Variational calculation of some S-states of Coulomb three-body systems

A generalized Hylleraas-type basis set with three nonlinear parameters is proposed to study three-body systems interacting via coulomb forces within the framework of non-relativistic quantum mechanics. This basis set improves the rate of convergence with respect to previous ones, specially for non-symmetric systems and excited states of two electron atoms. Accurate binding energies and other properties for S-states of helium-like ions, muonic molecules and the positronium negative ion are reported. Received 21 July 2000 and Received in final form 4 October 2000     

Hyperspherical Coulomb spheroidal basis in the Coulomb three-body problem

A hyperspherical Coulomb spheroidal (HSCS) representation is proposed for the Coulomb three-body problem. This is a new expansion in the set of well-known Coulomb spheroidal functions. The orthogonality of Coulomb spheroidal functions on a constant-hyperradius surface ρ = const rather than on a constant-internuclear-distance surface R = const, as in the traditional Born-Oppenheimer approach, is a distinguishing feature of the proposed approach. Owing to this, the HSCS representation proves to be consistent with the asymptotic conditions for the scattering problem at energies below the threshold for three-body breakup: only a finite number of radial functions do not vanish in the limit of ρ→∞, with the result that the formulation of the scattering problem becomes substantially simpler. In the proposed approach, the HSCS basis functions are considerably simpler than those in the well-known adiabatic hyperspherical representation, which is also consistent with the asymptotic conditions. Specifically, the HSCS basis functions are completely factorized. Therefore, there arise no problems associated with avoided crossings of adiabatic hyperspherical terms.     

On the three-body Coulomb scattering problem

We describe the smoothness properties and the asymptotic form of the Green's function (in configuration space) for three charged particles. We also discuss the integral equations and the boundary value problems for the Coulomb wavefunctions and we show that they form a complete set. Finally, we study the singularities of the Coulomb scattering operator, and we investigate the connection between the Dollard wave operators and the Coulomb wavefunctions.     

Integrable generalizations of oscillator and Coulomb systems via action-angle variables

Oscillator and Coulomb systems on N-dimensional spaces of constant curvature can be generalized by replacing their angular degrees of freedom with a compact integrable (N−1)-dimensional system. We present the action-angle formulation of such models in terms of the radial degree of freedom and the action-angle variables of the angular subsystem. As an example, we construct the spherical and pseudospherical generalization of the two-dimensional superintegrable models introduced by Tremblay, Turbiner and Winternitz and by Post and Winternitz. We demonstrate the superintegrability of these systems and give their hidden constant of motion.     

Calculation of the stability domain for Coulomb systems

A variational method is proposed for calculating the values of particle mass corresponding to the dissociation threshold of an atomic-molecular system. The calculations of the dissociation stability domain for three-particle Coulomb systems demonstrate the high effectiveness of the method as compared to the traditional variational principle for calculating energy.     

A solution of the Coulomb three-body problem

In this work, a final state wave function is constructed which represents a solution of the three-body Schr?dinger equation. The formulated wave function is superimposed of one basic analytical function with various parameters. The coefficients of these basic functions involved in final state wave function can be easily calculated from a set of linear equations. The coefficients depend only on incident energy of the system. The process can also be prolonged for application to the problems more than three bodies.     

How to relate the oscillator and Coulomb systems on spheres and pseudospheres?

Physics of Atomic Nuclei - We show that oscillators on a sphere and a pseudosphere are related, by the so-called Bohlin transformation, with Coulomb systems on a pseudosphere: even states of an...     

An analytic and parameter-free wavefunction for studying the stability of three-body systems

An analytic wavefunction is proposed for the ground state of general atomic three-body systems in which two light particles are negatively charged and the third (heavy) is positively charged. By construction the wavefunction (i) has the same analytical form for all systems; (ii) is parameter-free; (iii) is nodeless; (iv) satisfies all two-particle cusp conditions; and (v) yields reasonable ground state energies for several three-body systems, including the prediction of a bound state for H^??, D^??, T^?? and Mu^??. Simple polynomial fits are provided for certain important subcases, allowing for a rapid estimate of the ground state energy and of the stability of three-body systems.     

Parabolic sturmians approach to the three-body continuum Coulomb problem

The three-body continuum Coulomb problem is treated in terms of the generalized parabolic coordinates. Approximate solutions are expressed in the form of a Lippmann-Schwinger-type equation, where the Green’s function includes the leading term of the kinetic energy and the total potential energy, whereas the potential contains the non-orthogonal part of the kinetic energy operator. As a test of this approach, the integral equation for the (e ^?, e ^?, He⁺⁺) system has been solved numerically by using the parabolic Sturmian basis representation of the (approximate) potential. Convergence of the expansion coefficients of the solution has been obtained as the basis set used to describe the potential is enlarged.     

Periodic orbits and binary collisions in the classical three-body Coulomb problem

In the helium case of the classical three-body Coulomb problem in two dimensions with zero angular momentum, we develop a procedure to find periodic orbits applying two symbolic dynamics for one-dimensional and planar problems. Focusing our attention on binary collisions with these tools, a sequence of periodic orbits are predicted and are actually found numerically. A family of periodic orbits found has regularity in their actions. For this family of periodic orbits, it is shown that thanks to its regularity, a partial summation of the Gutzwiller trace formula with a daring approximation gives a Rydberg series of energy levels.     

On fermionic stability of Vlasov descriptions of finite Coulomb systems

We explore from a numerical point of view the validity of the Vlasov equation as a semi-classical approximation of time-dependent Hartree-Fock and time-dependent LDA theories, in terms of the survival of the Pauli principle. The fermionic properties are investigated by using a Na₉⁺ cluster as test case and solving the Vlasov equation with the test particle method. This allows to derive a time span for which the Vlasov equation provides an acceptable approximation.     

Modelling of a two-dimensional Coulomb friction oscillator

Xia proposes a model for investigating the stick-slip motion caused by dry friction of a two-dimensional oscillator under arbitrary excitations. Instead of the harmonic balance method used by most investigators, a numerical approach to investigate the system is provided. The concept of the friction direction angle is introduced to determine the components of the static and kinetic friction force vector and the hyperbolic secant function is introduced to deal with the transition of the friction force from the static to the kinetic state. The friction direction angle is determined by either relative velocities or input forces. With this method the switch conditions for stick state, slip state and stick-slip state can be easily derived. The orbits of the responses, which are either straight line segments, circular or elliptic are obtained. In the general case, the orbit of the response is a complex planar curve. Zero-stop, one-stop, two-stops and more than two-stops per cycle are also found.     

Solution of three-dimensional Faddeev equations for three-body Coulomb bound states

A new method of directly solving the three-dimensional Faddeev equations in the total-angular-momentum representation for the pure Coulomb bound-state problem is developed. The method is based on the tri-quintic Hermite spline expansion of the Faddeev components. The ground states of thee ^– e ^– e ⁺ system and thepp ^– mesic molecule are calculated.     

Classical dynamics near the triple collision in a three-body Coulomb problem

We investigate the classical motion of three charged particles with both attractive and repulsive interactions. The triple collision is a main source of chaos in such three-body Coulomb problems. By employing the McGehee scaling technique, we analyze here for the first time in detail the three-body dynamics near the triple collision in 3 degrees of freedom. We reveal surprisingly simple dynamical patterns in large parts of the chaotic phase space. The underlying degree of order in the form of approximate Markov partitions may help in understanding the global structures observed in quantum spectra of two-electron atoms.     

On account of Coulomb excitations of a target for the three-body break-up

The three-body break-up wavefunction asymptotics is reformulated to take into account only leading terms of the infinite number of Coulomb excitations of the target. The infinite tail of the Coulomb excitation contribution is summed up by the exact representation for the projection onto the subspace of Coulomb bound states.     

Reduced adiabatic hyperspherical basis in the Coulomb three-body bound state problem

A new version of the adiabatic hyperspherical approach (AHSA) is suggested which has significant advantages for the calculation of three-body states with total angular momentumJ> 0. The binding energies of all bound states of mesic molecules with normal parity are calculated by the suggested method. Comparison with results of variational calculations and the fast convergence of the method confirm its high efficiency.     

Faddeev approach to the three-body problem in total-angular-momentum representation

For a system of three charged particles the Faddeev equations are derived in the total-angular-momentum representation. They have the form of coupled sets of partial differential equations in three-dimensional space and can be used to develop new efficient numerical procedures to tackle the three-body Coulomb problem. The asymptotic conditions at large distances corresponding both to binary scattering and bound-state problems are presented. The behaviour of the Faddeev components near the triple and double collision points is studied.