ORTHOGONAL POLYNOMIAL EXPANSION OF THE SPECTRAL DENSITY OPERATOR AND THE CALCULATION OF BOUND-STATE ENERGIES AND EIGENFUNCTIONS

An orthogonal polynomial expansion method is presented, and illustrated with calculations, for calculating delta(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.