Schrödinger operators with symmetries

We consider the stationary Schrödinger operator H of a many-body system M with two-body rotation invariant interactions. The operator H is reduced with respect to the symmetries of permutation of identical particles, rotations and reflections, into a direct sum of operators H^τ̃, where τ̃ is an index of the irreducible representations of the symmetry group of the system.The spectra of the operators H^τ̃ were investigated in a series of papers of G.M. Zislin and A.G. Sigalov ([20], [21], [31]-[35]). In a recent paper [3] we have developed the spectral theory of these operators on the basis of the Weinberg equations.In the present work we complete and simplify this theory. In particular we treat in detail the case where the given system can be decomposed into two identical subsystems. For such systems there is a certain coupling between permutation and rotation-reflection symmetries, because a permutation, which interchanges the two subsystems, imposes a reflection on the relative position vector of the two centers of mass. This requires a modification of the theorem on essential spectrum as formulated in [3] in the case where such a division is not possible. The importance of this special case, as exemplified by diatomic molecules, fully justifies such a detailed treatment.This special case was treated by Zislin [34] under the assumption that the interactions are essentially multiplicative, relatively compact two-body interactions. Our method allows for general relatively compact two-body interactions, and can without difficulty be generalized to many-body interactions.Moreover, the method based on the Weinberg equation is suitable for a further analysis of the spectra of these operators.     

A study of even-parity states of the nuclei 19F, 21Ne and 23Na with the generator coordinate method and with mixing of projected Hartree-Fock determinants in comparison with complete diagonalization

The energies and electromagnetic properties of the even-parity states of the nuclei ¹⁹F, ²¹Ne and ²³Na are calculated with the generator coordinate method and with mixing of projected HartreeFock determinants, using the Kuo and the Preedom-Wildenthal two-body interactions. The two parameters β and ? of Nilsson's potential are chosen as generator coordinates and the subspace is enlarged with a few different configurations, i.e. one for A = 19 and three for A = 21 and 23 nuclei. Special care is taken in choosing the appropriate Hartree-Fock solution corresponding to possible occupied single-particle states. For both methods six basic functions are enough to obtain a good approximation to complete diagonalization within the same model space. The collective features of these nuclei are also investigated.     

Critical scattering and pionic instabilities in heavy ion collisions

We investigate the effect of collective instabilities in heavy ion collisions. The focus is on critical scattering phenomena associated with pionic instabilities. The decay rate Γ of excited many-body systems is calculated in RPA. Γ is shown to give the rate of spontaneous “phonon” pair production. We express Γ as a sum of a collective, Γ_col, and a scattering, Γ_scat, rate. Γ_col is the pair production rate of phonons in unstable states. In the case of pionic instabilities, Γ_col is the condensation rate of π⁺π^? and π⁰π⁰ pairs into unstable states. Γ_scat is the pair production rate of phonons in the particle-hole excitation region and gives the two-body scattering rate in the medium. An effective (density-dependent) two-body cross section is obtained. The difference between critical scattering of external particles in a system near equilibrium and that of constituents of systems far from equilibrium is investigated. A model calculation suggests the existence of pionic instabilities in heavy ion collisions. Growth rates of unstable modes and the effective cross sections displaying critical scattering are calculated. Finally, we estimate Γ_scat and Γ_col.     

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.     

Hartree-Fock calculations for finite nuclei with equivalent nonlocal potentials

The method of constructing equivalent regular two-body potentials by a unitary transformation of the two-body Hamiltonian has been generalized to spin-parity dependent nuclear potentials containing tensor- and spin-orbit terms. Starting from the Gammel-Christian-Thaler potential, which includes tensor forces, we obtained a class of equivalent regular, but nonlocal potentials depending on a parameterλ — the range of nonlocality. — These potentials have been used in a Hartree-Fock calculation for the closed-shell nuclei He⁴, C¹², O¹⁶, Si²⁸, S³², Ca⁴⁰. The calculated binding energies show a slowλ-variation with a minimum in the region of 0.7 f. The nuclear radii decrease with increasingλ and are in general too small. The sequence of single particle levels of the nuclei with closedl- shells is in agreement with that obtained with the usual nuclear shell model potential including spin-orbit coupling.     

Beyond the first order of the Hyperspherical Harmonic Expansion Method

In this paper we demonstrate the inadequacy of the first order of the Hyperspherical Harmonic Expansion Method, the L_m approximation, for the calculation of the binding energies, charge form factors and charge densities of doubly magic nuclei like ¹⁶O and ⁴⁰Ca. We then extend the Hyperspherical Expansion Method to many-fermion systems, consisting of an arbitrary number of fermions, and develop an exact formalism capable of generating the complete optimal subset of the hyperspherical harmonic basis functions. This optimal subset consists of those hyperspherical harmonic basis functions directly connected to the dominant first term in the expansion, the hyperspherical harmonic of minimal order L_m, through the total interaction between the particles. The required many-body coefficients are given using either the Gogny or Talmi-Moshinsky coefficients for the two-body operators. Using the two-body coefficients the weight function generating the orthogonal polynomials associated with the optimal subset is constructed.     

Coupled channel T-operator equations with connected kernels: (II). N coupled two-body channels

A time-independent theory of rearrangement collisions involving transitions between two-body states is presented. It is assumed that the system of interest consists of particles that may be partitioned into two-body systems in N ways, including interchanges of particle labels without changing the kind of channel. An infinite family of sets of N coupled T-operator equations is derived by use of the channel coupling array, as in previous work on the three-body problem. Specialization to the channel-permuting arrays guaranteeing connected (N?1)th iterates of the kernel of the coupled equations is made in the N-channel case (N > 3) and the nature of the solutions to the coupled equations is discussed. Various approximation schemes to be used with numerical calculations are suggested. Since the transition operators for all rearrangement channels are coupled together, no problems concerning non-orthogonality of the eigenstates of different channel Hamiltonians are encountered; also the presence of the outgoing wave boundary condition in all channels is made explicit. The close resemblance of the equations in matrix form to those of one-channel scattering is exploited by introducing Møller wave operators and associated channel scattering states, an optical potential formalism that leads to rearrangement channel optical potential operators, and a variational formulation of the coupled equations using a Schwinger-like variational principle. A brief comparison with other many-body formalisms is also given.     

U(3) → R(3) integrity-basis spectroscopy

The U(3) → R(3) algebra, widely used in nuclear spectroscopy studies, is revisited. The most general form of a U(3)-preserving interaction that is rotationally invariant and of given degree in the group generators is presented. Here the full purpose and beauty of the integrity-basis concept is realized. En route to the above it is shown that the structure of the U(3) → R(3) integrity basis can be deduced from a systematic counting of defining space matrix elements of p-shell, k-body scalar operators, k = 0, 1, 2?. The tensorial character of the so-called “missing label” operators and, more importantly, of the operators responsible for the splitting and inversion of rotational bands is obtained by relating integrity-basis multinomials through degree four in the U(3) generators to density operators of a standard U(3) → R(3) many-body spectroscopy. The results are used to show how K-band splitting as well as an [L(L + 1)]² factor in the energy can be realized within a single representation of SU(3) by two-body interactions of the ds and higher nuclear shells. Parameter sets of model interactions associated with both normal and inverted K-band structures are given, as well as the results of a “best fit” theory for the ground and gamma bands of ²⁴Mg.     

A nuclear fluiddynamical approximation with one-plus-two-body viscosity

A semiclassical fluiddynamical model based on an usual scaling approximation (SCA) is extended to investigate the role of one and two-body dissipation in the widths of nuclear collective modes. The competition between one and two-body viscosity in:i) the collisionless (elastic) limit;ii) the hydrodynamical case andiii) the general viscoelastic regime is examined over the whole range of nuclear collision time scales. Numerical solutions are investigated for the first magnetic 2^? twist mode in²⁰⁸Pb.     

Analytic expressions for energy centroids and widths in the microscopic collective model

Analytic formulae are given for the U(3) centroids of the collective Bohr-Mottelson potential in the microscopic collective model. In particular, formulae are reported for the centroids of the quadratic [Q · Q ～ β²] and cubic [Q · (Q × Q ～ β³cos 3γ] rotational scalars in the microscopic quadrupole operator. Favorable comparisons for ground-state intensities are achieved between shell-model diagonalizations and statistical predictions based upon the gaussian approximation to the energy density. These results suggest that statistical measures can be used reliably for truncation of the infinite-dimensional representation spaces of the microscopic symplectic collective theory.     

Quantum dynamics of atomic coherence in a spin-1 condensate: Mean-field versus many-body simulation

We analyse and numerically simulate the full many-body quantum dynamics of a spin-1 condensate in the single spatial mode approximation. Initially, the condensate is in a “ferromagnetic” state with all spins aligned along the y axis and the magnetic field pointing along the z axis. In the course of evolution the spinor condensate undergoes a characteristic change of symmetry, which in a real experiment could be a signature of spin-mixing many-body interactions. The results of our simulations are conveniently visualised within the picture of irreducible tensor operators.     

Dynamics of nuclear fluid: (I). Foundation

Starting with the time-dependent Hartree-Fock (TDHF) formulation of the many-body problem, we cast the equation into a set of conservation laws of classical type. Besides the equation of continuity, TDHF leads to an equation of motion which is analogous to the Euler equation in classical fluid dynamics. The forces do not come from the collective kinetic stress alone, but also from a density-dependent chemical potential, the surface tensional force which depends on density differences and the Coulomb interaction. With an assumed Navier-Stokes generalization of the stress tensor, such a set of differential equations provides a powerful tool for the study of complicated collective motions of nuclear systems such as those involved in heavy-ion reactions and nuclear fission. In the static case, the equation of motion leads to the Thomas-Fermi model of a finite nucleus as formulated by Bethe.     

Construction of a finite energy momentum tensor

A non-perturbative approach, postulating the existence of a family of Zimmermann normal products, certain linear relations among field operators, and the Wilson short distance expansion, is used to construct a finite energy momentum tensor. The dependence of the tensor on the field operators is made explicit by a suitable limit procedure. The calculations are performed in a scalar A⁴ model as an example. The results obtained are generalizations of the perturbation theory treatment of products of operators.     

Three-body forces and4He

Some two-pion exchange three-body forces are examined as effective two-body potentials in the framework of exp(S) many-body theory. Each effective potential is added to the Reid soft core potential and a fully self-consistent calculation carried out. The most reasonable three-body forces give remarkably good agreement with experiment.     

Reduction of two two-body operators in the effective hyperfine hamiltonian to one-body operators

The two two-body operators representing the effect of far configurations on the contact hyperfine interaction in annl ^N n′s configuration are shown to reduce to one-body operators. In such a configuration the operator for the contact hyperfine field is (a _nlS_nl +a_n,ss_n′s), where a_nl and a_n′s take the same values in all terms of the configuration. A discrepancy between expectation values of these two-body operators deduced from experiment and calculated values is explained.     

Interplay between two- and three-body interaction in light nuclei and nuclear matter

A recent study of different models of three-nucleon interaction (TNI) in ³He, ³H, ⁴He and nuclear matter is extended to study the influence of different choices of the accompanying two-body interaction. A new two-body potential, Argonne υ₁₄, is coupled with both the Tucson and isobar intermediate-state models of two-pion-exchange TNI, with a phenomenological intermediate-range repulsive TNI added to the latter. Variational calculations are carried out for these systems, and compared to the earlier work. We find that a stronger tensor component in the two-body potential, as typified by a larger deuteron D-state percentage, gives more attraction for the TNI, counteracting the saturation effect obtained when only two-body forces are considered.     

Shape variables and the shell model

The irreducible representation labelsλ andμ of the SU(3) shell model are related to the shape variablesβ andγ of the collective model by invoking a linear mapping between eigenvalues of invariant operators of the two theories. All but one parameter of the theory is fixed if the shell-model result is required to reproduce the collective-model geometry. And for one special value of the remaining free parameter there is a simple linear relationship between the eigenvalues, λ_α, of the quadrupole matrix of the collective model and the SU(3) representation labels: $$\lambda _1 = ( - \lambda + \mu )/3, \lambda _2 = ( - \lambda + 2\mu + 3)/3, \lambda _3 = (2\lambda + \mu + 3)/3.$$ The correspondence between hamiltonians that describe rotations in each theory is also given. Results are shown for two cases,²⁴Mg and¹⁶⁸Er, to demonstrate that the simplest mapping yields excellent results for both energies and transition rates. For λ and/or μ large, the (β, γ)?(λ,μ) correspondence introduced here reduces to the symplectic shell-model result.     

Boson-like quantum dynamics of association in ultracold Fermi gases

We study the collective association dynamics of a cold Fermi gas of 2N atoms in M atomic modes into a single molecular bosonic mode. When the atomic translational motion is slow compared to the atom-molecule conversion rate, the many-body fermionic problem for 2^M amplitudes is effectively reduced to a dynamical system of min{N, M} + 1 amplitudes, making the solution no more complex than the solution of a two-mode Bose-Einstein condensate and allowing realistic calculations with up to 10⁴ particles. The many-body dynamics is shown to be formally similar to the dynamics of the bosonic system under the mapping of boson particles to fermion holes, producing collective enhancement effects due to many-particle constructive interference.