One-Dimensional Algorithm for Finding Eigenbasis of the Schrödinger Operator
I. M. Livshits · SIAM Journal on Scientific Computing · 2008
Schrödinger equations are used to model numerous applications arising in quantum chemistry and physics. Most of these applications have no analytical solutions and need to be solved numerically, often an extremely challenging task. This paper offers an efficient multigrid/multiscale solver for the one-dimensional Schrödinger eigenvalue problem, a preliminary step toward developing solvers to application-rich two-dimensional problems. The solver employs a gradual multiscale eigenbasis representation that allows calculation and storage of only a small number of eigenfunction representatives on the expensive finest grids and a full eigenbasis representation and calculation on the coarsest grids. This structure not only allows calculation and storage of the entire eigenbasis in $O(N\log N)$ operations, but it is also beneficial for many applications. The algorithms will be eventually adapted for solving the Schrödinger equation as it appears in the Kohn–Sham formulation for calculating electronic structure, though in this paper the nonlinearity of the problem (dependence of potentials on low eigenfunctions) is not discussed.