On the calculation of absolute spectral densities

Abstract

A new method of calculating the absolute spectral density of a Hamiltonian operator is derived and discussed. The spectral density is expressed as the solution of an integral equation in which the kernel is a renormalized one-sided energy correlation function of the full microcanonical density operator and a microcanonical density operator for a reference Hamiltonian. The integral operator associated with this equation transforms a known spectral density function for the reference Hamiltonian into the spectral density of the full Hamiltonian. The integral equation, by virtue of its formulation in energy space, is inherently one-dimensional and offers no storage difficulties, and the elements of its kernel may be computed by applying the Lanczos algorithm to randomly selected eigenfunctions of the reference Hamiltonian. This spectral density correlation method offers a number of advantages over variational methods. In particular, it has the potential for overcoming the hitherto largely insurmountable problem of tracing over a multidimensional Hilbert space in order to compute the spectral density of a nonseparable molecular Hamiltonian.