Michel Hesse

Coupled-channel R-matrix method on a Lagrange mesh

Abstract The coupled-channel R -matrix method on a Lagrange mesh is a very simple approximation of the R -matrix method with a basis. The mesh points are zeros of shifted Legendre polynomials. Bound-state energies and scattering matrices are easily calculated with small numbers of potential values at mesh points. A test with an exactly solvable two-channel potential provides an excellent accuracy over a broad energy range with only 30 mesh points. The efficiency of the method is illustrated for a single channel on α + α scattering and for two channels on the deuteron ground-state energy and on nucleon-nucleon scattering.

Solving the resonating-group equation on a Lagrange mesh

Abstract The resonating-group method allows treating reactions in a fully microscopic way. The non-local resonating-group equation can be accurately solved on a Lagrange mesh involving few mesh points. This mesh technique is combined with either the R -matrix method or the Hulthen–Kohn method. The forbidden states can be eliminated by a special treatment. The accuracy of the technique of solution is illustrated on a solvable non-local potential. Phase shifts for the α +n and α +p scatterings are calculated with both variants of the resonating-group method on a Lagrange mesh and a comparison is performed between them and the equivalent generator-coordinate method.

Lagrange-mesh calculations of three-body atoms and molecules

The Lagrange-mesh numerical method has the simplicity of a mesh calculation and the accuracy of a variational calculation. A three-dimensional Lagrange-mesh method based on zeros of Laguerre polynomials is applied to the study of the ground states of the helium atom, the hydrogen and positronium negative ions and the hydrogen molecular ion. The calculations lead to big sparse matrices. Highly accurate energies are obtained, which essentially agree with recently published results or improve them. The accuracy is estimated by varying the mesh size and two nonlinear scale parameters. Different mean values of observables are obtained very simply with the Gauss quadrature approximation associated with the mesh.

Analysis of the R-matrix method on Lagrange meshes

The R-matrix method on a Lagrange mesh is a very simple approximation of the R-matrix method with a basis. By analysing an exactly solvable example, we observe that the mesh approximation does not reduce the accuracy of the R-matrix bound-state energies and phase shifts. This property is obtained with two different meshes, the shifted Legendre and shifted Jacobi meshes, which correspond to equivalent polynomial bases. Their comparison shows that the orthogonality of the Lagrange basis functions is not as crucial as was previously assumed: the Legendre mesh, which corresponds to a nonorthogonal Lagrange basis, is at least as accurate as the Jacobi mesh based on an orthogonal basis. We also emphasize the surprising origin of a known property of the R-matrix method: the results are much more accurate with basis functions without uniform boundary conditions because the quality of the matching is realized by a few highly excited eigenfunctions, with weak physical content, of the sum of the Hamiltonian and Bloch operators.

Lagrange-mesh calculations of excited states of three-body atoms and molecules

Energies of S and P bound states of the helium atom He, the hydrogen ion H-, the hydrogen molecular ion H2+ and the muonic molecule dtµ are calculated with the Lagrange-mesh numerical method. Interparticle distances and mass radii of these states are evaluated very simply with the Gauss quadrature approximation associated with the mesh. Except for weakly bound states, highly accurate values are obtained, which agree with or improve published results.

Helium atoms in a strong magnetic field studied with the Lagrange-mesh method

The helium atom in a strong magnetic field B ≤ 2.35 × 105 T is studied with the help of the Lagrange-mesh method, in the case of an infinitely heavy nucleus. An approximate variational calculation is performed with a basis formed by combining Wigner functions depending on Euler angles with Lagrange functions for the three internal degrees of freedom. The Lagrange functions are associated with a three-dimensional mesh in perimetric coordinates. The method is applied to the determination of the energies of the two lowest singlet and triplet helium states with a total orbital momentum projection M equal to zero and to the lowest singlet M = −1 state.

Lagrange-mesh calculations of the ground-state rotational bands of the H2+ and D2+ molecular ions

All the energies of the ground-state ?g rotational band of the H2+ molecular ion are calculated by solving the three-body Schr?dinger equation with the Lagrange-mesh numerical method in the perimetric coordinates system. This method provides the same accuracy as a variational calculation but is much simpler. Energies are obtained with an accuracy of 10?12 Hartree. The wavefunctions display a strongly dominant axial (K = 0) symmetry. Calculations including components up to K = 2 show that components beyond K = 1 are negligible at the 10?10 accuracy. Accurate analytical approximations of the wavefunctions are used to evaluate mean interparticle distances and electric quadrupole moments of these states. The proton?proton distances show a progressive increase accompanied by a corresponding increase of the radius of the electron orbital. For the three bound states of the ?u rotational band of H2+, the interproton distance takes values larger than 17 a0. A similar study of the D2+ ion is also carried out.

Supersymmetry in a three-body model of halo nuclei

Abstract Supersymmetric transformations allow eliminating forbidden bound states from a potential without affecting its scattering properties. They are applied to the core-neutron potential in a three-body model of the 6 He , 11 Li and 14 Be halo nuclei and compared with the pseudo-potential method. The supersymmetric approach provides larger binding energies and smaller radii for these three nuclei. When the core-neutron interaction is readjusted to fit the experimental binding energy, the radii of the halo nuclei obtained with the two methods become very close. The 14 Be wave function qualitatively differs in both approaches.

Simple and accurate calculations on a Lagrange mesh of the hydrogen atom in a magnetic field

Five easily reproducible calculations on a mesh of the hydrogen atom in a magnetic field using three different coordinate systems are described. The calculations are performed with the Lagrange-mesh method, an approximate variational method resembling a mesh method and providing very accurate energies and wavefunctions. Highly accurate energies agreeing with the series expansion results are obtained for the lowest m = 0 to −10 states in magnetic fields γ = 1, 10, 100 and 1000. These calculations are easily reproducible in short computation times. The root-mean-square radius and the quadrupole moment are computed with high accuracy and discussed. The first and second positive-parity excited states and the lowest negative-parity state are also considered for m = 0. In all cases, the accuracy decreases with increasing fields.

Electromagnetic transitions of the hydrogen atom in a magnetic field by the Lagrange-mesh method

Very accurate wavefunctions obtained with the Lagrange-mesh method are employed to calculate dipole strengths and the dominant quadrupole strength in the high-field domain (γ ≥ 1) within the infinite-proton-mass approximation. Calculations are performed in the semi-parabolic and spherical coordinate systems. The accuracy is tested by varying the number of mesh points and by comparison between both coordinate systems. Comparison between the position (or length) and velocity expressions of the dipole strength is found not to be a reliable test as it overestimates the accuracy. We confirm that published tables have in general a four-digit accuracy. The Lagrange-mesh technique provides a simple and fast way of obtaining similar results at arbitrary fields with γ ≤ 1000. Using the semi-parabolic coordinate system is simpler and more robust.