Discrete energy representation and generalized propagation of physical systems

This work discusses the discrete energy representation based on generalized propagation of a physical system. Here, the propagation is defined as a recursion scheme which generates a series of system states from a given initial state. Examples of such schemes include the time propagation and polynomial recursion. It is argued that each propagation determines a set of energy points, which form the discrete energy representation. A unitary transformation can be established between the discrete energy representation and the generalized time representation, much like the well-known transformation between the discrete variable representation and the finite basis representation. Such a collocation approach can be useful in calculating many properties that are local in the energy domain. Numerical examples are presented to demonstrate the utility in filter diagonalization.