TheN 7/5 law for charged bosons

Non-relativistic bosons interacting with Coulomb forces are unstable, as Dyson showed 20 years ago, in the sense that the ground state energy satisfiesE ₀–AN ^7/5. We prove that 7/5 is the correct power by proving thatE ₀–BN ^7/5. For the non-relativistic bosonic, one-component jellium problem, Foldy and Girardeau showed thatE ₀–CN^1/4. This 1/4 law is also validated here by showing thatE ₀–DN^1/4. These bounds prove that the Bogoliubov type paired wave function correctly predicts the order of magnitude of the energy.Work partially supported by U.S. National Science Foundation grant DMS 8600748Work partially supported by U.S. National Science Foundation grant PHY85-15288-A01Work supported by Alfred Sloan Foundation dissertation fellowship     

Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions

The ground state energy of an atom of nuclear chargeZe in a magnetic fieldB is exactly evaluated to leading order asZ in the following three regions:BZ ^4/3,BZ ^4/3 andZ ^4/3BZ³. In each case this is accomplished by a modified Thomas-Fermi (TF) type theory. We also analyze these TF theories in detail, one of their consequences being the nonintuitive fact that atoms are spherical (to leading order) despite the leading order change in energy due to theB field. This paper complements and completes our earlier analysis [1], which was primarily devoted to the regionsBZ ³ andBZ³ in which a semiclassical TF analysis is numerically and conceptually wrong. There are two main mathematical results in this paper, needed for the proof of the exactitude of the TF theories. One is a generalization of the Lieb-Thirring inequality for sums of eigenvalues to include magnetic fields. The second is a semiclassical asymptotic formula for sums of eigenvalues that isuniform in the fieldB.Work partially supported by U.S. National Science Foundation grant PHY90-19433 A02Work partially supported by U.S. National Science Foundation grant DMS 92-03829Work partially supported by the Heraeus Stiftung and the Research Fund of the University of Iceland.     

The stability and instability of relativistic matter

We consider the quantum mechanical many-body problem of electrons and fixed nuclei interacting via Coulomb forces, but with a relativistic form for the kinetic energy, namelyp ²/2m is replaced by (p ² c ²+m ² c ⁴)^1/2–mc ². The electrons are allowed to haveq spin states (q=2 in nature). For one electron and one nucleus instability occurs ifz>2/, wherez is the nuclear charge and is the fine structure constant. We prove that stability occurs in the many-body case ifz2/ and <1/(47q). For smallz, a better bound on is also given. In the other direction we show that there is a critical _c (no greater than 128/15) such that if > _c then instability always occurs forall positivez (not necessarily integral) when the number of nuclei is large enough. Several other results of a technical nature are also given such as localization estimates and bounds for the relativistic kinetic energy.Work partially supported by U.S. National Science Foundation grant PHY-85-15288-A02The author thanks the Institute for Advanced Study for its hospitality and the U.S. National Science Foundation for support under grant DMS-8601978     

A proof of part of Haldane's conjecture on spin chains

It has been argued that the spectra of infinite length, translation and U(1) invariant, anisotropic, antiferromagnetic spin s chains differ according to whether s is integral or 1/2 integral: There is a range of parameters for which there is a unique ground state with a gap above it in the integral case, but no such range exists for the 1/2 integral case. We prove the above statement for 1/2 integral spin. We also prove that for all s, finite length chains have a unique ground state for a wide range of parameters. The argument was extended to SU(n) chains, and we prove analogous results in that case as well.Work partially supported by U.S. National Science Foundation grant PHY80-19754 and by the A.P. Sloan Foundation.Work partially supported by U.S. National Science Foundation grant PHY-85-15288.     

Phase transitions in anisotropic lattice spin systems

A general method for proving the existence of phase transitions is presented and applied to six nearest neighbor models, both classical and quantum mechanical, on the two dimensional square lattice. Included are some two dimensional Heisenberg models. All models are anisotropic in the sense that the groundstate is only finitely degenerate. Using our method which combines a Peierls argument with reflection positivity, i.e. chessboard estimates, and the principle of exponential localization we show that five of them have long range order at sufficiently low temperature. A possible exception is the quantum mechanical, anisotropic Heisenberg ferromagnet for which reflection positivity isnot proved, but for which the rest of the proof is valid.Work partially supported by U.S. National Science Foundation grant no. MPS 75-11864Work partially supported by U.S. National Science Foundation grant no. MCS 75-21684 A01     

Infinite clusters in percolation models

The qualitative nature of infinite clusters in percolation models is investigated. The results, which apply to both independent and correlated percolation in any dimension, concern the number and density of infinite clusters, the size of their external surface, the value of their (total) surface-to-volume ratio, and the fluctuations in their density. In particular it is shown thatN ₀, the number of distinct infinite clusters, is either 0, 1, or and the caseN ₀= (which might occur in sufficiently high dimension) is analyzed.Alfred P. Sloan Research Fellow, Research supported in part by National Science Foundation grant No. MCS 77-20683 and by the U.S.-Israel Binational Science Foundation.Research supported in part by the U.S.Israel Binational Science Foundation.     

Lower bounds for resonance widths in potential and obstacle scattering

Explicit lower bounds are given for the size of the imaginary parts of resonances for Schrödinger operators with non-trapping or trapping potentials, and for the Dirichlet Laplacian in the exterior of a star-shaped obstacle, both acting in three dimensions.Work partially supported by DiUC/FONDECYT (Chile)Work partially supported by U.S. National Science Foundation grant DMS 8705610     

A Note on Polarization Vectors in Quantum Electrodynamics

A photon of momentum k can have only two polarization states, not three. Equivalently, one can say that the magnetic vector potential A must be divergence-free in the Coulomb gauge. These facts are normally taken into account in QED by introducing two polarization vectors (k) with {1,2}, which are orthogonal to the wave-vector k. These vectors must be very discontinuous functions of k and, consequently, their Fourier transforms have bad decay properties. Since these vectors have no physical significance there must be a way to eliminate them and their bad decay properties from the theory. We propose such a way here.Dedicated to Freeman Dyson on the occasion of his eightieth birthdayWork partially supported by U.S. National Science Foundation grant PHY 01-39984.Work partially supported by U.S. National Science Foundation grant DMS 03-00349. 2003 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.Acknowledgement We thank Herbert Spohn and Jakob Yngvason for many useful discussions about this work. After completing this work and submitting it to CMP it was brought to our attention that the last section, 10.3, of the paper [2] by Fröhlich, Griesemer and Schlein contains the same idea in the context of Rayleigh scattering in the dipole approximation. The three-component concept enables them to extend the results in the rest of their paper from scalar fields to vector fields, but, as we see here, the concept works in much greater generality.     

Asymptotics for Bosonic atoms

It was proved by Benguria and Lieb that for an atom where the electrons do not satisfy the exclusion principle, the critical electron number N _c, i.e., the maximal number of electrons the atom can bind, satisfies lim inf_zN_c/Z 1 + , where Z is the nuclear charge. Here is a positive constant derived from the Hartree model. We complete this result by proving that the correct asymptotics for N _c(Z) is indeed _zN_c/Z = 1 + .This work was done while the author was a graduate student at Princeton University supported by a Danish Research Academy fellowship and U.S. National Science Foundation grant PHY-85-15288-A03.     

Long range atomic potentials in Thomas-Fermi theory

We prove that the interaction among neutral atoms in Thomas-Fermi theory behaves, for large separationl, likel ^–7. The constant is independent of the atomic nuclear charges, but does depend on the relative positions of the nuclei. We also show that is not a simple sum of pair terms, i.e. in TF theory three and higher body terms persist into the asymptotic (inl) region.Work partially supported by U.S. National Science Foundation grant MCS 75 21684 A02     

Proof of an entropy conjecture of Wehrl

Wehrl has proposed a new definition of classical entropy,S, in terms of coherent states and conjectured thatS1. A proof of this is given. We discuss the analogous problem for Bloch coherent spin states, but in this case the conjecture is still open. An inequality for the entropy of convolutions is also given.Work partially supported by US National Science Foundation grant MCS 75-21684 A02     

Ground State Energy of the Two-Component Charged Bose Gas

We continue the study of the two-component charged Bose gas initiated by Dyson in 1967. He showed that the ground state energy for N particles is at least as negative as –CN^7/5 for large N and this power law was verified by a lower bound found by Conlon, Lieb and Yau in 1988. Dyson conjectured that the exact constant C was given by a mean-field minimization problem that used, as input, Foldys calculation (using Bogolubovs 1947 formalism) for the one-component gas. Earlier we showed that Foldys calculation is exact insofar as a lower bound of his form was obtained. In this paper we do the same thing for Dysons conjecture. The two-component case is considerably more difficult because the gas is very non-homogeneous in its ground state.Dedicated to Freeman J. Dyson on the occasion of his 80th birthday©2003 by the authors. This article may be reproduced in its entirety for non-commercial purposes.Work partially supported by NSF grant DMS-0111298, by EU grant HPRN-CT-2002-00277, by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation, and by grants from the Danish research council.Work partially supported by U.S. National Science Foundation grant PHY01 39984-A01.     

Rayleigh Scattering at Atoms with Dynamical Nuclei

Scattering of photons at an atom with a dynamical nucleus is studied on the subspace of states of the system with a total energy below the threshold for ionization of the atom (Rayleigh scattering). The kinematics of the electron and the nucleus is chosen to be non-relativistic, and their spins are neglected. In a simplified model of a hydrogen atom or a one-electron ion interacting with the quantized radiation field in which the helicity of photons is neglected and the interactions between photons and the electron and nucleus are turned off at very high photon energies and at photon energies below an arbitrarily small, but fixed energy (infrared cutoff), asymptotic completeness of Rayleigh scattering is established rigorously. On the way towards proving this result, it is shown that, after coupling the electron and the nucleus to the photons, the atom still has a stable ground state, provided its center of mass velocity is smaller than the velocity of light; but its excited states are turned into resonances. The proof of asymptotic completeness then follows from extensions of a positive commutator method and of propagation estimates for the atom and the photons developed in previous papers. The methods developed in this paper can be extended to more realistic models. It is, however, not known, at present, how to remove the infrared cutoff. Activities supported, in part, by a grant from the Swiss National Foundation. Work partially supported by U.S. National Science Foundation grant DMS 01-00160. Supported by a NSF postdoctoral fellowship.     

Toward a microscopic basis for the de Gennes narrowing

It is pointed out that recent work on the high-frequency behavior of the neutron scattering function S(k, ) in liquids indicates that at fixedgw there is a dip at a value of k very close to the maximum of S(k) (first diffraction peak). This dip is due to the configurational average over pairs of particles and scales as the inverse particle radius.Work supported in part by a grant from the National Science Foundation (No. GH 34890).     

Quantum coherent operators: A generalization of coherent states

We introduce a technique to compare different, but related, quantum systems, thereby generalizing the way that coherent states are used to compare quantum systems to classical systems in semiclassical analysis. We then use this technique to estimate the dependence of the free energy of the quantum Heisenberg model on the spin value, and to estimate the relation between the ferromagnetic and antiferromagnetic free energies.Work supported in part by the U.S. National Science Foundation grant PHY-9019433.Work supported in part by the U.S. National Science Foundation grant DMS-9002416.     

The comoving coordinate frame for differentially rotating,relativistic bodies

A first-order, ordinary differential equation is derived that can in principle be integrated for any assumed rotation law = (r) within a relativistic, perfect-fluid body. The solution of this equation reduces the number of unknown metric functions appearing in the line element for the spacetime of such differentially rotating bodies to three. In the case of rigid rotation this is equivalent to a result of Harrison.Work supported in part by the U.S. National Science Foundation under grant No. MPS 74-15524.     

The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics

Starting with a relativistic Schrödinger Hamiltonian for neutral gravitating particles, we prove that as the particle numberN and the gravitation constantG0 we obtain the well known semiclassical theory for the ground state of stars. For fermions, the correct limit is to fixGN ^2/3 and the Chandrasekhar formula is obtained. For bosons the correct limit is to fixGN and a Hartree type equation is obtained. In the fermion case we also prove that the semiclassical equation has a unique solution — a fact which had not been established previously.Dedicated to Walter Thirring on his 60^th birthdayWork partially supported by U.S. National Science Foundation grant PHY 85-15288-A01Work supported by Alfred Sloan Foundation dissertation Fellowship     

High-density percolation: Exact solution on a Bethe lattice

A new percolation problem is posed where the sites on a lattice are randomly occupied but where only those occupied sites with at least a given numberm of occupied neighbors are included in the clusters. This problem, which has applications in magnetic and other systems, is solved exactly on a Bethe lattice. The classical percolation critical exponents=gg=1 are found. The percolation thresholds vary between the ordinary percolation thresholdp _c (m=1)=l/(z – 1) andp _c(m=z) =[l/(z – 1)]¹/(z–1). The cluster size distribution asymptotically decays exponentially withn, for largen, p p _c .Supported in part by National Science Foundation grant DMR78-10813.     

Local Ward identities and the decay of correlations in ferromagnets

Using local Ward identities we prove a number of correlation inequalities forN-component, isotropically coupled, pair interacting ferromagnets; some for allN2 and some forN=2, 3, 4. These are used to prove a mass gap above the mean field temperature, for allN2. ForN=2, 3, 4 we prove an upper bound on a critical exponent, and a lower bound on the susceptability which diverges asm0.Research partially supported by US National Science Foundation under Grant PHY-7825390 A01.Research partially supported by US National Science Foundation under Grant MCS-78-01885.     

Hydrodynamical limit for a Hamiltonian system with weak noise

Starting from a general hamiltonian system with superstable pairwise potential, we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles. We prolve that, in the scaling limit, the time conserved quantities, energy, momenta and density, satisfy the Euler equation of conservation laws up to a fixed timet provided that the Euler equation has a smooth solution with a given initial data up to timet. The strength of the noise term is chosen to be very small (but nonvanishing) so that it disappears in the scaling limit.Research partially supported by U.S. National Science Foundation grants DMS 89001682, DMS 920-1222 and a grant from ARO, DAAL03-92-G-0317Research partially supported by U.S. National Science Foundation grants DMS-9101196, DMS-9100383, and PHY-9019433-A01, Sloan Foundation Fellowship and David and Lucile Packard Foundation Fellowship