Integrals of Lagrange functions and sum rules

Baye, Daniel Jean

doi:10.1088/1751-8113/44/39/395204

Cited by 11 publications

(17 citation statements)

References 25 publications

(70 reference statements)

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“…For α ̸ = 0, the matrix elements are antisymmetric as expected for functions vanishing at the origin and at infinity. The second derivatives read with (3.53)-(3.55) as [41,40] λ…”

Section: Values and Matrix Elementsmentioning

confidence: 99%

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The Lagrange-mesh method

2015

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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 timedependent Schrödinger equation, to the Gross-Pitaevskii equation and to Hartree-Fock calculations are also discussed as well as translations and rotations on a Lagrange mesh.

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“…For α ̸ = 0, the matrix elements are antisymmetric as expected for functions vanishing at the origin and at infinity. The second derivatives read with (3.53)-(3.55) as [41,40] λ…”

Section: Values and Matrix Elementsmentioning

confidence: 99%

“…for α > 0, its exact matrix elements are given after a rather heavy calculation [40] by (3.66) given in [4].…”

Section: Values and Matrix Elementsmentioning

confidence: 99%

The Lagrange-mesh method

2015

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“…Figure 1 illustrates the geometry of the studied system in the laboratory and the molecular frame of reference. Both the Ψ(t, r) = ∞ m=−∞ Ψ (m) (ξ, η; t)e imϕ / √ 2π (m ∈ Z) wave function (WF) and the Hamiltonian were expressed in the prolate spheroidal coordinate system: ξ = (r A +r B )/R, η = (r A −r B )/R, ϕ (the azimuthal angle); with R being the internuclear distance, and [36][37][38][39][40][41][42]. Then, they were represented on finite-element discrete variable representation (FE-DVR) grids [43][44][45][46] (for both ξ and η configuration subspaces), where the local basis functions [47] were built using rescaled Lagrange interpolation polynomials (DVR functions) built on top of the x α i (w α i ) Gauss-Lobatto quadrature points (weights).…”

Section: Moleculementioning

confidence: 99%

“…It is worth mentioning, that in the case when the molecular axis is parallel to the laser polarization vector, the coupling between the partial channels corresponding to different m values vanishes, since the second term in equation (7) is zero for θ R = 0. In order to obtain a symmetric form for the kinetic energy matrix T mm iji j , the matrix elements of theT ξ andT η operators were symmetrized by following the procedures outlined in [37,38]:…”

Section: Moleculementioning

confidence: 99%

Photoelectron holography of the H2+ molecule

et al. 2020

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We investigate the photoelectron spectrum of the H2+ target induced by few-cycle XUV laser pulses using first principle calculations. In the photoelectron spectrum, by performing calculations for different internuclear separations, we investigate how the structure of the target is influencing the spatial interference pattern. This interference pattern is created by the coherent superposition of electronic wave packets emitted at the same time, but following different paths. We find that the location of the interference minima in the spectra is dominantly determined by the target’s ionization energy, however, by comparing the H2+ results with model calculations with spherically symmetric potentials, clear differences were observed for the molecular potential relative to the central potentials. Next to the main feature (spatial interference) we have also identified the traces of the two-center interference in the photoelectron spectrum, however, these were mainly washed out due to the complex electronic wave packet dynamics that occurs during the interaction with the considered laser field. Graphical abstract

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“…To compare this convergence with that for the contact interaction, V g has been chosen as V 0 /( √ πa) so that both the contact and Gaussian interactions have the same volume integral. For N = 3, the non-vanishing hyperradial potentials are obtained by a direct calculation of (8) using basis (17) as…”

Section: Hh Expansion For Gaussian Two-body Interactionmentioning

confidence: 99%

Hyperspherical Harmonics Expansion on Lagrange Meshes for Bosonic Systems in One Dimension

Timofeyuk

Baye

2017

Few-Body Syst

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A one-dimensional system of bosons interacting with contact and single-Gaussian forces is studied with an expansion in hyperspherical harmonics. The hyperradial potentials are calculated using the link between the hyperspherical harmonics and the single-particle harmonic-oscillator basis while the coupled hyperradial equations are solved with the Lagrange-mesh method. Extensions of this method are proposed to achieve good convergence with small numbers of mesh points for any truncation of hypermomentum. The convergence with hypermomentum strongly depends on the range of the two-body forces: it is very good for large ranges but deteriorates as the range decreases, being the worst for the contact interaction. In all cases, the lowest-order energy is within 4.5% of the exact solution and shows the correct cubic asymptotic behaviour at large boson numbers. Details of the convergence studies are presented for 3, 5, 20 and 100 bosons. A special treatment for three bosons was found to be necessary. For single-Gaussian interactions, the convergence rate improves with increasing boson number, similar to what happens in the case of three-dimensional systems of bosons.

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Integrals of Lagrange functions and sum rules

Cited by 11 publications

References 25 publications

The Lagrange-mesh method

The Lagrange-mesh method

Photoelectron holography of the H2+ molecule

Hyperspherical Harmonics Expansion on Lagrange Meshes for Bosonic Systems in One Dimension

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