44 research outputs found
Integral representation of one dimensional three particle scattering for delta function interactions
The Schr\"{o}dinger equation, in hyperspherical coordinates, is solved in
closed form for a system of three particles on a line, interacting via pair
delta functions. This is for the case of equal masses and potential strengths.
The interactions are replaced by appropriate boundary conditions. This leads
then to requiring the solution of a free-particle Schr\"{o}dinger equation
subject to these boundary conditions. A generalized Kontorovich - Lebedev
transformation is used to write this solution as an integral involving a
product of Bessel functions and pseudo-Sturmian functions. The coefficient of
the product is obtained from a three-term recurrence relation, derived from the
boundary condition. The contours of the Kontorovich-Lebedev representation are
fixed by the asymptotic conditions. The scattering matrix is then derived from
the exact solution of the recurrence relation. The wavefunctions that are
obtained are shown to be equivalent to those derived by McGuire. The method can
clearly be applied to a larger number of particles and hopefully might be
useful for unequal masses and potentials.Comment: 18 pages, 2 figures, to be published in J. Math. Phy
Renormalisation-group analysis of repulsive three-body systems
A coordinate space approach, based on that used by Efimov, is applied to
three-body systems with contact interactions between pairs of particles. In
systems with nonzero orbital angular momentum or with asymmetric spatial wave
functions, the hyperradial equation contains a repulsive 1/r^2 potential. The
resulting wave functions are used in a renormalisation group analysis. This
confirms Griesshammer's power counting for short-range three-body forces in
these systems. The only exceptions are ones like the 4S channel for three
nucleons, where any derivatives needed in the interaction are found to be
already counted by the scaling with the cut-off.Comment: 5 pages, RevTe
The parabolic Sturmian-function basis representation of the six-dimensional Coulomb Green's function
The square integrable basis set representation of the resolvent of the
asymptotic three-body Coulomb wave operator in parabolic coordinates is
obtained. The resulting six-dimensional Green's function matrix is expressed as
a convolution integral over separation constants.Comment: 14 pages, 2 figure
How To Classify 3-Body Forces -- And Why
For systems with only short-range forces and shallow 2-body bound states, the
typical strength of any 3-body force in all partial-waves, including external
currents, is systematically estimated by renormalisation-group arguments in the
Effective Field Theory of Point-Like Interactions. The underlying principle and
some consequences in particular in Nuclear Physics are discussed.Comment: 7 pages LaTeX2e using FBSart-class (provided); 2 figures in 3 .eps
files included using graphicx; to appear in Few-Body System
Sturmian bases for two-electron systems in hyperspherical coordinates
We give a detailed account of an spectral approach
for the calculation of energy spectra of two active electron atoms in a system
of hyperspherical coordinates. In this system of coordinates, the Hamiltonian
has the same structure as the one of atomic hydrogen with the Coulomb potential
expressed in terms of a hyperradius and the nuclear charge replaced by an angle
dependent effective charge. The simplest spectral approach consists in
expanding the hyperangular wave function in a basis of hyperspherical
harmonics. This expansion however, is known to be very slowly converging.
Instead, we introduce new hyperangular sturmian functions. These functions do
not have an analytical expression but they treat the first term of the
multipole expansion of the electron-electron interaction potential, namely the
radial electron correlation, exactly. The properties of these new functions are
discussed in detail. For the basis functions of the hyperradius, several
choices are possible. In the present case, we use Coulomb sturmian functions of
half integer angular momentum. We show that, in the case of H, the accuracy
of the energy and the width of the resonance states obtained through a single
diagonalization of the Hamiltonian, is comparable to the values given by
state-of-the-art methods while using a much smaller basis set. In addition, we
show that precise values of the electric-dipole oscillator strengths for
transitions in helium are obtained thereby confirming the
accuracy of the bound state wave functions generated with the present method.Comment: 28 pages, 4 figure
Non-homogeneous solutions of a Coulomb Schrödinger equation as basis set for scattering problems
We introduce and study two-body Quasi Sturmian functions which are proposed as basis functions for applications in three-body scattering problems. They are solutions of a two-body non-homogeneous Schrödinger equation. We present different analytic expressions including asymptotic behaviors for the pure Coulomb potential with a driven term involving either Slater-type or Laguerre-type orbitals. The efficiency of Quasi Sturmian functions as basis set is numerically illustrated through a two-body scattering problem.Fil: del Punta, Jessica Adriana. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Ambrosio, Marcelo José. Consejo Nacional de Investigaciónes Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; ArgentinaFil: Gasaneo, Gustavo. Universidad Nacional del Sur. Departamento de Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Zaytsev, S.. Pacific National University; RusiaFil: Ancarani, L. U.. Universite de Lorraine; Franci
On Parity-Violating Three-Nucleon Interactions and the Predictive Power of Few-Nucleon EFT at Very Low Energies
We address the typical strengths of hadronic parity-violating three-nucleon
interactions in "pion-less" Effective Field Theory in the nucleon-deuteron
(iso-doublet) system. By analysing the superficial degree of divergence of loop
diagrams, we conclude that no such interactions are needed at leading order.
The only two linearly independent parity-violating three-nucleon structures
with one derivative mix two-S and two-P-half waves with iso-spin transitions
Delta I = 0 or 1. Due to their structure, they cannot absorb any divergence
ostensibly appearing at next-to-leading order. This observation is based on the
approximate realisation of Wigner's combined SU(4) spin-isospin symmetry in the
two-nucleon system, even when effective-range corrections are included.
Parity-violating three-nucleon interactions thus only appear beyond
next-to-leading order. This guarantees renormalisability of the theory to that
order without introducing new, unknown coupling constants and allows the direct
extraction of parity-violating two-nucleon interactions from three-nucleon
experiments.Comment: 20 pages LaTeX2e, including 9 figures as .eps file embedded with
includegraphicx. Minor modifications and stylistic corrections. Version
accepted for publication in Eur. Phys. J.
Low-Energy Universality in Atomic and Nuclear Physics
An effective field theory developed for systems interacting through
short-range interactions can be applied to systems of cold atoms with a large
scattering length and to nucleons at low energies. It is therefore the ideal
tool to analyze the universal properties associated with the Efimov effect in
three- and four-body systems. In this "progress report", we will discuss recent
results obtained within this framework and report on progress regarding the
inclusion of higher order corrections associated with the finite range of the
underlying interaction.Comment: Commissioned article for Few-Body Systems, 47 pp, 16 fig