Confined helium on Lagrange meshes.

The Lagrange-mesh method has the simplicity of a calculation on a mesh and can have the accuracy of a variational method. It is applied to the study of a confined helium atom. Two types of confinement are considered. Soft confinements by potentials are studied in perimetric coordinates. Hard confinement in impenetrable spherical cavities is studied in a system of rescaled perimetric coordinates varying in [0,1] intervals. Energies and mean values of the distances between electrons and between an electron and the helium nucleus are calculated. A high accuracy of 11 to 15 significant figures is obtained with small computing times. Pressures acting on the confined atom are also computed. For sphere radii smaller than 1, their relative accuracies are better than 10(-10). For larger radii up to 10, they progressively decrease to 10(-3), still improving the best literature results.

Tensor force manifestations in ab initio study of the 2H(d,γ)4He, 2H(d,p)3H, and 2H(d,n)3He reactions.

The (2)H(d,p)(3)H, (2)H(d,n)(3)He, and (2)H(d,γ)(4)He reactions are studied at low energies in a multichannel ab initio model that takes into account the distortions of the nuclei. The internal wave functions of these nuclei are given by the stochastic variational method with the AV8' realistic interaction and a phenomenological three-body force included to reproduce the two-body thresholds. The obtained astrophysical S factors are all in very good agreement with the experiment. The most important channels for both transfer and radiative capture are identified by comparing to calculations with an effective central force. They are all found to dominate thanks to the tensor force.

Confined hydrogen atom by the Lagrange-mesh method: energies, mean radii, and dynamic polarizabilities.

The Lagrange-mesh method is an approximate variational calculation which resembles a mesh calculation because of the use of a Gauss quadrature. The hydrogen atom confined in a sphere is studied with Lagrange-Legendre basis functions vanishing at the center and surface of the sphere. For various confinement radii, accurate energies and mean radii are obtained with small numbers of mesh points, as well as dynamic dipole polarizabilities. The wave functions satisfy the cusp condition with 11-digit accuracy.

Sixth-order factorization of the evolution operator for time-dependent potentials.

The evolution operator of a quantum system in a time-dependent potential is factorized in unitary exponential operators at order 6. This expression is derived with the time-ordering method. It is compared with lower-order factorizations on several simple one-dimensional examples. Better accuracies are reached at sixth order for a given time step than at lower orders. Due to a significant increase of computation duration per time step, the sixth-order approximation is mainly useful when high accuracies are required.

The unexplained accuracy of the Lagrange-mesh method.

The Lagrange-mesh method is an approximate variational calculation which resembles a mesh calculation because of the use of a Gauss quadrature. In order to analyze its accuracy, four different Lagrange-mesh calculations based on the zeros of Laguerre polynomials are compared with exact variational calculations based on the corresponding Laguerre basis. The comparison is performed for three solvable radial potentials: the Morse, harmonic-oscillator, and Coulomb potentials. The results show that the accuracies of the energies obtained for different partial waves with the different mesh approximations are very close to the variational accuracy, even in the presence of the centrifugal singularity. The same property holds for the approximate wave functions. This striking accuracy remains unexplained.

Semirelativistic Lagrange mesh calculations.

The Lagrange mesh method is a very powerful procedure to compute eigenvalues and eigenfunctions of nonrelativistic Hamiltonians. The trial eigenstates are developed in a basis of well-chosen functions and the computation of Hamiltonian matrix elements requires only the evaluation of the potential at grid points. It is shown that this method can be used to solve semirelativistic two-body eigenvalue equations. As in the nonrelativistic case, it is very accurate, fast, and very simple to implement.

Resolution of the Gross-Pitaevskii equation with the imaginary-time method on a Lagrange mesh.

The Lagrange-mesh method is an approximate variational calculation which has the simplicity of a mesh calculation. Combined with the imaginary-time method, it is applied to the iterative resolution of the Gross-Pitaevskii equation. Two variants of a fourth-order factorization of the exponential of the Hamiltonian and two types of mesh (Lagrange-Hermite and Lagrange-sinc) are employed and compared. The accuracy is checked with the help of these comparisons and of the virial theorem. The Lagrange-Hermite mesh provides very accurate results with short computing times for values of the dimensionless parameter of the nonlinear term up to 104. For higher values up to 107, the Lagrange-sinc mesh is more efficient. Examples are given for anisotropic and nonseparable trapping potentials.

15C-15F Charge symmetry and the 14C(n,gamma)15C reaction puzzle.

The low-energy reaction 14C(n,gamma)15C provides a rare opportunity to test indirect methods for the determination of neutron capture cross sections by radioactive isotopes versus direct measurements. It is also important for various astrophysical scenarios. Currently, puzzling disagreements exist between the 14C(n,gamma)15C cross sections measured directly, determined indirectly, and calculated theoretically. To solve this puzzle, we offer a strong test based on a novel idea that the amplitudes for the virtual 15C-->14C + n and the real 15F -->14O + p decays are related. Our study of this relation, performed in a microscopic model, shows that existing direct and some indirect measurements strongly contradict charge symmetry in the 15C and 15F mirror pair. This brings into question the experimental determinations of the astrophysically important (n,gamma) cross sections for short-lived radioactive targets.

Collisions of halo nuclei within a dynamical eikonal approximation.

The dynamical eikonal approximation unifies the semiclassical time-dependent and eikonal methods. It allows calculating differential cross sections for elastic scattering and breakup in a quantal way by taking into account interference effects. Good agreement is obtained with experiment for 11Be breakup on 208Pb. Dynamical effects are weak for elastic scattering.

Lagrange meshes from nonclassical orthogonal polynomials.

The Lagrange-mesh numerical method has the simplicity of a mesh calculation and the accuracy of a variational calculation. A flexible general procedure for deriving an infinity of new Lagrange meshes related to orthogonal or nonorthogonal bases is introduced by using nonclassical orthogonal polynomials. As an application, different Lagrange meshes based on shifted Gaussian functions are constructed. A simple quantum-mechanical example shows that the Lagrange-mesh method may become more accurate than the original variational calculation with a nonorthogonal basis.