Systems with separable many-particle interactions. I

In this paper we give an exact evaluation of the free energy per particle for systems with separable many-particle interactions described by a hamiltonian of the type ? = ∑_kT(k) + NP (N^-1 ∑_kV(k)), where P is an arbitrary polynomial. In the proof use is made of a fundamental theorem due to Bogoliubov Jr. for ferromagnetic quadratic operators. The free energy can be obtained from a trial hamiltonian, which is linear in the operators T and V.     

Convex-envelope formulation for separable many-particle interactions

The exact result for the free energy per particle in systems described by a hamiltonian of the type NP(V_N), where NV_N denotes a set of short-range operators, is reformulated in terms of a convex-envelope construction. A comparison is given with results obtained for classical systems with interactions of the so-called Kac-type.     

Asymptotic completeness for long-range many-particle systems with Stark effect. II

We prove the existence and the asymptotic completeness of the Dollard-type modified wave operators for many-particle Stark Hamiltonians with long-range potentials.     

On the free energy in systems with separable interactions. III

In this paper we extend the calculation of the free energy in systems with separable interactions of the ferromagnetic type as given in paper I of this series to a more general class of systems characterized by a hamiltonian, in which the one-particle operator Σ_kT(k) is replaced by a more general operator T which may contain interactions as well. This extension enables us to simplify a great deal the derivation given in paper II of the lower bound to the free energy for systems with both ferromagnetic and antiferromagnetic interactions. We also discuss in some detail under which conditions the upper and the lower bounds can be equal. (The conditions include in particular the hamiltonian treated in paper II.)     

Statistical properties of many-particle spectra. II. Two-point correlations and fluctuations

The two-point correlation function for complex spectra described by the Gaussian Orthogonal Ensemble (GOE) is calculated, and its essential simplicity displayed, by an elementary procedure which derives from orthogonal invariance and the dominance of intrinsic binary correlations. The resultant function is used for an approximate calculation of the standard fluctuation measures. Good agreements are found with exact results where these are available, this incidentally demonstrating that the measures are, for the most part, two-point measures. It is shown that they vary slowly over the spectrum, a result which is in agreement both with experiment and with Monte Carlo calculations. The same technique can be used for higher-order correlation functions, and possibly also for more complicated ensembles in which case the results would be relevant to the question why GOE fluctuations give a good account of experimental results.     

Evidence of many-particle interactions in two-dimensional charge-stabilized colloidal crystals

Electrostatic interaction in two-dimensional colloidal crystals obeying the nonlinear Poisson-Boltzmann equation is studied numerically. The force constants which are the coefficients of the energy quadratic form of the crystal are obtained. Significant discrepancy between the results of computer experiments and predictions of the harmonic crystal theory based on the idea of pairwise interaction has been detected. Two simple parameters are introduced for uniform quantitative estimation of many-particle effects and validity of the approximation of nearest-neighbor interaction. It is shown that the contribution of many-particle interaction into the total electrostatic interaction in colloidal crystals is significant for a broad range of crystal lattice constants and particles' sizes.     

Integral equations for four identical particles with separable two-particle interactions

The problem of four identical spinless particles with separable two-particle interactions is considered. The integral equations of motion for four particles in the bound states and in the continuum states with two non-ihteracting subsystems in the initial channel are formulated. The angular-momentum reduction of the obtained equations is given.     

A remark on different norms and analyticity for many-particle interactions

We compare a recent result of Dobrushin and Martirosyan with previous results by Gallavotti and Miracle-Sole and by Israel and point out that the analytic behavior at high temperatures for many-particle interactions is different depending on whether the interactions are weighted with a lattice-gas or Ising norm or, on the other hand, with the supremum norm.     

Pauli-correct separable potentials for composite particle interactions

Experimental totalβ-decay energies for short-lived isotopes of gallium and germanium are presented. The sources were produced as mass-separated fission products at the OSIRIS on-line separator at Studsvik. By applyingβ-γ-coincidence methodsQ _α-values were determined for^80,81Ga and^79,81,82Ge. For these nuclei on or close to the shellN =50 and in the vicinity of ther-process “seed”, the atomic mass excess is derived and compared with predictions from mass formulae.     

Solutions of the density matrix equation with separable interactions in Liouvillespace

The masterequation for the density matrix is solved with new techniques in Liouvillespace for two examples: 1. The stationary equation for the Lasersystem, for which the resolvent of the Liouvillean ?₀ is explicitly known, is solved by a variational procedure and 2. the exact solution of a general quantum system with simple coupling to reservoirs can be obtained, as the interaction ?₁ is separable in this case.     

Local potentials phase equivalent to separable two-nucleon interactions

Local energy-dependent potentials have been constructed phase equivalent to members of a family of phase-equivalent separable two-nucleon potentials in the¹ S ₀-state. It has been shown that these potentials obey the known off-shell constraints in the¹ S ₀-state and can therefore not be regarded as unrealistic in this sense. They have the same shape as the energy-independent local Kermode potentials. However we also find that the off-shell behaviour of a separable¹ S ₀ potential and its local equivalent can differ considerably.     

An algorithm for separable representations of realistic interactions

A general method for expanding in a separable form a local interaction is developed. The algorithm makes use of the Gamow vectors and momenta, calculated from an eigenvalue problem, and preserves the analytic properties of the original S-matrix. Applications are presented for realistic nucleon-nucleon interactions and phenomenological optical potentials. The agreement between the separable and exact scattering matrices is quite good on and off the energy shell, for a wide range of energies.     

Fast computation of many-particle hydrodynamic and electrostatic interactions in a confined geometry

An O(N) method is presented for calculation of hydrodynamic or electrostatic interactions between N point particles in a confined geometry. This approach splits point forces or sources into a local contribution for which rapidly decaying free-space analytical solutions to the Stokes or Poisson equations are used, and a global contribution whose effect is determined numerically using a fast iterative method. The scheme is applied to Brownian dynamics simulations of flowing confined polymer solutions, and the effects of concentration on hydrodynamically induced migration phenomena are illustrated.     

Effective theory of NN interactions in a separable representation

We consider the effective field theory of the NN system in a separable representation. The pionic part of the effective potential is included nonperturbatively and approximated by a separable potential. The use of a separable representation allows for the explicit solution of the Lippmann-Schwinger equation and a consistent renormalization procedure. The phase shifts in the ¹ S ₀ channel are calculated to subleading order.     

A separable potential approach to nucleon-nucleus scattering with exact treatment of Coulomb interactions

A suitable, physically reasonable choice of separable nucleon-nucleus potential is made, which leads to very simple analytic expressions for the form factors in the Coulomb-state representation. This allows one to take account of Coulomb effects in an exact and simple way. The model is applied to the analysis of the elastic scattering of nucleons by ⁴He, ¹²C and ¹⁶O targets. A good fit of the experimental phase shifts is obtained with almost the same values of the potential parameters for neutron-nucleus and proton-nucleus elastic scattering.     

Studies of many-particle correlations in proton-nucleus interactions using distributions of rapidity-gaps between particles

Many-particle correlations in proton-Emulsion interactions at 400 GeV have been investigated using distributions of rapidity-gaps between produced particles. We have defined the normalized semi-inclusive rapidity-gap correlation function which is shown to have advantages over the normalized two-particle correlation function. Small, but significant, deviations from zero-correlation are observed in the data.     

Exact statements about many-particle systems with higher symmetries

A new method of calculating the energy spectrum of a system of A identical Fermi particles with translationally invariant interaction is developed under the assumption that there exists a high symmetry in the 3A-dimensional space of particle coordinates. For a special class of symmetries the many-body problem is split exactly into two sets of equations: one containing only totally symmetric combinations of the particle coordinates which are called “collective variables” and the other equation taking essentially into account the requirements of the Pauli principle and connected symmetry properties. In several cases it is possible to obtain the excitation spectra exactly showing qualitatively new features. They depend on “many-particle quantum numbers” varying independently of each other in an interval which sometimes depends on A. For special high symmetries the collective variables obey equations which are very similar to one-particle equations providing a new explanation of the “Independent-Particle Model” for arbitrary strength and form of the interaction potential. A manifold of unknown up to now excitation spectra of many-particle systems is obtained and discussed.     

Simple separable expansions for calculating matrix elements of two-body local interactions with harmonic oscillator functions

A simple method is proposed to calculate the matrix elements of two-body local interactions using a harmonic oscillator basis (HOF). Using the properties of HOF, it is shown that any local potential can be replaced by a simple series for the purpose of calculating matrix elements. This series can be reduced to a finite sum when evaluating a matrix element. Its terms are separable functions of the coordinates of the two particles; hence the advantage of the method. In the present article we treat the most important components of the two-body interaction, namely central, two-body spin-orbit, and tensor forces. As a representation we have chosen spherical harmonic oscillator functions expressed with spherical coordinates. This technique appears to be very well adapted to and efficient for Hartree-Fock calculations in any representation of the HOF. A very interesting feature of this formulation is that it can be easily extended to calculations employing generalized HOF as defined by Wong.