Orthogonal polynomial expansion of the spectral density operator and the calculation of bound state energies and eigenfunctions

Abstract

An orthogonal polynomial expansion method is presented, and illustrated with calculations, for calculating δ( E– H), the spectral density operator (SDO), the projection operator that projects out of any L 2 wavepacket the eigenstate (s) of H having energy E. If applied to an L 2 wavepacket which overlaps the interaction, it yields either scattering-type (improper) eigenstates or proper bound eigenstates. For negative energies, the exact SDO yields zero away from an eigenvalue, and yields the energy eigenstate (times a constant) when E equals an eigenvalue. The finite orthogonal polynomial expansion of the SDO, acting on an L 2 wavepacket, yields approximately zero for E not equal to an eigenvalue, and becomes nonzero in the neighborhood of an eigenvalue.