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

Abstract

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 Lagrange-mesh bases size increases. The Lagrange-mesh solutions converge to the exact solution when the parameters of the matrix elements are chosen appropriately.