Integrals for Lower Bounds to the Exact Energy

Abstract

The analytical solutions to quantum few-body systems are rarely found meaning that, for atoms and molecules, stationary states are found by numerical computations. If high precision is required then the computations can be expensive and, in contrast to experiment, it can be difficult to attach rigorous error bounds to results. The exact energy of quantum states can be estimated using the variational method which provides an upper bound to the exact energy. Achieving lower bounds of comparable quality to upper bounds would then provide an energy range within which the exact energy eigenvalue would reside. The formal theory of lower bounds is well established, but the numerical application to systems occurs less frequently compared to that of upper bounds.