Stability of a Model of Relativistic Quantum Electrodynamics

The relativistic “no pair” model of quantum electrodynamics uses the Dirac operator, D(A) for the electron dynamics together with the usual self-energy of the quantized ultraviolet cutoff electromagnetic field A– in the Coulomb gauge. There are no positrons because the electron wave functions are constrained to lie in the positive spectral subspace of some Dirac operator, D, but the model is defined for any number, N, of electrons, and hence describes a true many-body system. In addition to the electrons there are a number, K, of fixed nuclei with charges ≤Z. If the fields are not quantized but are classical, it was shown earlier that such a model is always unstable (the ground state energy E=−∞) if one uses the customary D(0) to define the electron space, but is stable (E > − const.(N+K)) if one uses D(A) itself (provided the fine structure constant α and Z are not too large). This result is extended to quantized fields here, and stability is proved for α= 1/137 and Z≤ 42. This formulation of QED is somewhat unusual because it means that the electron Hilbert space is inextricably linked to the photon Fock space. But such a linkage appears to better describe the real world of photons and electrons. Received: 8 September 2001 / Accepted: 18 March 2002     

Segregation in the Falicov--Kimball Model

The Falicov–Kimball model is a simple quantum lattice model that describes light and heavy electrons interacting with an on-site repulsion; alternatively, it is a model of itinerant electrons and fixed nuclei. It can be seen as a simplification of the Hubbard model; by neglecting the kinetic (hopping) energy of the spin up particles, one gets the Falicov–Kimball model. We show that away from half-filling, i.e. if the sum of the densities of both kinds of particles differs from 1, the particles segregate at zero temperature and for large enough repulsion. In the language of the Hubbard model, this means creating two regions with a positive and a negative magnetization. Our key mathematical results are lower and upper bounds for the sum of the lowest eigenvalues of the discrete Laplace operator in an arbitrary domain, with Dirichlet boundary conditions. The lower bound consists of a bulk term, independent of the shape of the domain, and of a term proportional to the boundary. Therefore, one lowers the kinetic energy of the itinerant particles by choosing a domain with a small boundary. For the Falicov- Kimball model, this corresponds to having a single “compact” domain that has no heavy particles. Received: 21 June 2001 / Accepted: 4 January 2002     

Lifetime study of high-spin states in 104,105In

In the neutron deficient isotopes _104,105In lifetimes of high spin states in the range from 0.5 ps to 600 ps have been measured in a coincidence recoil distance Doppler shift (RDDS) experiment. By combining the Doppler shift attenuation (DSA) and RDDS methods at few μm flight distances, the problem of delayed feeding has been avoided and very short lifetimes in the range 0.5–0.8 ps have been determined. Shell model calculations with strong restrictions for the neutron orbitals reveal good agreement with experimental level energies and still fair agreement for most measured B(M1) and B(E2) values. Received: 30 June 1998     

Lieb–Thirring Inequalities for Schrödinger Operators with Complex-valued Potentials

Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrödinger operator with a complex-valued potential.     

Valence bond ground states in isotropic quantum antiferromagnets

Haldane predicted that the isotropic quantum Heisenberg spin chain is in a massive phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exactSO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.Supported in part by N.S.F. Grant PHY-80-19754. Fellow of the A.P. Sloan Foundation and the Canadian Institute for Advanced ResearchN.S.F. Post-doctoral FellowSupported in part by N.S.F. Grant PHY-85-15288-A01     

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     

Approximate neutrality of large-Z ions

LetN(Z) denote the number of electrons which a nucleus of chargeZ can bind in non-relativistic quantum mechanics (assuming that electrons are fermions). We prove thatN(Z)/Z1 asZ.Research partially supported by the NSERC under Grant NA7901 and by the USNSF under Grants DMS-8416049 and PHY 85-15288-A01     

The Sialons MLn[Si4−xAlxOxN7−x] with M = Eu,Sr, Ba and Ln = Ho‐Yb – Twelve Substitution Variants with the MYb[Si4N7] Structure Type

The oxonitridoalumosilicates (so‐called sialons) MLn[Si_4?xAl_xO_xN_7?x] with M = Eu, Sr, Ba and Ln =Ho, Er, Tm, Yb were obtained by the reaction of the respective lanthanoid metal, the alkaline earth carbonates or europium carbonate, resp., AlN, “Si(NH)₂” and MCl₂ as a flux in a radiofrequency furnace at temperatures around 2100 °C. The compounds MLn[Si_4?xAl_xO_xN_7?x] are relevant for the investigation of substitutional effects on the materials properties due to their ability of tolerating a comparatively large phase width up to x ≈ 2.0(5). The crystal structures of the twelve compounds were refined from X‐ray single crystal data and X‐ray powder data and are found to be isotypic to the MYb[Si₄N₇] structure type. The compounds crystallize in space group P6₃mc (no. 186, hexagonal) and are made up of chains of so‐called starlike units [N^[4](SiN₃)₄] or [N^[4]((Si,Al)(O,N)₃)₄], respectively. These units are formed by four (Si,Al)(N/O)₄ tetrahedra sharing a common central nitrogen atom. The structure refinement was performed utilizing an O/N‐distribution model according to Paulings rules, i.e. nitrogen was positioned on the four‐fold bridging site and nitrogen and oxygen were distributed equally on both of the two‐fold bridging sites, resulting in charge neutrality of the compound. The Si and Al atoms were distributed equally on their two crystallographic sites, referring to their elemental proportion in the compound, due to being poorly distinguishable by X‐ray methods. The chemical compositions of the compounds were derived from electron probe micro analyses (EPMA).