Stability of Matter
from atoms in the microscopic world to stars in astrophysical scale
Updated on 25 August 2017
This web site will be soon moved to http://www.cc.matsuyama-u.ac.jp/~dan/stability/
We investigate the mathematical analysis of quantum mechanics for many-body systems, a large number of elements interacting with each other.
Our research project will contribute
to understand structure of atoms and molecules as quantum particles,
to compute the mechanism of beautiful H2 O crystal formation of snow,
to estimate the effect of complicated molecules in organic chemistry or computational pharmaceutical developments,
even to apply our results to insulators, semiconductors, transistors, and quantum computers, for instance.
Analysis Seminar
References
Satosi Watanabe,
"Über die Anwendung thermodynamischer Begriffe auf den Normalzustand des Atomkerns ,"
Zeitschrift für Physik, July 1939, Volume 113, Issue 7, pp 482–513.
Harold W. Kuhn,
"Linear inequalities and the Pauli principle,"
Proc. Sympos. Appl. Math., Vol. 10 pp. 141–147. (1960)
A. J. Coleman,
"Structure of Fermion Density Matrices ,"
Rev. Mod. Phys. 35, 668 – Published 1 July 1963.
A. J. Coleman,
'Discussion on "Structure of Fermion Density Matrices" ,'
Rev. Mod. Phys. 35, 687 – Published 1 July 1963.
Tsuyoshi Ando,
"Properties of Fermion Density Matrices ,"
Rev. Mod. Phys. 35, 690 – Published 1 July 1963
R. L. Frank, E. H. Lieb, R. Seiringer and H. Siedentop;
"Müller’s exchange-correlation energy in density-matrix-functional theory ,"
Phys. Rev. A, 76 (2007), 052517.
R. L. Frank, P. T. Nam, and H. Van Den Bosch;
"The maximal excess charge in Müller density-matrix-functional theory ,"
preprint arXiv:1608.05625 (2016).
Christoph Kehle,
"The maximal excess charge for a family of density-matrix-functional theories including Hartree-Fock and Müller theories ,"
Journal of Mathematical Physics 58 , 011901 (2017).
Monday Friday Seminar
Textbook
Elliott H. Lieb and Robert Seiringer,
The Stability of Matter in Quantum Mechanics ,
Cambridge University Press (2009).
See also Errata .
Seminar Notes
Slater determinant is the simplest example of antisymmetric functions
The Space of Wave Function (24 Oct. 2016)
Density Matrices (31 Oct. 2016)
Trace of Operators (21 Nov. 2016)
Density Matrices (21 Nov. 2016)
Reduced Density Matrices (28 Nov. 2016)
Admissible One-Body Density Matrices (5 Dec. 2016)
Admissible One-Body Density Matrices for Fermions (12 Dec. 2016)
Under consideration (19 Dec. 2016)
Under construction (16 Jan. 2017)
Sufficiency (15 Feb. 2017)
A completion of the proof (24 Feb. 2017)
Many Body Hamiltonian (3 Mar. 2017)
Gamma function (30 Mar. 2017)
Coulomb Potential (2 May 2017)
Basic Electrostatic Inequality (12 May 2017)
Baxter's Inequality (19 May 2017)
Refined Electrostatic Inequality (26 May 2017)
The Method of Image Charge (2 June 2017)
Stability of Non-Relativistic Matter (23 June 2017)
Alternative Proof (30 June 2017)
Extensivity and Bosonic Instability (21 July 2017)
The N7/5 Law (28 July 2017) [New! ]
Prologue
Introduction to Elementary Quantum Mechanics and Stability of the First Kind
Introduction
Brief Outline of the Book
Many-Particle Systems and Stability of the Second Kind
Lieb-Thirring and Related Inequalities
Electrostatic Inequalities
An Estimation of the Indirect Part of the Coulomb Energy
Stability of Non-Relativistic Matter
Stability of Relativistic Matter
Magnetic Fields and the Pauli Operator
The Dirac Operator and the Brown-Ravenhall Model
Quantized Electromagnetic Fields and Stability of Matter
The Inonization Problem, and the Dependance of the Energy on N and M Separately
Gravitational Stability of White Dwarfs and Neutron Stars
The Thermodynamic Limit for Coulomb Systems
Research Review around PDE
"Lieb-Thirring inequalities for eigenvalues of Schrödinger and Dirac operators,"
Seminar on Spectral Theory at Gakushuin University, Japan
"The approach to Lieb-Thirring inequalities for Schrödinger operators,"
Talk at Gakushuin University, Japan
References
Stability of Matter
Yukimi Goto, "Absence of a Ground State for Bosonic Coulomb Systems with Critical Charge,"
arXiv:1605.02949 .
M. Klaus, B. Simon,
"Coupling constant thresholds in nonrelativistic quantum mechanics II. Two cluster thresholds in N-body systems ,"
Communications in Mathematical Physics, December 1980, Volume 78, Issue 2, pp 153–168.
M. Klaus, B. Simon,
"Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case ,"
Annals of Physics, Volume 130, Issue 2, December 1980, Pages 251-281.
E. H. Lieb,
"Residual Entropy of Square Ice,"
Physical Review. 162 (1): 162–172. doi:10.1103/PhysRev.162.162 .
(1967)
Ice-type model
Elliott H. Lieb,
"Residual Entropy of Square Ice ,"
Phys. Rev. 162, 162 – Published 5 October 1967
Linus Pauling,
"The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement ,"
J. Am. Chem. Soc., 1935, 57 (12), pp 2680–2684
J. F. Nagle, M. Mille, and H. J. Morowitz,
"Theory of hydrogen bonded chains in bioenergetics ",
The Journal of Chemical Physics 72, 3959 (1980)
Elliott H. Lieb,
"Exact Solution of the Problem of the Entropy of Two-Dimensional Ice ,"
Phys. Rev. Lett. 18, 692 – Published 24 April 1967
J. C. Slater,
"Theory of the Transition in KH2 PO4 ,"
The Journal of Chemical Physics 9, 16 (1941)
The ionization conjecture
Entropy and Entanglement
VIDEO
VIDEO
Recent upcomming results
Phan Thành Nam,
"Lieb-Thirring inequality with semiclassical constant and gradient error term ."
Sergey Morozov and David Müller,
"Lower bounds on the moduli of three-dimensional Coulomb-Dirac operators via fractional Laplacians with applications ."
Sergey Morozov and David Müller,
"On the virtual levels of positively projected massless Coulomb-Dirac operators ."
Sergey Morozov and David Müller,
"Lieb-Thirring and Cwickel-Lieb-Rozenblum inequalities for perturbed graphene with a Coulomb impurity ."
Norihiro Someyama,
"Number of Eigenvalues of Non-selfadjoint Schrödinger Operators with Dilation Analytic Complex Potentials ."
Elliott H. Lieb, Nicolas Rougerie and Jakob Yngvason,
"Rigidity of the Laughlin liquid ."
Julien Sabin,
"Littlewood-Paley decomposition of operator densities and application to a new proof of the Lieb-Thirring inequality ."
Rupert L. Frank,
"Eigenvalue bounds for the fractional Laplacian: A review ."
Thomas Moser and Robert Seiringer,
"Stability of a fermionic N+1 particle system with point interactions ."
Yul Otani and Yoh Tanimoto,
"Towards entanglement entropy with UV cutoff in conformal nets ."
Sven Bachmann, Wojciech De Roeck, Martin Fraas;
"The Adiabatic Theorem for Many-Body Quantum Systems ."
Density Matrices and N-Representability Problem
A. J. Coleman,
"Structure of Fermion Density Matrices ,"
Rev. Mod. Phys. 35, 668 – Published 1 July 1963.
A. J. Coleman,
'Discussion on "Structure of Fermion Density Matrices" ,'
Rev. Mod. Phys. 35, 687 – Published 1 July 1963.
Tsuyoshi Ando,
"Properties of Fermion Density Matrices ,"
Rev. Mod. Phys. 35, 690 – Published 1 July 1963
J. K. Percus,
"The role of model systems in the few-body reduction of the N-fermion problem ,"
Int. J. Quantum Chem., 13: 89–124. (1978)
R. M. Erdahl,
"Representability ,"
Int. J. Quantum Chem., 13: 697–718. (1978)
Eyvind H. Wichmann,
"Density Matrices Arising from Incomplete Measurements ,"
Journal of Mathematical Physics 4 , 884 (1963).
Samuel A. Ocko, Xie Chen, Bei Zeng, Beni Yoshida, Zhengfeng Ji, Mary Beth Ruskai, and Isaac L. Chuang,
"Quantum Codes Give Counterexamples to the Unique Preimage Conjecture of the N-Representability Problem ,"
Phys. Rev. Lett. 106, 110501 – Published 14 March 2011.
Jianxin Chen, Zhengfeng Ji, Mary Beth Ruskai, Bei Zeng, and Duan-Lu Zhou,
"Comment on some results of Erdahl and the convex structure of reduced density matrices ,"
J. Math. Phys. 53, 072203 (2012).
W. Stephan,
"Continuity of the Maximum-Entropy Inference ,"
Commun. Math. Phys. (2014) 330: 1263.
M. Krein and D. Milman,
"On extreme points of regular convex sets ,"
Studia Mathematica 9.1 (1940): 133-138.
Alexander A Klyachko,
"Quantum marginal problem and N-representability ,"
J. Phys.: Conf. Ser. 36 , 72 (2006).
Harold W. Kuhn,
"Solvability and Consistency for Linear Equations and Inequalities ,"
The American Mathematical Monthly,
Vol. 63, No. 4 (Apr., 1956), pp. 217-232.
Mark L. Yoseloff and Harold W. Kuhn,
"Combinatorial Approach to the N‐Representability of P‐Density Matrices" ,"
Journal of Mathematical Physics, 10 , 703. (1969)
Mark Laurance Yoseloff,
"A Combinatorial Approach to the Diagonal N-Representability Problem ,"
Transactions of the American Mathematical Society, Vol. 190 (Apr., 1974), pp. 1-41.
Satosi Watanabe,
"Über die Anwendung thermodynamischer Begriffe auf den Normalzustand des Atomkerns ,"
Zeitschrift für Physik, July 1939, Volume 113, Issue 7, pp 482–513.
Murat Altunbulak and Alexander Klyachko,
"The Pauli Principle Revisited ,"
Communications in Mathematical Physics, September 2008, Volume 282, Issue 2, pp 287–322
R E Borland and K Dennis,
"The conditions on the one-matrix for three-body fermion wavefunctions with one-rank equal to six ,"
Journal of Physics B: Atomic and Molecular Physics, Volume 5, Number 1
Mary Beth Ruskai,
"Connecting N-representability to Weyl's problem: the one-particle density matrix for N = 3 and R = 6 ,"
Journal of Physics A: Mathematical and Theoretical, Volume 40, Number 45 (2007)
Christian Schilling and Rolf Schilling,
"Duality of reduced density matrices and their eigenvalues ,"
Journal of Physics A: Mathematical and Theoretical, Volume 47, Number 41 (2014).
M. Laczkovich,
"Solvability and consistency for infinite systems of linear inequalities ,"
Periodica Mathematica Hungarica, March 1978, Volume 9, Issue 1, pp 63–70.
Harold W. Kuhn,
"Linear inequalities and the Pauli principle,"
Proc. Sympos. Appl. Math., Vol. 10 pp. 141–147. (1960)
Allen A. Goldstein AND Ward Cheney,
"A finite algorithm for the solution of consistent linear equations and inequalities and for the Tchebycheff approximation of inconsistent linear equations ,"
pacific J. Math, 8 , 3 (1958).
S. Kh. Samvelyan,
"N-Representability of Diagonal Elements of Second-Order Reduced Density Matrices ,"
International Journal of Quantum Chemistry, Vol. 65, 127-142. (1997)
Löwdin,
"Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction ."
Coleman, A.J., Yukalov, V.I.,
Reduced Density Matrices, Coulson’s Challenge ,
Springer. (2000)
Maho Nakata, et. al. ,
"Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm ,"
The Journal of Chemical Physics 114 , 8282 (2001).
Maho Nakata, Mituhiro Fukuda and Katsuki Fujisawa,
"Variational approach for the electronic structure calculation on the second-order reduced density matrices and the N-representability problem ".
Stability of Matter
Michael Loss;
Stability of Matter
Elliott H. Lieb and Michael Loss,
Analysis , Second Edition,
American Mathematical Society, Providence, Phode Island (2001).
See also Supplementary Material .
Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej and Jakob Yngvason,
The Mathematics of the Bose Gas and its Condensation ,
Birkhaeuser Basel (2005).
See also Slightly revised and extended version .
Elliott H. Lieb and Robert Seiringer,
The Stability of Matter in Quantum Mechanics ,
Cambridge University Press (2009).
See also Supplementary Material .
Freeman J. Dyson and A. Lenard,
"Stability of Matter. I ,"
J. Math. Phys. 8 , 423-434 (1967);
ibid. II , 9 , 698 (1968).
Freeman J. Dyson,
"Ground‐State Energy of a Finite System of Charged Particles ,"
J. Math. Phys. 8 , 1538 (1967).
Paul Federbush,
"A new approach to the stability of matter problem. I ,"
J. Math. Phys. 16 , 347-351 (1975);
ibid. II , 16 , 706-709 (1975).
Lars Onsager,
"Electrostatic Interaction of Molecules ,"
J. Phys. Chem., 43 (2), pp 189-196 (1939).
Jürg Fröhlich, Elliott H. Lieb, and Michael Loss,
"Stability of Coulomb systems with magnetic fields. I. The one-electron atom ,"
Comm. Math. Phys. 104 (1986), no. 2, 251-270.
Elliott H. Lieb and Michael Loss,
"Stability of Coulomb systems with magnetic fields. II. The many-electron atom and the one-electron molecule ,"
Comm. Math. Phys. 104 (1986), no. 2, 271-282.
Michael Loss and Horng-Tzer Yau,
"Stabilty of Coulomb systems with magnetic fields. III. Zero energy bound states of the Pauli operator ,"
Comm. Math. Phys., 104 , no. 2, 283-290 (1986).
C. Fefferman,
"Stability of Coulomb systems in a magnetic field ,"
Proc. Nat. Acad. Sci. U.S.A. 92 (1995), no. 11, 5006–5007.
Marius C. Lemm,
"Stability of Matter ."
M. Loss,
"Stability of Matter ."
Claude Garrod and Jerome K. Percus,
"http://scitation.aip.org/content/aip/journal/jmp/5/12/10.1063/1.1704098 ,"
J. Math. Phys. 5, 1756 (1964)
J. K. Percus,
"The role of model systems in the few-body reduction of the N-fermion problem ,"
Int. J. Quantum Chem., 13 : 89–124. (1978)
A. J. Coleman,
"Structure of Fermion Density Matrices ,"
Rev. Mod. Phys. 35 , 668 – Published 1 July 1963
T. Kato, Schrödinger operators with singular potentials.
Charles Fefferman,
"Stability of Coulomb systems in a magnetic field ,"
PNAS May 23, 1995 vol. 92 no. 11 5006-5007.
Charles Fefferman,
"On Electrons and Nuclei in a Magnetic Field ,"
Advances in Mathematics,
Volume 124, Issue 1, 1 December 1996, Pages 100-153
Elliott H. Lieb, Heinz Siedentop and Jan Philip Solovej,
"Stability and instability of relativistic electrons in classical electromagnetic fields ,"
Journal of Statistical Physics,
October 1997, Volume 89, Issue 1, pp 37–59
Elliott H. Lieb, Michael Loss,
"Stability of a Model of Relativistic Quantum Electrodynamics ,"
The Stability of Matter: From Atoms to Stars,
pp 723-750
Elliott H. Lieb, Michael Loss, Jan Philip Solovej,
"Stability of Matter in Magnetic Fields ,"
The Stability of Matter: From Atoms to Stars
pp 437-441
Max Plank, Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum
Elliott H. Lieb,
"The stability of matter ,"
Rev. Mod. Phys. 48, 553 – Published 1 October 1976.
Elliott H. Lieb,
"The stability of matter: from atoms to stars ,"
Bull. Amer. Math. Soc. (N.S.)
Volume 22, Number 1 (1990), 1-49.
M. Loss,
"Stability of Matter - Georgia Tech ."
Elliott H. Lieb and Stephen Oxford,
"Improved lower bound on the indirect Coulomb energy,"
Int. J. Quantum Chem., 19: 427–439. (1981)
doi:10.1002/qua.560190306
Lieb, E. H. and Oxford, S. (1981), Improved lower bound on the indirect Coulomb energy. Int. J. Quantum Chem., 19: 427–439. doi:10.1002/qua.560190306
Benguria, R. D., Bley, G. A. and Loss, M. (2012), A new estimate on the indirect Coulomb energy. Int. J. Quantum Chem., 112: 1579–1584. doi:10.1002/qua.23148
On Relativistic Wave Equations for Particles of Arbitrary Spin in an Electromagnetic Field
M. Fierz, W. Pauli
Proc. R. Soc. Lond. A 1939 173 211-232; DOI: 10.1098/rspa.1939.0140. Published 28 November 1939
Rupert L. Frank, Mathieu Lewin, Elliott H. Lieb, and Robert Seiringer,
"A positive density analogue of the Lieb–Thirring inequality ,"
Duke Math. J., Volume 162, Number 3 (2013), 435-495.
Combinatorial Analysis
Water H2 O Molecule and Snow Crystals
Aung, S., Pitzer, R.M, Chan, S.I.,
“Approximate Hartree-Fock
Wavefunctions, One-Electron
Properties, and Electronic Structure
of the Water Molecule.” J. Chem.
Phys. 49, 2071 (1968)
Andriessen, J. “A Calculation of The
Ground State of H2O Using a
Minimal Basis Set.”
Chemical Physics Letters
Volume 3, Issue 4, April 1969, Pages 257-258
Quantum Pigeonhole Principle
Physics
C. Patrignani et al. (Particle Data Group),
The Review of Particle Physics (2016) ,
Chin. Phys. C, 40 , 100001 (2016).
A. Sommerfeld,
"Zur Quantentheorie der Spektrallinien ."
Annalen der Physik 356 (17): 1-94. (1916)
Alberto Galindo and Pedro Pascual,
Quantum Mechanics I ,
Springer Texts and Monographs in Physics (1990);
ISBN: 978-3-642-83856-9 (Print), 978-3-642-83854-5 (Online).
Alberto Galindo and Pedro Pascual,
Quantum Mechanics II ,
Springer Texts and Monographs in Physics (1991);
ISBN: 978-3-642-84131-6 (Print), 978-3-642-84129-3 (Online).
Arno Bohm,
Quantum Mechanics: Foundations and Applications ,
Springer Texts and Monographs in Physics, Second Edition (1986);
ISBN: 978-3-540-13985-0 (Print), 978-3-662-01168-3 (Online).
Third Edition (1993);
ISBN 978-0-387-95330-4 .
Other materials