Solution of the Faddeev equations for Coulombic systems

Many important processes in muon catalyzed fusion cycle involve the reaction of three charged particles. Solving the Faddeev equations is one of the most direct approaches to the scattering of these charged particles. Difficulties encountered previously in solving such problems have been resolved. We report here a number of our initial calculations for bound as well as low energy scattering states.     

A new form of three-body Faddeev equations in the continuum

We propose a novel approach to solve the three-nucleon (3N) Faddeev equation which avoids the complicated singularity pattern going with the moving logarithmic singularities of the standard approach. In this new approach the treatment of the 3N Faddeev equation becomes essentially as simple as the treatment of the two-body Lippmann-Schwinger equation. Very good agreement of the new and old approaches in the application to nucleon-deuteron elastic scattering and the breakup reaction is found.     

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.     

Algorithm for solving the Faddeev equations in the three-body continuum under avoidance of moving logarithmic singularities

A new method is presented for solving the Faddeev equations in the three-body continuum, which avoids the moving logarithmic singularities present in momentum space methods used up to now. The new algorithm leads to a simple structure of the Faddeev integral kernel, what simplifies significantly the numerical realization. Its application in nuclear physics is, however, still plagued by the presence of the virtual-state pole in the nucleon-nucleon¹S₀ channel. Omitting that channel in calculations with the Bonn-B potential we demonstrate excellent agreement between three-nucleon observables obtained with the new and a former method. Since the codes are quite different, this can be considered as a convincing test.     

Configuration-space Faddeev calculations for bound states of three identical particles using the tensor method

The three-body problem is solved at negative energies using the Faddeev-Noyes equations. The latter are reduced to a matrix equation by spline approximation and orthogonal collocation. This matrix equation is solved using a method that is based on the tensor structure of the matrices. High-accuracy results obtained with this method for systems of three identical particles are presented, for various values of the total angular momentum and for various potential strengths and shapes.Dedicated to Profs. Erich Schmid and Ivo laus on the occasion of their 60th birthdays     

Calculation of bound states and resonances in perturbed Coulomb models

We calculate accurate bound states and resonances of two interesting perturbed Coulomb models by means of the Riccati-Padé method. This approach is based on a rational approximation to a modified logarithmic derivative of the eigenfunction and produces sequences of roots of Hankel determinants that converge rapidly towards the eigenvalues of the equation.     

Kinetic equations for a quantum gas with bound states

The kinetic theory for dense gases is modified to take into account the existence of bound states. A molecular chaos condition is used which corresponds to the division of the two- and three-particle Hubert spaces into scattering and boundstate subspaces. A kinetic stage results from a long-time limit which converges to yield time-independent functionals for the two- and three-particle density matrices, as functionals of the density matrices for atoms and molecules. Coupled kinetic equations are obtained which describe the gas as a reacting mixture of atoms and diatomic molecules. These include the effects of scattering and rearrangement collisions between the atom and the molecule, and of molecular formation and dissociation.     

Some properties of bound states in the three-body problem with separable potentials

A system of three identical bosons with the pair separable S-interaction is considered. The monotonic (Yamaguchi potential) and oscillating pair potentials for the three-alpha model of¹²C have been investigated. In the case of the Yamaguchi potential many excited states (second, third, etc.) have been found.The author is thankful to Prof. V. G. Neudatchin and Prof. Yu. F. Smirnov for valuable discussions on the problems concerned.     

The Faddeev equations for the Heisenberg ferromagnet

The Faddeev equations for the three-magnon T-matrix of the Heisenberg ferromagnet with nearest neighbour interactions are derived for the cubic lattice in arbitrary dimensions. The extreme case of spin $\text{1}\text{2}$ is considered and the kinematical restriction, that only one spin deviation per site is possible, has been taken into account rigorously. Hence the T-matrix is unitary and suited for the study of bound state as well as scattering state properties. The analytic solution of the homogeneous Faddeev equations in one dimension is given.     

Number of bound states of three-body systems and Efimov's effect

Using the variational approach we obtain lower bounds for the number of eigenvalues of three-particle Hamiltonians with short-range potentials close to critical ones and construct corresponding trial functions. A new proof of the infiniteness of the number of eigenvalues for critical potentials (Efimov's effect) is given.     

Quantum kinetic equations in systems with bound states

Starting from the quantum mechanical BBGKY-hierarchy kinetic equations in systems with two particles bound states are given in this paper. With this equation it is possible to consider transport properties in nonideal gases with three particles reactions.     

Flow equations forN point functions and bound states

Zeitschrift für Physik C Particles and Fields - We discuss the exact renormalization group or flow equation for the effective action and its decomposition into one particle irreducibleN point...     

库仑势加新环形势的相对论束缚态

在标量势等于矢量势的条件下，获得了库仑势加新环形势的Klein-Gordon方程和Dirac方程的束缚态的精确解. 对于Klein-Gordon方程，获得了精确的能谱方程和归一化的波函数. 对于Dirac方程，给出了精确的能谱方程和归一化的旋量波函数. 关键词： 库仑势加新环形势 束缚态 精确解     

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.     

A lower bound for Coulomb energies

The indirect Coulomb energy for particles of charge e is usually approximated by E ≈ -Ce^2/3 ∫ (x)^4/3d(x), where (x) is the single-particle charge density. We prove that, for a suitable constant C, this form for E is actually a universal lower bound.     

Bound-state calculations of the three-dimensional Yakubovsky equations with the inclusion of three-body forces

The four-body Yakubovsky equations in a three-dimensional approach with the inclusion of the three-body forces are proposed. The four-body bound state with two- and three-body interactions is formulated in the three-dimensional approach for identical particles as a function of vector Jacobi momenta, specifically, the magnitudes of the momenta and the angles between them. The modified three-dimensional Yakubovsky integral equations are successfully solved with the scalar two-meson exchange three-body force, where the Malfliet-Tjon-type two-body force is implemented. The three-body force effects on the energy eigenvalue and the four-body wave function, as well as accuracy of our numerical calculations are presented.     

New approach for numerical solution of configuration-space Faddeev equations

A new computational scheme for solving the bound state configuration-space Faddeev equations is applied. The scheme is based on the spline-approximation and the adiabatic limit of Faddeev equations. An ordering of variables being in agreement with the limit was chosen. As a result the matrix of the eigenvalue problem has a sparse block structure. Calculations of the bound states of µpp, µdd, µtt mesic molecules and ¯pdd, ¯ptt antiprotonic ones, were performed. To check the method, calculations of the binding energies for such systems as the positronium ion Ps_–,³H and³He were carried out. The results are compared with the best results of other authors.