DIRECT CALCULATION OF ENERGY EIGENVALUE SPECTRA

Author Institution: University of Michigan, Ann ArborAn arbitrary function $\Psi (q)$, obeying the analyticity and boundary conditions appropriate to the eigenfunctions of some dynamical system, may be regarded as the initial configuration of a non-stationary state. The formal solution of the time-dependent $Schr\""{o}$dinger equation, which describes the time-evolution of this hypothetical state, is given by $\begin{equation}\Psi (q, t) =\sum\limits^{\infty}_{n=0}\frac{(-it)}{k!}H^{k} \Psi (q)\end{equation}$ which is equivalent to the Fourier development $\begin{equation}\Psi (q, t) = \sum^{\infty}_{n=0}=c_{n}\phi_{n}(q)e^{-1 \omega_{n}^{t}}+\int^{\infty}_{0}d \omega c(\omega, q)e^{- \omega t}\end{equation}$ where $\phi_{n} (q), \phi(\omega, q)$ are the discrete and continuum eigenstates, respectively. The Fourier transform of $F (t) = $ exhibits, in principle, the entire eigenvalue spectrum: $\begin{equation}G (\omega) = \int^{\infty}_{-\infty}dt\ e^{-i \omega t} F (t)=2 \pi \sum^{\infty}_{n=0} |c_{n}|^{2}\delta(\omega + \omega_{n})+2 \pi|c(-\omega)|^{2}\end{equation}$ A practical method based on analogous transformation of a truncated summation (1) has been developed. The spectral function analogous to (3) may be shown to converge only if the expansion coefficients $c_{a}, c(\omega)$ decrease with $\omega$ as $e-(\omega/\lambda)^{2}$ or faster. Applications have been made to spin-Hamiltonian systems