Mathematical and Numerical Methods for Complex Quantum Systems

Solving the Schroedinger equation using Smolyak interpolants

Tucker Carrington

Queen's University


I present a new collocation method for solving the Schroedinger equation. Col- location has the advantage that is obviates integrals. Previous collocation methods have, however, all had the crucial disadvantage that they require solving a generalized eigenvalue problem. By combining Lagrange-like functions with a Smolyak interpolant we device a collocation method that does not require solving a generalized eigenvalue problem. We exploit the structure of the basis and the grid to develop an efficient algorithm for evaluating the matrix-vector products required to compute energy levels and wavefunctions. Energies systematically converge as the number of points and basis functions is increased.