Precise numerical calculations of bound states of three-atomic Helium cluster are performed. The modern techniques of solution of Faddeev equations are combined to obtain an efficient numerical scheme. Binding energies and other observables for ground and excited states are calculated. Geometric properties of the clusters are discussed.

The binding energy and the corresponding wave function of excitons in GaAs-based finite square quantum wells (QWs) are calculated by the direct numerical solution of the three-dimensional Schrödinger equation. The precise results for the lowest exciton state are obtained by the Hamiltonian discretization using the high-order finite-difference scheme. The microscopic calculations are compared with the results obtained by the standard variational approach. The exciton binding energies found by two methods coincide within 0.1 meV for the wide range of QW widths. The radiative decay rate is calculated for QWs of various widths using the exciton wave functions obtained by direct and variational methods. The radiative decay rates are confronted with the experimental data measured for high-quality GaAs/AlGaAs and InGaAs/GaAs QW heterostructures grown by molecular beam epitaxy. The calculated and measured values are in good agreement, though slight differences with earlier calculations of the radiative decay rate are observed.

A three-body scattering process in the presence of Coulomb interaction can be decomposed formally into a two-body single channel, a two-body multichannel and a genuine three-body scattering. The corresponding integral equations are coupled Lippmann-Schwinger and Faddeev-Merkuriev integral equations. We solve them by applying the Coulomb-Sturmian separable expansion method. We present elastic scattering and reaction cross sections of the e + + H system both below and above the H(n = 2) threshold. We found excellent agreements with previous calculations in most cases. PACS number(s): 34.10.+x, 34.85.+x, 21.45.+v, 03.65.Nk, 02.30.Rz, 02.60.Nm The three-body Coulomb scattering problem is one of the most challenging long-standing problems of nonrelativistic quantum mechanics. The source of the difficulties is related to the long-range character of the Coulomb potential. In the standard scattering theory it is supposed that the particles move freely asymptotically. That is not the case if Coulombic interactions are involved. As a result the fundamental equations of the three-body problems, the Faddeev-equations, become illbehaved if they are applied for Coulomb potentials in a straightforward manner.The first, and formally exact, approach was proposed by Noble [1]. His formulation was designed for solving the nuclear three-body Coulomb problem, where all Coulomb interactions are repulsive. The interactions were split into short-range and long-range Coulomb-like parts and the long-range parts were formally included in the "free" Green's operator. Therefore the corresponding Faddeev-Noble equations become mathematically wellbehaved and in the absence of Coulomb interaction they fall back to the standard equations. However, the associated Green's operator is not known. This formalism, as presented at that time, was not suitable for practical calculations.In Noble's approach the separation of the Coulomb-like potential into short-range and long-range parts were carried out in the two-body configuration space. Merkuriev extended the idea of Noble by performing the splitting in the three-body configuration space. This was a crucial development since it made possible to treat attractive Coulomb interactions on an equal footing as repulsive ones. This theory has been developed using integral equations with connected (compact) kernels and transformed into configuration-space differential equations with asymptotic boundary conditions [2]. In practical calculations, so far only the latter version of the theory has been considered. The primary reason is that the more complicated structure of the Green's operators in the kernels of the Faddeev-Merkuriev integral equations has not yet allowed any direct solution. However, use of integral equations is a very appealing approach since no boundary conditions are required.Recently, one of us has developed a novel method for treating the three-body problem with repulsive Coulomb interactions in three-potential picture [3]. In this approach a three-body Coulomb scattering process can ...

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