A unitary correlation operator method

The short range repulsion between nucleons is treated by a unitary correlation operator which shifts the nucleons away from each other whenever their uncorrelated positions are within the repulsive core. By formulating the correlation as a transformation of the relative distance between particle pairs, general analytic expressions for the correlated wave functions and correlated operators are given. The decomposition of correlated operators into irreducible n-body operators is discussed. The one- and two-body-irreducible parts are worked out explicitly and the contribution of three-body correlations is estimated to check convergence. Ground state energies of nuclei up to mass number A = 48 are calculated with a spin-isospin-dependent potential and single Slater determinants as uncorrelated states. They show that the deduced energy- and mass-number-independent correlated two-body Hamiltonian reproduces all “exact” many-body calculations surprisingly well.     

“Exact” shell model calculations and linked nuclear effective interactions

It is shown that so-called “exact” shell model calculations of spectra (diagonalizations) contain important unlinked terms which are due to space truncation. These terms render less interesting a comparison between the “exact” calculation of spectra and model space calculations with effective interactions containing linked terms only. The spectral calculations were performed in mass 6. The lowest order unlinked terms which arise in the “exact” calculation were calculated in mass 6 and mass 18.     

On the solution of a “2D Coulomb + Aharonov-Bohm” problem: oscillator strengths in the discrete spectrum and scattering

In this paper we present an exact analytic solution of the Schrödinger equation both inthe discrete and continuous spectra for the combination of a 2D Coulomb potential and theAharonov-Bohm flux. We analyze the influence of the Aharonov-Bohm flux on the energyspectrum of such a system and show that its presence leads to the broadening of theelectron density in the bound states with the given value of the principal quantum number.We have shown that the scattering phase shift, which determines theS-matrix, can be represented as a sum of the Aharonov-Bohm scatteringphase, first obtained by Henneberger, and a “modified” 2D Coulomb phase. We have noticed,that the Aharonov-Bohm scattering phase has a full analogy with the “quantum defect” forsuch a system. We have shown also, that the presence of the Aharonov-Bohm flux affects theradiation spectrum of the electron in this case, and this fact is demonstrated bycalculations of the corresponding oscillator strengths. The explicit analytic expressionfor the scattering cross section on such a system is found in the frame of the eikonalapproach. Obtained formula contains the two exact limiting cases, namely, the “pure” 2DCoulomb scattering as well as the “pure” Aharonov-Bohm effect. The mutual influence of a2D Coulomb potential and the Aharonov-Bohm flux is also discussed.     

Nonequilibrium statistical mechanics in the general theory of relativity I. A general formalism

This is the first in a series of papers, the overall objective of which is the formulation of a new covariant approach to nonequilibrium statistical mechanics in classical general relativity. The object here is the development of a tractable theory for self-gravitating systems. It is argued that the “state” of an N-particle system may be characterized by an N-particle distribution function, defined in an 8N-dimensional phase space, which satisfies a collection of N conservation equations. by mapping the true physics onto a fictitious “background” spacetime, which may be chosen to satisfy some “average” field equations, one then obtains a useful covariant notion of “evolution” in response to a fluctuating “gravitational force.” For many cases of practical interest, one may suppose (i) that these fluctuating forces satisfy linear field equations and (ii) that they may be modeled by a direct interaction. In this case, one can use a relativistic projection operator formalism to derive exact closed equations for the evolution of such objects as an appropriately defined reduced one-particle distribution function. By capturing, in a natural way, the notion of a dilute gas, or impulse, approximation, one is then led to a comparatively simple equation for the one-particle distribution. If, furthermore, one treats the effects of the fluctuating forces as “localized” in space and time, one obtains a tractable kinetic equation which reduces, in the newtonian limit, to the standard Landau equation.     

Non-equilibrium statistical mechanics in the general theory of relativity: II. Linear fields in a kinetic approximation

The first paper in this series introduced a new, manifestly covariant approach to non-equilibrium statistical mechanics in classical general relativity. The object of this second paper is to apply that formalism to the evolution of a collection of particles that interact via linear fields in a fixed curved background spacetime. Given the viewpoint adopted here, the fundamental objects of the theory are a many-particle distribution function, which lives in a many-particle phase space, and a many-particle conservation equation which this distribution satisfies. By viewing a composite N-particle system as interacting one- and (N ? 1)-particle subsystems, one can derive exact coupled equations for appropriately defined reduced one- and (N ? 1)-particle distribution functions. Alternatively, by treating all the particles on an identical footing, one can extract an exact closed equation involving only the one-particle distribution. The implementation of plausible assumptions, which constitute straightforward generalizations of standard non-relativistic “kinetic approximations”, then permits the formulation of an approximate kinetic equation for the one-particle distribution function. In the obvious non-relativistic limit, one recovers the well-known Vlasov-Landau equation. The explicit form for the relativistic expression is obtained for three concrete examples, namely, interactions via an electromagnetic field, a massive scalar field, and a symmetric second rank tensor field. For a large class of interactions, of which these three examples are representative, the kinetic equation will admit a relativistic Maxwellian distribution as an exact stationary solution; and, for these interactions, an H-theorem may be proved.     

Single-particle properties in an exactly solvable A-body system

A recent theorem states that for quantum many-body systems with short-range interactions the following property holds: the single-particle overlap functions, spectroscopic factors and separation energies of bound eigenstates of the (A ? 1)-particle system are fully determined by the one-body density matrix of the A-particle system in its ground state. We confirm this property, by explicit construction, for the case of a schematic, exactly solvable system.     

Exchange and correlation energy of an inhomogeneous electron gas

A convenient expression is derived for the coefficient, B_xc(n), which determines the first gradient corrections to the exchange and correlation energy of an inhomogeneous electron gas. The result is exact to all orders in $\text{e}$² and is expressed in terms of the single particle progagator. A method of approximation, which is exact at high density, is given for the explicit evaluation of B_xc. Numerical results are given for B_xc in the metallic density range.     

Precocious scaling and nonperturbative effects in QCD

The onset of nonperturbative effects in QCD is studied in two ways: (1) by means of the nonvanishing vacuum expectation value of 〈g ² F ²〉 introduced by Shifman, Vainshtein and Zakharov; (2) using a finite energy sum rule (FESR) for the renormalization group function β. Our considerations are based upon the recently proposed “physical” β_MMOM, which is gauge invariant and shows explicit mass-decoupling.     

Multiplicative and additive cluster expansions for the evolution of quantum spin systems in the ground state

It is shown that for the v-dimensional quantum Ising model in the high temperature region e^?tH in the GNS representation admits a “multiplicative” N-particle cluster expansion and H admits an “additive” N-particle cluster expansion.     

A semi-empirical correlation energy for nuclei

A semi-empirical interaction is used to calculate higher order corrections to the binding energies of even—even nuclei close to the line of stability. These corrections are taken to come from two phonon configurations and are treated as a perturbation with respect to the BCS nuclear ground state which is obtained from applying the energy density method to finite nuclei. The overall correspondence between theory and experiment for the 60 nuclei calculated between A =52 and A =234 is good, with excellent agreement for the non-deformed nuclei situated within the regions A = 72 to 144 and A = 200 to 212. The large correction enegies (several MeV per nucleus on the average) indicate that these correlations are of importance for explaining nuclear binding energies and that it is necessary to include them within energy functional itself. The fact that these correlations come almost exclusively from nucleons close to the fermi surface is also discussed.     

Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function

A general theory of monodromy preserving deformation is developed for a system of linear ordinary differential equations $\text{d}\text{Y}\text{d}\text{x}\text{=A(x)Y}$, where A(x) is a rational matrix. The non-linear deformation equations are derived and their complete integrability is proved. An explicit formula is found for a 1-form ω, expressed rationally in terms of the coefficients of A(x), that has the property dω=0 for each solution of the deformation equations. Examples corresponding to the “soliton” and “rational” solutions are discussed.     

Hydrodynamic parameters and correlation functions of superfluid 3He

In this paper we present the first of two closely related studies devoted to the connection between the phenomenological hydrodynamics and microscopic theories of superfluid ³He. In this first part, we express in a systematical way all the phenomenological parameters appearing in the hydrodynamic equations in terms of microscopically well defined correlation functions. The method used for this purpose goes back to the work of Kadanoff and Martin on normal fluids and has recently been formulated in the framework of Mori's projector formalism by Forster. Apart from the assumptions about the structure of the superfluid phases of ³He and certain regularity assumptions about the collision-dominated part of the correlation functions, which have to be satisfied if the hydrodynamic limit exists at all, but which remain unproven, this part of our work is nearly exact. The only approximation made is the perturbative treatment of the very small magnetic dipole energy. The k → O limit of various correlation functions is considered. Novel consequences of the directional long range order for the transverse momentum density correlation function of the A phase, which are unique to that phase, are obtained. The instantaneous reactive parameters of the system are expressed as equal time commutators and evaluated rigorously. Some reactive parameters, e.g. the one leading to “orbit waves” are shown to have a collisional part, which has not yet been evaluated in a microscopic theory. NMR is considered as the k → O limit of spin wave resonances with an energy gap due to the magnetic dipole-dipole interaction at k = 0. The NMR linewidth is thereby fixed by the same transport parameter, which determines the spin wave damping in the theory where the gap is neglected. The relevant transport parameter is expressed along with all others by the rigorous Kubo formulae of the theory in which the magnetic dipole energy is neglected. The close analogy of the NMR linewidth parameter to transport parameters like the shear viscosity, which are measured in quite different experiments, is thereby elucidated. Further, approximate evaluations of the Kubo relations of these parameters, within a common microscopic approach will be given in a second related study.     

The Tc-SSR correlation in A3B(A15-type) compounds

The 28A₃B(A15-type) compounds, for which both the phase equilibrium diagram and the superconducting critical temperature, T_c, are known, can be divided into two groups. The first group consists of compounds whose SSR (solid solution range) is either zero or extends only to the A-rich side of the stoichiometric composition. The second group consists of the compounds whose SSR extends to both sides or only to the B-rich side. The first group of compounds generally has high T_c's and follow the T_c-SSR correlation proposed earlier.Both the grouping and the T_c-SSR correlation are directly relatable to “A-chain integrity” and thus lend further support to the thesis that “A-chain integrity” is of fundamental importance in the superconductivity of these compounds.     

Model three-body problem in the molecular mass limit

The bound states of a three-body molecule composed of two identical heavy nuclei and a light “electron” interacting through short-range s-wave potentials are studied. The spectrum of three-body bound states grows as the mass ratio m between the heavy and light particles increases, and presents a remarkable vibration rotation structure that can be fitted with the usual empirical energy formulas of molecular spectroscopy. The results of the exact three-body calculation for the binding energy and bound-state wavefunction are compared with the predictions of the Born-Oppenheimer method for the same system. We find that for m > 30, the Born-Oppenheimer approximation yields very good results for both the binding energies and wavefunctions. For smaller m (1 <m < 30) the Born-Oppenheimer results are still surprisingly good and this is shown to be related to the range of the two-body interactions.     

Strength functions for fragmented doorway states

Coupling a strongly excited “doorway state” to weak “hallway states” distributes its strength into micro-resonances seen in differential cross sections taken with very good energy resolution. The distribution of strength is shown to be revealed by reduced widths of the K-matrix rather than by the imaginary part of poles of the S-matrix. Different strength functions (SF) constructed by averaging the K-matrix widths are then investigated to determine their dependences on energy and on parameters related to averages of microscopic matrix elements. A new sum rule on the integrated strength of these SF is derived and used to show that different averaging procedures actually distribute the strength differently. Finally, it is shown that the discontinuous summed strength defines spreading parameters for the doorway state only in strong coupling, where it approximates the indefinite integral of the continuous SF of MacDonald-Mekjian-Kerman-De Toledo Piza. A new method of “parametric continuation” is used to relate a discontinuous sliding box-average, or a finite sum, of discrete terms to a continuous function.     

Adiabatic similarity and dependence of the energy of molecular systems on the masses of particles

The dependence of the energy of three-particle molecules on their masses is examined. It is shown that such molecules with the same values of the ratio of the reduced masses for motion in a “fast” and “slow” Jacobi coordinates have the property of adiabatic similarity: In the adiabatic approximation, their energies are proportional to the reduced masses. This allows information on the energy of molecules symmetric in the masses of particles to be extended to asymmetric molecules adiabatically similar to the symmetric molecules. For molecules with arbitrary masses of the particles, an analytic expression for the adiabatic energy and a formula approximating the exact energy are constructed using the principle of adiabatic similarity. Along with the adiabatic energy, which is the lower bound of the exact energy, a simple procedure is considered for determining the upper bound of the energy of asymmetric molecules from the energy of their symmetric counterparts. Based on these results, values of the lower and upper energy bounds are calculated and an approximation of the exact energy is obtained for 43 three-particle molecular systems.     

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the ‘cubication’ of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this Letter predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used.