Formal Relationships Between the Bound States of Spatially Confined and Unconfined Quantum Systems

Formal results about the properties of confined and unconfined quantum systems are discussed. The results provide a rigorous theoretical foundation to many numerical results (that consider from one-dimensional problems to atoms and molecules) which show that the properties of a system confined into a box Ω with impenetrable walls, converge to those of the unconfined system as Ω increases, independently of the shape of Ω. The analysis starts from the Schroedinger equation so that its extrapolation to density functional theory is immediate. The similarity between the properties of confined and unconfined systems provides a powerful scheme to compute the bound states of the latter. This approach solves in a natural way several problems posed by the standard numerical methods to compute the eigenstates of a Schroedinger operator