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The Lagrange-mesh method is known to be an efficient tool for evaluating the bound states of various three-body atomic and molecular systems. By combining it with the complex scaling method, resonances can also be studied. In this paper, this approach is used for evaluating several S-wave resonances of the helium atom and of the negative positronium ion in vacuum and in Debye plasmas. In spite of its simplicity, the Lagrange-mesh method provides resonance energies and widths more accurate than the best literature results., SCOPUS: ar.j, info:eu-repo/semantics/published

The original two-variable integrodifferential equation for few-body systems is mod ified by introducing boundary conditions in the radial and angular domains. The accuracy of the adiabatic approximation in solving this two-variable modified few body integrodifferential equation is investigated. In this approximation the inte grodifferential equation is decoupled into two single-variable equations for the ra dial motion and angular motion. The two equations are solved using the Lagrange-mesh methods. Ground-state energies of systems of particles interacting through realistic nucleon-nucleon and alpha-alpha interacting potentials and constituted by various numbers of particles are considered. The ground-state energies obtained are compared with those from the solution of the original two-variable integrodifferential equation as well as those obtain by other methods reported in the literature.

We present a program to compute polarizabilities of nominal one-electron systems using the Lagrange-mesh method (LMM) (Baye, 2015), that was used by Filippin et al. (2018). A semiempirical-core-potential approach is implemented, ultimately solving a Dirac-like equation by diagonalizing the corresponding Hamiltonian matrix. In order to build the core potential, the core orbitals are obtained from independent calculations using the GRASP2018 package (Fischer et al. 2019). Therefore we provide an easy-to-use interface between the GRASP2018 package and the LMM complete finite basis, allowing to switch easily from one one-electron basis to the other. Program summary: Program Title: POLALMM CPC Library link to program files: http://dx.doi.org/10.17632/6mw5gdwfkt.1 Licensing provisions: MIT license Programming language: Fortran90 Nature of problem: Determination of the dipole and quadrupole polarizabilities. Solution method: We combine a semiempirical-core-potential approach with the numerical Lagrange-mesh method to solve a Dirac-like one-electron equation [2]. The building of the core potential requires the prior knowledge of core orbitals provided by GRASP [3]. Two free parameters are optimized by fitting the computed single-electron valence energies to their experimental reference value. References: [1] The Lagrange-mesh method, D. Baye, Phys. Rep. 565 (2015) 1-107 [2] Relativistic semiempirical-core-potential calculations in Ca+, Ba+ and Sr+ ions on Lagrange meshes, L. Filippin, S. Schiffmann, J. Dohet-Eraly, D. Baye and M. Godefroid, Phys. Rev. A 97 (2018) 012506 [3] GRASP2018 - A Fortran 95 version of the General Relativistic Atomic Structure Package, C. Froese Fischer, G. Gaigalas, P. Jönsson and J. Bieroń, Comput. Phys. Commun. 237 (2019) 184-187, SCOPUS: ar.j, info:eu-repo/semantics/published

The problem of a particle confined in a spherical cavity is studied with the Dirac equation. A hard confinement is obtained by forcing the large component to vanish at the cavity radius. It is shown that the small component cannot vanish simultaneously at this radius. In the case of a confined hydrogen atom, the energies are given by an implicit equation. For some values of the radius, explicit analytical expressions of the energy exist like in the nonrelativistic case. Very accurate energies and wave functions are obtained with the Lagrange-mesh method with few mesh points. To this end, two differently regularized Lagrange-Jacobi bases associated with the same mesh are used for the large and small components. The importance of relativistic effects is discussed for hydrogenlike ions. The validity of this definition of hard confinement is discussed with a soft-confinement model studied with the R-matrix method., SCOPUS: ar.j, info:eu-repo/semantics/published

The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid thanks to the use of a Gauss-quadrature approximation. The variational basis related to this Gauss quadrature is composed of Lagrange functions which are infinitely differentiable functions vanishing at all mesh points but one. This method is quite simple to use and, more importantly, can be very accurate with small number of mesh points for a number of problems. The accuracy may however be destroyed by singularities of the potential term. This difficulty can often be overcome by a regularization of the Lagrange functions which does not affect the simplicity and accuracy of the method.The principles of the Lagrange-mesh method are described, as well as various generalizations of the Lagrange functions and their regularization. The main existing meshes are reviewed and extensive formulas are provided which make the numerical calculations simple. They are in general based on classical orthogonal polynomials. The extensions to non-classical orthogonal polynomials and periodic functions are also presented.Applications start with the calculations of energies, wave functions and some observables for bound states in simple solvable models which can rather easily be used as exercises by the reader. The Dirac equation is also considered. Various problems in the continuum can also simply and accurately be solved with the Lagrange-mesh technique including multichannel scattering or scattering by non-local potentials. The method can be applied to three-body systems in appropriate systems of coordinates. Simple atomic, molecular and nuclear systems are taken as examples. The applications to the time-dependent Schrödingerequation, to the Gross-Pitaevskii equation and to Hartree-Fock calculations are also discussed as well as translations and rotations on a Lagrange mesh., SCOPUS: re.j, info:eu-repo/semantics/published