S. I. Vinitsky

Calculation of a hydrogen atom photoionization in a strong magnetic field by using the angular oblate spheroidal functions

A new efficient method for calculating the photoionization of a hydrogen atom in a strong magnetic field is developed based on the Kantorovich approach to the parametric boundary problems in spherical coordinates using the orthogonal basis set of angular oblate spheroidal functions. The progress as compared with our previous paper (Dimova M G, Kaschiev M S and Vinitsky S I 2005 J. Phys. B: At. Mol. Opt. Phys. 38 2337–52) consists of the development of the Kantorovich method for calculating the wavefunctions of a continuous spectrum, including the quasi-stationary states imbedded in the continuum. Resonance transmission and total reflection effects for scattering processes of electrons on protons in a homogenous magnetic field are manifested. The photoionization cross sections found for the ground and excited states are in good agreement with the calculations by other authors and demonstrate correct threshold behavior. The estimates using the calculated photoionization cross section show that due to the quasi-stationary states the laser-stimulated recombination may be enhanced by choosing the optimal laser frequency.

Channeling problem for charged particles produced by confining environment

Channeling problem produced by confining environment that leads to resonance scattering of charged particles via quasistationary states imbedded in the continuum is examined. Nonmonotonic dependence of physical parameters on collision energy and/or confining environment due to resonance transmission and total reflection effects is confirmed that can increase the rate of recombination processes. The reduction of the model for two identical charged ions to a boundary problem is considered together with the asymptotic behavior of the solution in the vicinity of pair-collision point and the results of R-matrix calculations. Tentative estimations of the enhancement factor and the total reflection effect are discussed.

The application of adiabatic method for the description of impurity states in quantum nanostructures

In the framework of effective mass approximation the application of adiabatic method for the description of impurity states in quantum dots, wires and wells with parabolic confinement potential as well as rectangular infinitely-high potential is presented. A rate of convergence of the method and efficiency of the proposed program complex for solving a boundary value problem, realized by the finite element method, is demonstrated on examples of calculation of spectral and optical characteristics of the considered quantum nanostructures.

A symbolic-numerical algorithm for the computation of matrix elements in the parametric eigenvalue problem

A symbolic-numerical algorithm for the computation of the matrix elements in the parametric eigenvalue problem to a prescribed accuracy is presented. A procedure for calculating the oblate angular spheroidal functions that depend on a parameter is discussed. This procedure also yields the corresponding eigenvalues and the matrix elements (integrals of the eigenfunctions multiplied by their derivatives with respect to the parameter). The efficiency of the algorithm is confirmed by the computation of the eigenvalues, eigenfunctions, and the matrix elements and by the comparison with the known data and the asymptotic expansions for small and large values of the parameter. The algorithm is implemented as a package of programs in Maple-Fortran and is used for the reduction of a singular two-dimensional boundary value problem for the elliptic second-order partial differential equation to a regular boundary value problem for a system of second-order ordinary differential equations using the Kantorovich method.

Adiabatic description of nonspherical quantum dot models

Within the effective mass approximation an adiabatic description of spheroidal and dumbbell quantum dot models in the regime of strong dimensional quantization is presented using the expansion of the wave function in appropriate sets of single-parameter basis functions. The comparison is given and the peculiarities are considered for spectral and optical characteristics of the models with axially symmetric confining potentials depending on their geometric size, making use of the complete sets of exact and adiabatic quantum numbers in appropriate analytic approximations.

Resonant tunneling of a few-body cluster through repulsive barriers

A model for quantum tunnelling of a cluster comprised of A identical particles, interacting via oscillator-type potential, through short-range repulsive barrier potentials is introduced for the first time in symmetrized-coordinate representation and numerically studied in the s-wave approximation. A constructive method for symmetrizing or antisymmetrizing the (A − 1)-dimensional harmonic oscillator basis functions in the new symmetrized coordinates with respect to permutations of coordinates of A identical particles is described. The effect of quantum transparency, manifesting itself in nonmonotonic resonance-type dependence of the transmission coefficient upon the energy of the particles, their number A = 2, 3, 4 and the type of their symmetry, is analyzed. It is shown that the total transmission coefficient demonstrates the resonance behavior due to the existence of barrier quasi-stationary states, embedded in the continuum.

Analytical and numerical calculations of spectral and optical characteristics of spheroidal quantum dots

In the effective mass approximation for electronic (hole) states of a spheroidal quantum dot with and without external fields the perturbation theory schemes are constructed in the framework of the Kantorovich and adiabatic methods. The eigenvalues and eigenfunctions of the problem, obtained in both analytical and numerical forms, were applied for the analysis of spectral and optical characteristics of spheroidal quantum dots in homogeneous electric fields.

Adiabatic Approach to the Problem of a Quantum Well with a Hydrogen-Like Impurity *

An adiabatic method is presented for solving a boundary discrete spectrum problem for a parabolic quantum well and a rectangular quantum well with infinitely-high walls in the presence of a hydrogen-like impurity. The upper and lower bounds for the energy of the ground state of the systems are obtained under the conditions of the shift of the Coulomb center in a given range of the parameter with respect to earlier variational estimates. The comparison of the rate of convergence of the adiabatic expansion of the solution in parametric bases in the cylindrical and spherical coordinates is carried out.

Three identical particles on a line: comparison of some exact and approximate calculations

The three-body scattering problem is formulated in the adiabatic representation as a multi-channel spectral problem for a set of coupled one-dimensional integral equations. New stable variational-iteration schemes are developed to calculate the Hamiltonian eigenfunctions and energy eigenvalues, as well as the reaction matrix in the eigenphase shift representation, with prescribed accuracy. The convergence and efficiency of the method are demonstrated in the vicinity of the three-body threshold in the exactly solvable model of three identical particles fixed on a line and coupled with pair-repulsive or attractive zero-range potentials.

Explicit Magnus expansions for solving the time-dependent Schrödinger equation

The symmetric implicit operator-difference multi-layer schemes for solving the time-dependent Schr?dinger equation based on decomposition of the evolution operator via the explicit Magnus expansion up to the sixth order of accuracy with respect to the time step are presented. Reduced schemes for solving the set of coupled time-dependent Schr?dinger equations with respect to the hyper-radial variable are devised by using the Kantorovich expansion of the wave packet over a set of appropriate parametric basis angular functions. Further discretization of the resulting problem with symmetric operators is implemented by means of the finite-element method. The convergence and efficiency of the numerical schemes are demonstrated in benchmark calculations of the exactly solvable models of a one-dimensional time-dependent oscillator, a two-dimensional oscillator in time-dependent electric field by using the conventual angular basis, and the inexactly solvable model of a three-dimensional kicked hydrogen atom in a magnetic field by using a parametric basis of the angular oblate spheroidal functions developed in our previous paper (Chuluunbaatar O et al 2007 J. Phys. A: Math. Theor. 40 11485?524).