Abstract. A model for quantum tunnelling of a cluster comprising A identical particles, coupled b... more Abstract. A model for quantum tunnelling of a cluster comprising A identical particles, coupled by oscillator-type potential, through short-range repulsive potential barriers is introduced for the first time in the new symmetrized-coordinate representation and studied within the s-wave approximation. The symbolic-numerical algorithms for calculating the effective potentials of the close-coupling equations in terms of the cluster wave functions and the energy of the barrier quasistationary states are formulated and implemented using the Maple computer al-gebra system. The effect of quantum transparency, manifesting itself in nonmonotonic resonance-type dependence of the transmission coefficient upon the energy of the particles, the number of the particles A = 2, 3, 4, and their symmetry type, is analyzed. It is shown that the resonance behavior of the total transmission coefficient is due to the existence of barrier quasistationary states imbedded in the continuum. 5. 1
Abstract. The quantum model of a cluster, consisting of A identical particles, coupled by the int... more Abstract. The quantum model of a cluster, consisting of A identical particles, coupled by the internal pair interactions and affected by the external field of a target, is considered. A symbolic-numerical algorithm for generating A−1-dimensional oscillator eigenfunctions, symmetric or antisymmetric with respect to permutations of A identical particles in the new symmetrized coordinates, is formulated and implemented using the MAPLE computer algebra system. Examples of generating the sym-metrized coordinate representation for A−1 dimensional oscillator func-tions in one-dimensional Euclidean space are analyzed. The approach is aimed at solving the problem of tunnelling the clusters, consisting of sev-eral identical particles, through repulsive potential barriers of a target. 6. 1
A 3C-type two-centre wavefunction with four parameters that we will call a modified two-centre co... more A 3C-type two-centre wavefunction with four parameters that we will call a modified two-centre continuum wavefunction (MTCC) is proposed. This function is constructed in a closed analytical form by solving the Schrödinger equation of an electron (with wave vector k and position vector r) in the Coulomb field of two fixed charged nuclei. The obtained solution fulfils the correct boundary conditions asymptotically up to the order O((kr) −2 ). The MTCC function is applied to calculations of the seven-fold differential cross section of the dissociative ionization of the simplest homo-nuclear diatomic system H + 2 by fast electrons. The good agreement of the obtained results with exact ones shows that the MTCC function could be a very elegant and useful compromise in the description of slow electrons emerging from diatomic targets.
We present program GITAN for symbolic computation of the class of polynomial Hamiltonians and for... more We present program GITAN for symbolic computation of the class of polynomial Hamiltonians and formal integrals with the help of ordinary and inverse Birkhoff–Gustavson normalization based on the algorithm ANFER using a conventional pseudocode. The corresponding algorithm of the program QUANTGIT for a semiclassical quantization of the BGNF is described too. Typical examples for a hydrogen atom in external fields demonstrating the runs of the above algorithms and programs as input and output data are given. A comparison of the obtained semiclassical spectrum and its quantum counterpart calculated by the POINTFIELD program is shown.
Saratov Fall Meeting 2019: Laser Physics, Photonic Technologies, and Molecular Modeling
The computational scheme and calculation results of bound, metastable and Rydberg states of atomi... more The computational scheme and calculation results of bound, metastable and Rydberg states of atomic and molecular systems important for laser spectroscopy are presented. The solution to the problem is performed using the authors' software package (see program libraries of the Computer Physics Communications journal and of the Joint Institute for Nuclear Research) that implement the high-accuracy finite element method. The FORTRAN procedure of matching tabulated potential functions with van der Waals asymptotic potential using interpolation Hermite polynomials which provides continuity of both the function itself and its derivative is presented. The efficiency of the proposed approach is demonstrated by calculated for the first time sharp metastable states with complex eigen-energies in a diatomic beryllium molecule and weakly bound Rydberg states of antiprotonic helium atom.
The mathematical model of quantum tunnelling of diatomic molecules through repulsive barriers is ... more The mathematical model of quantum tunnelling of diatomic molecules through repulsive barriers is formulated in the s-wave approximation. The 2D boundary-value problem in polar coordinates is reduced to a 1D one by means of Kantorovich expansion over the set of parametric basis functions. The algorithm for calculating the asymptotic form of the parametric basis functions at large values of the parameter (radial variable) is presented. The solution is sought by matching the numerical solution in one of the subintervals with the analytical solution in the adjacent one. The efficiency of the algorithm is shown by comparison of the calculated solutions with those of the parametric eigenvalue problem obtained by applying the finite element method in the entire domain of definition at large values of the parameter.
The exactly soluble model of a train of zero-duration electromagnetic pulses interacting with a 1... more The exactly soluble model of a train of zero-duration electromagnetic pulses interacting with a 1D atom with short-range interaction potential modelled by a δ-function is considered. The model is related to the up-to-date laser techniques providing the duration of pulses as short as a few attoseconds and the intensities higher than 1014 W/cm2.
We have developed a symbolic-numeric algorithm implemented in Wolfram Mathematica to compute the ... more We have developed a symbolic-numeric algorithm implemented in Wolfram Mathematica to compute the orthonormal non-canonical bases of symmetric irreducible representations of the \(\text {O(5)}\times \text {SU(1,1)}\) and \(\overline{\text {O(5)}}\times \overline{\text {SU(1,1)}}\) partner groups in the laboratory and intrinsic frames, respectively. The required orthonormal bases are labelled by the set of the number of bosons N, seniority \(\lambda \), missing label \(\mu \) denoting the maximal number of boson triplets coupled to the angular momentum \(L=0\), and the angular momentum (L, M) quantum numbers using the conventional representations of a five-dimensional harmonic oscillator in the laboratory and intrinsic frames. The proposed method uses a new symbolic-numeric orthonormalization procedure based on the Gram–Schmidt orthonormalization algorithm. Efficiency of the elaborated procedures and the code is shown by benchmark calculations of orthogonalization matrix O(5) and \(\o...
We propose benchmark calculations of the boundary-value problem for a systems of ODEs of large di... more We propose benchmark calculations of the boundary-value problem for a systems of ODEs of large dimension with help of KANTBP program using a finite element method. In practice, for solving problems with the long-range potential and a large number of open channels there is a need for solving boundary value problems of the large-scale systems of differential equations that require further investigation of convergence and stability of the algorithms and programs. With this aim we solve here the eigenvalue problem for an elliptic differential equation in a two-dimensional domain with Dirichlet boundary conditions. The solution is sought in the form of Kantorovich expansion over the basis functions of one of the independent variables with the second variable treated as a parameter. The basis functions are calculated in an analytical form as solutions of the parametric Sturm-Lioville problem for an ordinary second-order differential equation. As a result, the two-dimensional problem is re...
A review of some recently developed methods of calculating multiple differential cross-sections o... more A review of some recently developed methods of calculating multiple differential cross-sections of photoionization and electron impactionization of atoms and molecules having two active electrons is presented. The methods imply original approaches to calculating three-particle Coulomb wave functions. The external complex scaling method and the formalism of the Schroedinger equation with a source in the right-hand side are considered. Efficiency of the time-dependent approaches to the scattering problem, such as the paraxial approximation and the time-dependent scaling, is demonstrated. An original numerical method elaborated by the authors for solving the 6D Schroedinger equation for an atom with two active electrons, based on the Chang-Fano transformation and the discrete variable representation, is formulated. Basing on numerical simulations, the threshold behavior of angular distributions of two-electron photoionization of the negative hydrogen ion and helium atom, and multiple d...
Saratov Fall Meeting 2020: Computations and Data Analysis: from Molecular Processes to Brain Functions
A mathematical model of COVID-19 pandemic based on a two-parameter nonlinear first-order ordinary... more A mathematical model of COVID-19 pandemic based on a two-parameter nonlinear first-order ordinary differential equation with retarded time argument is proposed. The model uses two parameters: the time of possible dissemination of infection by an individual virus carrier and the probability of contamination of a healthy population member in a contact with an infected one per unit time. The parameters can be given functions of time, which is particularly important in describing multi-peak pandemic. The model is applicable to any community (country, city, etc.) and provides an optimal balance between the adequate description of a pandemic inherent in the known SIR model and the relative simplicity for practical estimates. Examples of the model application are in qualitative agreement with the dynamics of COVID-19 pandemic.
Abstract. A model for quantum tunnelling of a cluster comprising A identical particles, coupled b... more Abstract. A model for quantum tunnelling of a cluster comprising A identical particles, coupled by oscillator-type potential, through short-range repulsive potential barriers is introduced for the first time in the new symmetrized-coordinate representation and studied within the s-wave approximation. The symbolic-numerical algorithms for calculating the effective potentials of the close-coupling equations in terms of the cluster wave functions and the energy of the barrier quasistationary states are formulated and implemented using the Maple computer al-gebra system. The effect of quantum transparency, manifesting itself in nonmonotonic resonance-type dependence of the transmission coefficient upon the energy of the particles, the number of the particles A = 2, 3, 4, and their symmetry type, is analyzed. It is shown that the resonance behavior of the total transmission coefficient is due to the existence of barrier quasistationary states imbedded in the continuum. 5. 1
Abstract. The quantum model of a cluster, consisting of A identical particles, coupled by the int... more Abstract. The quantum model of a cluster, consisting of A identical particles, coupled by the internal pair interactions and affected by the external field of a target, is considered. A symbolic-numerical algorithm for generating A−1-dimensional oscillator eigenfunctions, symmetric or antisymmetric with respect to permutations of A identical particles in the new symmetrized coordinates, is formulated and implemented using the MAPLE computer algebra system. Examples of generating the sym-metrized coordinate representation for A−1 dimensional oscillator func-tions in one-dimensional Euclidean space are analyzed. The approach is aimed at solving the problem of tunnelling the clusters, consisting of sev-eral identical particles, through repulsive potential barriers of a target. 6. 1
A 3C-type two-centre wavefunction with four parameters that we will call a modified two-centre co... more A 3C-type two-centre wavefunction with four parameters that we will call a modified two-centre continuum wavefunction (MTCC) is proposed. This function is constructed in a closed analytical form by solving the Schrödinger equation of an electron (with wave vector k and position vector r) in the Coulomb field of two fixed charged nuclei. The obtained solution fulfils the correct boundary conditions asymptotically up to the order O((kr) −2 ). The MTCC function is applied to calculations of the seven-fold differential cross section of the dissociative ionization of the simplest homo-nuclear diatomic system H + 2 by fast electrons. The good agreement of the obtained results with exact ones shows that the MTCC function could be a very elegant and useful compromise in the description of slow electrons emerging from diatomic targets.
We present program GITAN for symbolic computation of the class of polynomial Hamiltonians and for... more We present program GITAN for symbolic computation of the class of polynomial Hamiltonians and formal integrals with the help of ordinary and inverse Birkhoff–Gustavson normalization based on the algorithm ANFER using a conventional pseudocode. The corresponding algorithm of the program QUANTGIT for a semiclassical quantization of the BGNF is described too. Typical examples for a hydrogen atom in external fields demonstrating the runs of the above algorithms and programs as input and output data are given. A comparison of the obtained semiclassical spectrum and its quantum counterpart calculated by the POINTFIELD program is shown.
Saratov Fall Meeting 2019: Laser Physics, Photonic Technologies, and Molecular Modeling
The computational scheme and calculation results of bound, metastable and Rydberg states of atomi... more The computational scheme and calculation results of bound, metastable and Rydberg states of atomic and molecular systems important for laser spectroscopy are presented. The solution to the problem is performed using the authors' software package (see program libraries of the Computer Physics Communications journal and of the Joint Institute for Nuclear Research) that implement the high-accuracy finite element method. The FORTRAN procedure of matching tabulated potential functions with van der Waals asymptotic potential using interpolation Hermite polynomials which provides continuity of both the function itself and its derivative is presented. The efficiency of the proposed approach is demonstrated by calculated for the first time sharp metastable states with complex eigen-energies in a diatomic beryllium molecule and weakly bound Rydberg states of antiprotonic helium atom.
The mathematical model of quantum tunnelling of diatomic molecules through repulsive barriers is ... more The mathematical model of quantum tunnelling of diatomic molecules through repulsive barriers is formulated in the s-wave approximation. The 2D boundary-value problem in polar coordinates is reduced to a 1D one by means of Kantorovich expansion over the set of parametric basis functions. The algorithm for calculating the asymptotic form of the parametric basis functions at large values of the parameter (radial variable) is presented. The solution is sought by matching the numerical solution in one of the subintervals with the analytical solution in the adjacent one. The efficiency of the algorithm is shown by comparison of the calculated solutions with those of the parametric eigenvalue problem obtained by applying the finite element method in the entire domain of definition at large values of the parameter.
The exactly soluble model of a train of zero-duration electromagnetic pulses interacting with a 1... more The exactly soluble model of a train of zero-duration electromagnetic pulses interacting with a 1D atom with short-range interaction potential modelled by a δ-function is considered. The model is related to the up-to-date laser techniques providing the duration of pulses as short as a few attoseconds and the intensities higher than 1014 W/cm2.
We have developed a symbolic-numeric algorithm implemented in Wolfram Mathematica to compute the ... more We have developed a symbolic-numeric algorithm implemented in Wolfram Mathematica to compute the orthonormal non-canonical bases of symmetric irreducible representations of the \(\text {O(5)}\times \text {SU(1,1)}\) and \(\overline{\text {O(5)}}\times \overline{\text {SU(1,1)}}\) partner groups in the laboratory and intrinsic frames, respectively. The required orthonormal bases are labelled by the set of the number of bosons N, seniority \(\lambda \), missing label \(\mu \) denoting the maximal number of boson triplets coupled to the angular momentum \(L=0\), and the angular momentum (L, M) quantum numbers using the conventional representations of a five-dimensional harmonic oscillator in the laboratory and intrinsic frames. The proposed method uses a new symbolic-numeric orthonormalization procedure based on the Gram–Schmidt orthonormalization algorithm. Efficiency of the elaborated procedures and the code is shown by benchmark calculations of orthogonalization matrix O(5) and \(\o...
We propose benchmark calculations of the boundary-value problem for a systems of ODEs of large di... more We propose benchmark calculations of the boundary-value problem for a systems of ODEs of large dimension with help of KANTBP program using a finite element method. In practice, for solving problems with the long-range potential and a large number of open channels there is a need for solving boundary value problems of the large-scale systems of differential equations that require further investigation of convergence and stability of the algorithms and programs. With this aim we solve here the eigenvalue problem for an elliptic differential equation in a two-dimensional domain with Dirichlet boundary conditions. The solution is sought in the form of Kantorovich expansion over the basis functions of one of the independent variables with the second variable treated as a parameter. The basis functions are calculated in an analytical form as solutions of the parametric Sturm-Lioville problem for an ordinary second-order differential equation. As a result, the two-dimensional problem is re...
A review of some recently developed methods of calculating multiple differential cross-sections o... more A review of some recently developed methods of calculating multiple differential cross-sections of photoionization and electron impactionization of atoms and molecules having two active electrons is presented. The methods imply original approaches to calculating three-particle Coulomb wave functions. The external complex scaling method and the formalism of the Schroedinger equation with a source in the right-hand side are considered. Efficiency of the time-dependent approaches to the scattering problem, such as the paraxial approximation and the time-dependent scaling, is demonstrated. An original numerical method elaborated by the authors for solving the 6D Schroedinger equation for an atom with two active electrons, based on the Chang-Fano transformation and the discrete variable representation, is formulated. Basing on numerical simulations, the threshold behavior of angular distributions of two-electron photoionization of the negative hydrogen ion and helium atom, and multiple d...
Saratov Fall Meeting 2020: Computations and Data Analysis: from Molecular Processes to Brain Functions
A mathematical model of COVID-19 pandemic based on a two-parameter nonlinear first-order ordinary... more A mathematical model of COVID-19 pandemic based on a two-parameter nonlinear first-order ordinary differential equation with retarded time argument is proposed. The model uses two parameters: the time of possible dissemination of infection by an individual virus carrier and the probability of contamination of a healthy population member in a contact with an infected one per unit time. The parameters can be given functions of time, which is particularly important in describing multi-peak pandemic. The model is applicable to any community (country, city, etc.) and provides an optimal balance between the adequate description of a pandemic inherent in the known SIR model and the relative simplicity for practical estimates. Examples of the model application are in qualitative agreement with the dynamics of COVID-19 pandemic.
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