Lagrange-mesh calculations in momentum space

Lacroix, Gwendolyn; Semay, Claude; Buisseret, Fabien

doi:10.1103/physreve.86.026705

Cited by 6 publications

(13 citation statements)

References 36 publications

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“…Numerical solutions of the Schrödinger equation in momentum space have been studied with the LMM by Semay and coworkers [137,138]. The basic motivation of working in momentum space is the possibility of using kinetic operators T (p 2 ) more complicated than the non-relativistic operator T NR (p 2 ) = p 2 /2m.…”

Section: Lagrange-mesh Methods In Momentum Spacementioning

confidence: 99%

“…The basic motivation of working in momentum space is the possibility of using kinetic operators T (p 2 ) more complicated than the non-relativistic operator T NR (p 2 ) = p 2 /2m. For example, working in momentum space [137,138] allows a simpler treatment of the semirelativistic kinetic expression…”

Section: Lagrange-mesh Methods In Momentum Spacementioning

confidence: 99%

“…Another interesting case, not considered in Refs. [137,138] but based on some of their results in §III.E, occurs when the Fourier transform of a potential V (r) is not available. It is based on the expression r = i∇ p .…”

Section: Lagrange-mesh Methods In Momentum Spacementioning

confidence: 99%

“…It is based on the expression r = i∇ p . Then the matrix elements ⟨f i |V (r)|f j ⟩ between scaled Lagrange functions in p space can be obtained [138] from Ground-state energy of the Yukawa potential (5.60) with g = 10 on a regularized Laguerre mesh: in r space (h = 0.1) and in p space with Eq. (5.56) (h = 10).…”

Section: Lagrange-mesh Methods In Momentum Spacementioning

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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Section: Lagrange-mesh Methods In Momentum Spacementioning

confidence: 99%

Section: Lagrange-mesh Methods In Momentum Spacementioning

confidence: 99%

Section: Lagrange-mesh Methods In Momentum Spacementioning

confidence: 99%

Section: Lagrange-mesh Methods In Momentum Spacementioning

confidence: 99%

See 2 more Smart Citations

The Lagrange-mesh method

2015

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“…This method is a variational method that employs the Lagrange basis functions defined on numerical quadratures to construct variational solutions to quantum mechanical equations. Compared to other numerical methods, the method was shown to generate simpler, faster and more accurate solutions for different quantum mechanical systems [8][9][10][11], and is not restricted to one-dimensional or separable problems. In the present work the Lagrange-mesh method is explored for the numerical solution of the Schrödinger equation in more than three dimensions in generalized spherical coordinates.…”

Section: Introductionmentioning

confidence: 99%

Lagrange-mesh solution of the Schrödinger equation in generalized spherical coordinates

Rampho

2018

J. Phys. Commun.

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The solution of the multi-dimensional Schrödinger equation in the generalized spherical coordinates is constructed in the Lagrange-mesh method. Laguerre and Jacobi meshes are used to construct matrix elements for the generalized Hamiltonian. The matrix elements are functions of quadrature abscissas and involve few free parameters for each angular dimension. A ring-shaped non-central separable potential, and a system of four linearly coupled anharmonic oscillators are used for illustrations of the efficiency and accuracy of the method. The numerical solutions involving eigenfunctions of the kinetic energy operator display the typical slow convergence with the accuracy improving as the Lagrangemesh bases size increases. The Lagrange-mesh solutions converge to the exact solution when the parameters of the matrix elements are chosen appropriately.

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Trigonometric Lagrange-Jacobi functions

Rampho

2024

Phys. Scr.

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This paper presents a class of trigonometric Lagrange-mesh functions constructed from the Lagrange-Jacobi functions. These functions extend the set of trigonometric Lagrange-mesh functions to beyond only those constructed directly from the sine and cosine functions. Trigonometric Lagrange-Chebyshev functions obtained from the presented trigonometric Lagrange-Jacobi functions are equivalent to known trigonometric Lagrange-Chebyshev functions constructed directly from the sine and cosine functions and generate identical matrix elements for the kinetic energy operator.

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Lagrange-mesh calculations in momentum space

Cited by 6 publications

References 36 publications

The Lagrange-mesh method

The Lagrange-mesh method

Lagrange-mesh solution of the Schrödinger equation in generalized spherical coordinates

Trigonometric Lagrange-Jacobi functions

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