This is a very short presentation regarding developments in the theory of nuclear many-body problems, as seen and experienced by the author during the past 60 years with particular emphasis on the contributions of Gerry Brown and his research-group.
Much of his work was based on Brueckners formulation of the nuclear many-body problem. It is reviewed briefly together with the Moszkowski-Scott separation method that was an important part of his early work. The core-polarisation and his work related to effective interactions in general are also addressed.
Linear response functions are calculated for symmetric nuclear matter of normal density by time-evolving two-time Greens functions with conserving self-energy insertions, thereby satisfying the energy-sum rule. Nucleons are regarded as moving in a me
an field defined by an effective mass. A two-body effective (or residual) interaction, represented by a gaussian local interaction, is used to find the effect of correlations in a second order as well as a ring approximation. The response function S(e,q) is calculated for 0.2<q<1.2 fm^{-1}. Comparison is made with the nucleons being un-correlated, RPA+HF only.
Using Gaussian integral transform techniques borrowed from functional-integral field theory and the replica trick we derive a version of the coherent-potential approximation (CPA) suited for describing ($i$) the diffusive (hopping) motion of classica
l particles in a random environment and ($ii$) the vibrational properties of materials with spatially fluctuating elastic coefficients in topologically disordered materials. The effective medium in the present version of the CPA is not a lattice but a homogeneous and isotropic medium, representing an amorphous material on a mesoscopic scale. The transition from a frequency-independent to a frequency-dependent diffusivity (conductivity) is shown to correspond to the boson peak in the vibrational model. The anomalous regimes above the crossover are governed by a complex, frequency-dependent self energy. The boson peak is shown to be stronger for non-Gaussian disorder than for Gaussian disorder. We demonstrate that the low-frequency non-analyticity of the off-lattice version of the CPA leads to the correct long-time tails of the velocity autocorrelation function in the hopping problem and to low-frequency Rayleigh scattering in the wave problem. Furthermore we show that the present version of the CPA is capable to treat the percolative aspects of hopping transport adequately.
The Busch-formula relates the energy-spectrum of two point-like particles interacting in a 3-D isotropic Harmonic Oscillator trap to the free scattering phase-shifts of the particles. This formula is used to find an expression for the it shift rm in
the spectrum from the unperturbed (non-interacting) spectrum rather than the spectrum itself. This shift is shown to be approximately $Delta=-delta(k)/pitimes dE$, where $dE$ is the spacing between unperturbed energy levels. The resulting difference from the Busch-formula is typically 1/2% except for the lowest energy-state and small scattering length when it is 3%. It goes to zero when the scattering length $rightarrow pm infty$. The energy shift $Delta$ is familiar from a relatedproblem, that of two particles in a spherical infinite square-well trap of radius $R$ in the limit $Rrightarrow infty$. The approximation ishowever as large as 30% for finite values of $R$, a situation quite different from the Harmonic Oscillator case. The square-well results for $Rrightarrow infty$ led to the use ofin-medium (effective) interactions in nuclear matter calculations that were $propto Delta$ and known as the it phase shift approximation rm.Our results indicate that the validity of this approximation depends on the trapitself, a problem already discussed by DeWitt more than 50 years ago for acubical vs spherical trap.
In scattering theory, the unitary limit is defined by an infinite scattering-length and a zero effective range, corresponding to a phase-shift pi/2, independent of energy. This condition is satisfied by a rank-1 separable potential V(k,k)=-v(k)v(k) w
ith v^{2}(k)=(4pi)^{2}(Lambda^{2}-k^{2})^{-1/2}, Lambda being the cut-off in momentum space.Previous calculations using a Pauli-corrected ladder summation to calculate the energy of a zero temperature many body system of spin 1/2 fermions with this interaction gave xi=0.24 (in units of kinetic energy) independent of density and with Lambda---->infinity. This value of xi is appreciably smaller than the experimental and that obtained from other calculations, most notably from Monte Carlo, which in principle would be the most reliable. Our previous work did however also show a strong dependence on effective range r_0 (with r_0=0 at unitarity). With an increase to r_0=1.0 the energy varied from xi~0.38 at k_f=0.6 1/fm to ~0.45 at k_f=1.8 1/fm which is somewhat closer to the Monte-Carlo results. These previous calculations are here extended by including the effect of the previously neglected mean-field propagation, the dispersion correction. This is repulsive and found to increase drastically with decreasing effective range. It is large enough to suggest a revised value of xi~0.4 <--> ~0.5 independent of r_0. Off-shell effects are also investigated by introducing a rank-2 (phase-shift equivalent) separable potential. Effects of 10% or more in energy could be demonstrated for r_0>0. It is pointed out that a computational cut-off in momentum-space brings in another scale in the in otherwise scale-less unitary problem.
Time-dependent electron transport through a quantum dot and double quantum dot systems in the presence of polychromatic external periodic quantum dot energy-level modulations is studied within the time evolution operator method for a tight-binding Ha
miltonian. Analytical relations for the dc-current flowing through the system and the charge accumulated on a quantum dot are obtained for the zero-temperature limit. It is shown that in the presence of periodic perturbations the sideband peaks of the transmission are related to combination frequencies of the applied modulations. For a double quantum dot system under the influence of polychromatic perturbations the quantum pump effect is studied in the absence of source-drain and static bias voltages. In the presence of spatial symmetry the charge is pumped through the system due to broken generalized parity symmetry.
We investigate the quantum ratchet effect under the influence of weak dissipation which we treat within a Floquet-Markov master equation approach. A ratchet current emerges when all relevant symmetries are violated. Using time-reversal symmetric driv
ing we predict a purely dissipation-induced quantum ratchet current. This directed quantum transport results from bath-induced superpositions of non-transporting Floquet states.
Two strongly coupled quantum dots are theoretically and experimentally investigated. In the conductance measurements of a GaAs based low-dimensional system additional features to the Coulomb blockade have been detected at low temperatures. These regi
ons of finite conductivity are compared with theoretical investigations of a strongly coupled quantum dot system and good agreement of the theoretical and the experimental results has been found.
