Fourier grid Hamiltonian method and Lagrange-mesh calculations

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Abstract
Bound state eigenvalues and eigenfunctions of a Schrödinger equation or a spinless Salpeter equation can be simply and accurately computed by the Fourier grid Hamiltonian (FGH) method. It requires only the evaluation of the potential at equally spaced grid points, and yields the eigenfunctions at the same grid points. The Lagrange-mesh (LM) method is another simple procedure to solve a Schrödinger equation on a mesh. It is shown that the FGH method is a special case of a LM calculation in which the kinetic energy operator is treated by a discrete Fourier transformation. This gives a firm basis for the FGH method and makes possible the evaluation of the eigenfunctions obtained with this method at any arbitrary values.