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.