org/notices/199805/lieb.pdf. See no. 266. This paper received the American Mathematical Society 2002 Levi Conant prize for “the best expository paper published in either the No­tices of the AMS or the Bulletin of the AMS in the preceding five years”.

$248. (with D. Hundertmark and L.E. Thomas) A Sharp Bound for an Eigenvalue Moment of the One-Dimensional Schroedinger Operator, Adv. Theor. Math. Phys. 2, 719- 731 (1998). arXiv math-ph/9806012, mp_arc 98-753.

$249. (with E. Carlen) A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy, in Amer. Math. Soc. Transl. (2), 189, 59-69 (1999).

250. (with J. Yngvason) The Physics and Mathematics of the Second Law of Ther­modynamics, Physics Reports 310, 1-96 (1999). arXiv cond-mat/9708200, mp_arc 97–457.

251. Some Problems in Statistical Mechanics that I would like to see Solved, 1998 IUPAP Boltzmann prize lecture, Physica A 263,491–499 (1999). (with P. Schupp) Ground State Properties of a Fully Frustrated Quantum Spin System, Phys. Rev. Lett. 83, 5362-5365 (1999). arXiv math-ph/9908019, mp_arc 99-304.

253. (with P. Schupp) Singlets and Reflection Symmetric Spin Systems, Physica A 279, 378-385 (2000). arXiv math-ph/9910037, mp_arc 99-404.

*254. (with R. Seiringer and J.Yngvason) Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional, Phys. Rev A 61, #043602-1-13 (2000). arXiv math-ph/9908027, mp_arc 99-312.

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255. (with J. Yngvason) The Ground State Energy of a Dilute Bose Gas, in Dif­ferential Equations and Mathematical Physics, University of Alabama, Birm­ingham, 1999, R. Weikard and G. Weinstein, eds., 295-306, Internat. Press (2000). arXiv math-ph/9910033, mp_arc 99-401.

*256. (with M. Loss) Self-Energy of Electrons in Non-perturbative QED, in Dif­ferential Equations and Mathematical Physics, University of Alabama, Birm­ingham, 1999, R. Weikard and G. Weinstein, eds. 279-293, Amer. Math. Soc./Internat. Press (2000). arXiv math-ph/9908020, mp_arc 99-305.

257. (with R. Seiringer and J. Yngvason) The Ground State Energy and Density of Interacting Bosons in a Trap, in Quantum Theory and Symmetries, Goslar, 1999, H.-D. Doebner, V.K. Dobrev, J.-D. Hennig and W. Luecke, eds., pp. 101-110, World Scientific (2000). arXiv math-ph/9911026, mp_arc 99-439.

258. (with J. Yngvason) The Ground State Energy of a Dilute Two-dimensional Bose Gas, J. Stat. Phys. 103, 509-526 (2001). arXiv math-ph/0002014, mp_arc 00-63. (with J. Yngvason) A Fresh Look at Entropy and the Second Law of Thermo­dynamics, Physics Today 53, 32-37 (April 2000). arXiv math-ph/0003028, mp_arc 00-123. See also 53,11-14,106 (October 2000).

260. Lieb-Thirring Inequalities, in Encyclopaedia ofMathematics, Supplementvol. 2, pp. 311-313, Kluwer (2000). arXiv math-ph/0003039, mp_arc 00-132.

261. Thomas-FermiTheory, in Encyclopaedia of Mathematics, Supplementvol. 2, pp. 455–457, Kluwer (2000). arXiv math-ph/0003040, mp_arc 00-131.

262. (with H. Siedentop) Renormalization of the Regularized Relativistic Electron-Positron Field, Commun. Math. Phys. 213, 673-684 (2000). arXiv math-ph/0003001 mp_arc 00-98.

263. (with R. Seiringer and J. Yngvason) A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-dimensional Bose Gas, Commun. Math. Phys. 224,17-31 (2001). arXivcond-mat/0005026,mp_arc 00-203.

*264. (with M. Griesemer and M. Loss) Ground States in Non-relativistic Quan­tum Electrodynamics, Invent. Math. 145, 557-595 (2001). arXiv math-ph/0007014, mp_arc 00-313.

*265. (with J.P. Solovej) Ground State Energy of the One-Component Charged Bose Gas, Commun. Math. Phys. 217,127-163(2001). Errata 225, 219-221 (2002). arXiv cond-mat/0007425, mp_arc 00-303.

266. (with J. Yngvason) The Mathematics of the Second Law of Thermodynamics, in Visions in Mathematics, Towards 2000, A. Alon, J. Bourgain, A. Connes, M. Gromov and V. Milman, eds., GAFA 2000, no. 1, Birkhauser, p. 334-358 (2000). See no. 247. mp_arc 00-332.

267. The Bose Gas: A Subtle Many-Body Problem, in Proceedings of the XIII International Congress on Mathematical Physics, London, A. Fokas, et al. eds. International Press, pp. 91-111, 2001. arXiv math-ph/0009009, mp_arc 00-351.

268. (with J. Freericks and D. Ueltschi) Segregation in the Falicov-Kimball Model, Commun. Math. Phys. 227, 243-279 (2002). arXiv math-ph/0107003, mp_arc 01-243.

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269. (with G.K. Pedersen) Convex Multivariable Trace Functions, Reviews in Math. Phys. 14,1-18(2002). arXiv math.OA/0107062. (with J. Freericks and D. Ueltschi) Phase Separation due to Quantum Me­chanical Correlations, Phys. Rev. Lett. 88, #106401-1-4 (2002). arXiv cond-mat/0110251.

*271. (with M. Loss) Stability of a Model of Relativistic Quantum Electrodynamics, Commun. Math. Phys. 228, 561-588 (2002). arXiv math-ph/0109002, mp_arc 01-315.

*272. (with M. Loss) A Bound on Binding Energies and Mass Renormalization in Models of Quantum Electrodynamics, J. Stat. Phys. 108,1057-1069(2002). arXiv math-ph/0110027.

*273. (with R. Seiringer) Proof of Bose-Einstein Condensation for Dilute Trapped Gases, Phys. Rev. Lett. 88, #170409-1-4 (2002). arXiv math-ph/0112032, mp_arc02-115.

274. (with M. Loss) Stability of Matter in Relativistic Quantum Mechanics, in Mathematical Results in Quantum Mechanics,Proceedings of QMath8,Taxco, Amer. Math. Soc. Contemporary Mathematics series, pp. 225-238, 2002.

275. (with J. Yngvason) The Mathematical Structure of the Second Law of Ther­modynamics, in Current Developments in Mathematics 2001, International Press, de Jong et al eds. 2002, pp. 89-129. arXiv math-ph/0204007.

276. (with R. Seiringer, J. P. Solovej and J. Yngvason) The Ground State of the Bose Gas, in Current Developments in Mathematics 2001, de Jong et al eds., International Press, 2002, pp. 131-178. arXiv math-ph/0204027, mp_arc 02-183.

277. (with R. Seiringer and J. Yngvason) Poincaré Inequalities in Punctured Do­mains, Annals of Math. 158,1067-1080 (2003). arXiv math.FA/0205088.

*278. (with R. Seiringer and J. Yngvason) Superfluidity in Dilute Trapped Bose Gases, Phys. Rev. B 66, #134529-1-6 (2002). arXiv cond-mat/0205570, mp_arc 02-339.

279. (with F.Y Wu) The one-dimensional Hubbard model: A reminiscence, Phys-ica A 321,1-27 (2003). arXiv cond-mat/0207529.

280. (with E. Eisenberg) Polarization of interacting bosons with spin, Phys. Rev. Lett. 89, #220403-1-4 (2002), mp_arc 02-446. arXiv cond-mat/0207042.

281. The Stability of Matter and Quantum Electrodynamics, in Proceedings of the Heisenberg symposium, Munich, Dec. 2001, Fundamental Physics -Heisenberg and Beyond, G. Buschhorn and J. Wess, eds., pp. 53-68, Springer (2004). arXiv math-ph/0209034. A modified, up-dated version appears in the Milan Journal of Mathematics 71, 199-217 (2003). A further modification appearsintheJahresbericht of the German Math. Soc. JB 106,93-110(2004) (Teubner). arXiv math-ph/0401004.

282. (with R. Seiringer and J. Yngvason) Two-dimensional Gross-Pitaevskii The­ory, in: A.G. Litvak (editor). Progress in Nonlinear Science, Proceedings of the International Conference Dedicated to the 100th Anniversary of A.A. Andronov, Volume II, pp 582-590, Nizhny Novgorod, Institute of Applied Physics, University of Nizhny Novgorod, 2002.

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283. (with R. Seiringer) Equivalent forms of the Bessis-Moussa-Villani Conjecture, J. Stat. Phys. 115, 185-190(2004). math-ph/0210027.

*284. (with R. Seiringer and J. Yngvason) One-dimensional Bosons in Three-dimensional Traps, Phys. Rev. Lett. 91, #150401-1-4 (2003). arXiv cond-mat/0304071.

285. (with R. Seiringer and J. Yngvason) One-dimensional Behavior of Dilute, Trapped Bose Gases. Commun. Math. Phys. 244, 347-393 (2004). arXiv math-ph/0305025.

286. (with R. Seiringer) Bose-Einstein Condensation of Dilute Gases in Traps, in Differential Equations and Mathematical Physics proceedings of a conference at the Univ. of Birmingham, 2002, Amer. Math. Soc. Contemporary Math. series, Y. Karpeshina, G. Stolz, R. Weikard, Y Zeng, eds., 327, 239-250 (2003).

*287. (with M. Loss) Existence of Atoms and Molecules in Non-Relativistic Quan­tum Electrodynamics, Adv. Theor. Math. Phys. 7, 667-710 (2003). arXiv math-ph/0307046.

*288. (with J.P. Solovej) Ground State Energy of the Two-Component Charged Bose Gas. Commun. Math. Phys. DOI: 10.1007/s00220-004-1144-1 (in press). arXiv math-phys/0311010, mp_arc 03-490.

289. (with J. Yngvason) The Entropy of Classical Thermodynamics, in Entropy, the proceedings of a conference in Dresden, 2000, A. Greven, G. Keller and G. Warnecke eds., Princeton University Press, pp. 147-196 (2003).

290. (with M. Loss) A Note on Polarization Vectors in Quantum Electrodynamics. Commun. Math. Phys. (in press) arXiv math-ph/0401016. (with M. Aizenman, R. Seiringer, J.P. Solovej and J. Yngvason) Bose-Einstein Quantum Phase Transition in an Optical Lattice Model. Phys. Rev. A 70, 023612-(1-12) (2004). arXiV cond-mat/0403240.

292. (with E. Carlen) Some Matrix Rearrangement Inequalities, Annali Matem. Pura Appl., (in press). arXiv math.OA/0402239. (with R. Seiringer, J.P. Solovej and J. Yngvason) The Quantum-Mechani­cal Many-Body Problem: The Bose Gas, in Proceedings of the conference ‘Perspectives in Analysis’, KTH Stockholm, 2003, M. Benedicks, P. Jones and S. Smirnov eds., Springer (in press). arXiv math-ph/0405004.

294. (with M. Loss) The Thermodynamic Limit for Matter Interacting with Coulomb Forces and with the Quantized Electromagnetic Field: I. The Lower Bound. arXiv math-ph/0408001.

295. (with E. Carlen and M. Loss) A Sharp analog of Young’s Inequality on S and Related Entropy Inequalities, Jour. Geom. Anal. 14, 487-520 (2004). arXiv math.FA/0408030.

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