Weakly Bound States in Light Hypernuclei

From the viewpoint of critical stability, we discuss the three- and four-body structure of ⁶He, ⁷He, ⁴He, and ⁴H. With the + + N three-body model, ⁶He is found to have a three-layer structure of the matter distribution: core, a skin and neutron halo. Also the level structure of ⁷He with the three-body model of ⁵He + n + n is predicted. This stimulates a new study of neutron-rich and proton-rich hypernuclei. By performing a four-body calculation with both NNN and NNN channels and with both NN and NNN channels, we show that the N-N and -N couplings are very important in critical stability of few-body hypernuclear systems.     

Lattice dynamics of sapphire (Al2O3)

Using inelastic neutron scattering we have determined all the dispersion branches of the ₁, ₂ and ₃ representations along the three-fold axis as well as the 2 times 15 branches of ₁ and ₂ symmetry along the -A-direction plus some branches along the -D-direction. The experimental data are analyzed using various rigid ion, polarizable ion and shell models. The shell models give a very satisfactory account of the dispersion curves as well as the scattering intensities. Special attention is given to the investigation of dielectric constants and equilibrium conditions.     

On the asymptotics of occurrence times of rare events for stochastic spin systems

We consider translation-invariant attractive spin systems. LetT _,x ^v be the first time that the average spin inside the hypercube reaches the valuex when the process is started from an invariant measure with density smaller thanx. We obtain sufficient conditions for (1) ¦¦^–1 logT _,x ^v (x) in distribution as ¦¦ , and ¦¦^–1 logT _,x ^v (x) as ¦¦ , where (x):= –lim ¦¦^–1 log {(average spin inside ) x. And (2)T _,x ^v /ET _,x ^v converges to a unit mean exponential random variable as ¦¦ . Both (1) and (2) are proven under some type of rapid convergence to equilibrium. (1) is also proven without extra conditions for Ising models with ferromagnetic pair interactions evolving according to an attractive reversible dynamics; in this case is a thermodynamic function. We discuss also the case of finite systems with boundary conditions and what can be said about the state of the system at the timeT _,x ^v .On leave from São Paulo University.     

On the equivalence of boundary conditions

We show that ifb andb are two boundary conditions (b.c.) for general spin systems on ^d such that the difference in the energies of a spin configuration in ^d is uniformly bounded, |H _,b ()–H _,b()|C < , then any infinite-volume Gibbs states and obtained with these b.c. have the same measure-zero sets. This implies that the decompositions of and into extremal Gibbs states are equivalent (mutually absolutely continuous). In particular, if is extremal,=. Application of this observation yields in an easy way (among other things) (a) the uniqueness of the Gibbs states for one-dimensional systems with forces that are not too long-range; (b) the fact that various b.c. that are natural candidates for producing non-translation-invariant Gibbs states cannot lead to such an extremal Gibbs state in two dimensions.Supported in part by NSF Grant PHY 78–15920 and by the Swiss National Foundation For Scientific Research.     

Particle polarizations in decays of5HeΛ and4HΛhypernuclei

Nucleon polarizations are examined in the hypernucleus decays⁵He, ⁴He + p + _–,⁴H, ³H + p + _–, and⁴H, ³He + n + _–. The effect of the strong interaction in the final state is taken into account. A significant role is predicted for the resonance interaction in the p-⁴He system in the formation of the proton polarization in the⁵He decay. It is also shown that in the⁴H, ³H + p + _– decay the proton-polarization components should attain significant magnitudes, while in the charge-exchange decay⁴H, ³He + n + _–, the neutron polarization is expected to be not too large.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 100–103, October, 1988.     

Nontrivial extensions of a representation of the Poincaré group with mass and helicity zero by its tensorial product

Let U(a, ) be a representation of the Poincaré group with mass and helicity zero, realized in the space of C -functions with compact support on ³, without the origin. Let U ⁽²⁾(a, ) denote the tensorial product of U(a, ) by itself. We explicitly determine the cocycles of extension of U(a, ) by U ⁽²⁾(a, ) and we prove that the nontrivial cohomology is indexed by (u(),),u()D ¹ _]0,3\,.     

Large Deviations in Rarefied Quantum Gases

The probability of observing a large deviation (LD) in the number of particles in a region in a dilute quantum gas contained in a much larger region V is shown to decay as exp[–||F], where || is the volume of and F is the change in the appropriate free energy density, the same as in classical systems. However, in contrast with the classical case, where this formula holds at all temperatures and chemical potentials our proof is restricted to rarefied gases, both for the typical and observed density, at least for Bose or Fermi systems. The case of Boltzmann statistics with a bounded repulsive potential can be treated at all temperatures and densities. Fermions on a lattice in any dimension, or in the continuum in one dimension, can be treated at all densities and temperatures if the interaction is small enough (depending on density and temperature), provided one assumes periodic boundary conditions.     

Meyer's concept of quasicrystal and quasiregular sets

This paper relates two mathematical concepts of long-range order of a set of atoms , each of which is based on restrictions on the set of interatomic distances –. A set in ⁿ is aMeyer set if is a Delone set and there is a finite setF such that . Meyer proposed that such sets include the possible structures of quasicrystals. He obtained a structure theory for such sets, which reformulates results that he obtained in harmonic analysis around 1970, and which relates these sets to cut-and-project sets. In geometric crystallography V.I. Galiulin introduced the concept ofquasiregular set, which is a set such that both and – are Delone sets. This paper shows that these two concepts are identical.     

Diffractive dissociation reaction np → Λ0K0p for incident neutron momenta between 30 and 70 GeV/c BIS-2 collaboration

We present results of the exclusive diffractive reaction np ⁰K⁰p at 0·08 < |t|< 0·53 GeV²/c² and momenta between 30–70 GeV/c at the Serpukhov neutron beam. Thet-dependence is exp (Bt) withB= 11·4± 2·3 (GeV/c)^–2 and it exhibits the diffractive minimum at ¦t¦ 0·28 GeV²/c² for ⁰K⁰ massesM < 1·8 GeV/c².B decreases with increasingM. Correlation with the beam direction is observed in the rest frame of the ⁰K⁰ system: the cross section rises for the forward-going ⁰. The rise is more pronounced for higherM. The cross section of reaction np ⁰K⁰p for 0<|t|< 0·53 GeV²/c² is 10·3 ± 2·5 ± 2·8 b.     

Debye screening for two-dimensional Coulomb systems at high temperatures

The grand canonical ensemble of a two-dimensional Coulomb system with±1 charges is proved to have screening phenomena in its high-temperature region. The Coulomb potential in a finite region is assumed to be (–)^–1, where is the Laplacian with zero boundary conditions on. The hard-core condition is not assumed. The model is set up by separating (–)^–1 into a shortrange part and a long-range part depending on a parameter. The self-energies are subtracted only for the short-range part and therefore a choice of is a choice of subtraction of self-energies. The method of proof is in general the same as that of Brydges-Federbush Debye screening, except that here a modification for the short-range part of the potentials is needed.     

Double hypernuclei and the Λ-Λ interaction

The binding energy of the double hypernucleus Be¹⁰ analyzed using an -particle model. A relationship is found between B, the binding energy of two -particlcs, and ¯V, the - potential integral. For the experimental value B = 17.5 Me V ¯V varies from 250 to 850 MeV-F³ according to the form of the - potential used.In conclusion, I wish to thank S. N. Kryuchkova for carrying out these calculations on a Minsk-2 electronic computer. I also wish to thank my senior colleague, V. A. Filimonov, for his support in the completion of the present work.     

Local fluctuation of the spectrum of a multidimensional Anderson tight binding model

We consider the Anderson tight binding modelH=–+V acting inl ²(Z ^d ) and its restrictionH to finite hypercubes Z ^d . HereV={V _x ;xZ ^d } is a random potential consisting of independent identically distributed random variables. Let {E _j ()} _j be the eigenvalues ofH , and let _j (,E)=||(E _j ()–E),j1, be its rescaled eigenvalues. Then assuming that the exponential decay of the fractional moment of the Green function holds for complex energies nearE and that the density of statesn(E) exists atE, we shall prove that the random sequence { _j (,E)} _j , considered as a point process onR ¹, converges weakly to the stationary Poisson point process with intensity measuren(E)dx as gets large, thus extending the result of Molchanov proved for a one-dimensional continuum random Schrödinger operator. On the other hand, the exponential decay of the fractional moment of the Green function was established recently by Aizenman, Molchanov and Graf as a technical lemma for proving Anderson localization at large disorder or at extreme energy. Thus our result in this paper can be summarized as follows: near the energyE where Anderson localization is expected, there is no correlation between eigenvalues ofH if is large.     

Inflation, Extra Dimensions, and ΩΛ ≈ 1

An (N + 1)-dimensional quantum mechanical model for the origin of the universeresults in a 58e-fold inflation and a cosmological constant/vacuum energy densitywith 1.     

Charge Fluctuations in the Two-Dimensional One-Component Plasma

We study, via computer simulations, the fluctuations in the net electric charge in a two-dimensional, one component plasma (OCP) with uniform background charge density –e in a region inside a much larger overall neutral system. Setting e=1, this is the same as the fluctuations in N , the number of mobile particles of charge e. As expected, the distribution of N has, for large , a Gaussian form with a variance which grows only as ^||, where || is the length of the perimeter of . The properties of this system depend only on the coupling parameter =kT, which is the same as the reciprocal temperature in our units. Our simulations show that when the coupling parameter increases, ^() decreases to an asymptotic value ^()^(2)/2 which is equal (or very close) to that obtained for the corresponding variance of particles on a rigid triangular lattice. Thus, for large , the characteristic length _L=2^/ associated with charge fluctuations behaves very differently from that of the Debye length, _D1/ , which it approaches as 0. The pair correlation function of the OCP is also studied.     

Superstability estimates for anharmonic systems

We consider an anharmonic crystal described by variablesS _x ,x ^d ,S _x , with one-body interaction ¦S _x ¦ and nearest neighbor (n.n.) two body interaction ¦S _x –S _y ¦. We prove that, for ^d bounded, , where is the correlation function for the free boundary condition Gibbs state in ,>0 and are suitable constants independent of and . This generalizes previous results obtained in the case.Research partially supported by Consiglio Nazionale delle Ricerche.     

Critical behavior of self-avoiding walks on fractals

Using a new graph counting technique suitable for self-similar fractals, exact 18th-order series expansions for SAWs on some Sierpinski carpets are generated. From them, the critical fugacityx _c and critical exponents _SAW and _SAW are obtained. The results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied. We find for some carpets with low lacunarity that _SAW<0.75, thus violating the relation _SAW(fractal) > _SAW (d) for SAWs on the fractals which are embedded in ad-dimensional Euclidean space.     

Second order large deviation estimates for ferromagnetic systems in the phase coexistence region

We consider thed-dimensional Ising model with ferromagnetic nearest neighbor interaction at inverse temperature . Let be the magnetization inside ad-dimensional hyper cube , ₊ be the+Gibbs state andm*() be the spontaneous magnetization. For such thatm*()>0 we find a sufficient condition (easily verified to hold for large ) for ₊({M [a,b]}) to decay exponentially with ||^(d–1)/d when –m*<b<m*, –1a<b. Ford=2 this sufficient condition is the exponential decay of a connectivity function. We also prove a partial converse to this result, obtain a sharper result for the magnetization ond–1 dimensional cross sections of the model and prove a similar result ford=2, –m*<a<b<m*, and large, when free boundary conditions are chosen outside .Work partly supported by the U.S. Army Research Office     

Hypernuclear spectroscopy

Conclusions The main difference between the- nucleus and the nucleon-nucleus interaction is in the spin dependence. The spin-spin and spin-orbit interaction is an order of magnitude weaker for the-nucleus than for the nucleon-nucleus system. If we want to emphasize the difference between hypernuclei and nuclei in an oversimplifying manner, we may say that the particle in the nucleus behaves like a spinless neutron.This radical difference in the behaviour of particles and nucleons in nuclear matter presents a sensitive test for the models of nuclear matter. Hypernuclear spectroscopy if studied in detail, a program just begun, promises to give valuable information on the effective properties of the quasiparticles in the nucleus and their interactions, both basic ingredients of nuclear models.Rather surprisingly, hypernuclei live long enough to observe individual states, as can be deduced from the berylium results. These results should be verified as soon as possible. If they are confirmed, this will challenge experimentalists to determine the -nucleus interaction in the same way as done for particles and the theorists to explain why the strong decay of particles in the nucleus is hindered.Invited talk at the symposium Mesons and Light Nuclei, Liblice, Czechoslovakia, June 1981.     

Dimensions of invariant sets of expanding maps

We consider a compact invariant set of an expanding map of a manifoldM and give upper and lowerbounds for the Hausdorff Dimension dim _H (), and box dimensionsdim _B () and dim _B (). These bounds are given in terms of the topological entropy, topological pressure, and uniform Lyapunov exponents of the map.A measure-theoretic version of these results is also included.Part of this work was done when I was in the Department of Mathematics, University of Arizona.     

Interaction theory: Relativistic hydrogen atom and the Lamb shift

The Dirac relativistic equation for the hydrogen atom as augmented by dual interaction terms is solved exactly and the eigenvalues for the bound states are determined. These areE _n =[1 + ²/ (n + s)²]^–1/2, withs ²=k ²– ²(1±)², where is a constant which is a measure of the strength of the dual interaction relative to the standard interaction ^µ A _µ . It is shown that the ratios of the experimental values for the Lamb shifts of various energy levels in hydrogen and singly ionized helium are correctly given by the theory. The origin of the anomalous magnetic moment and, in fact, the operator for the total magnetic moment is given.