A paradox of grid-based representation techniques: accurate eigenvalues from inaccurate matrix elements

Several approximately variational grid-based representation techniques devised to solve the time-independent nuclear-motion Schrödinger equation share a similar behavior: while the computed eigenpairs, the only results which are of genuine interest, are accurate, many of the underlying Hamiltonian matrix elements are inaccurate, deviating substantially from their values in a variational basis representation. Examples are presented for the discrete variable representation and the Lagrange-mesh approaches, demonstrating that highly accurate eigenvalues and eigenfunctions can be obtained even if some or even all of the Hamiltonian matrix elements in these grid-based representations are inaccurate. It is shown how the apparent contradiction of obtaining accurate eigenpairs with far less accurate individual matrix elements can be resolved by considering the unitary transformation between the representations. Furthermore, the relations connecting orthonormal bases and the corresponding Lagrange bases are generalized to relations connecting nonorthogonal, regularized bases and the corresponding nonorthogonal, regularized Lagrange bases.