Moment expansion and eigenvalues

Jay D. Mancini, Yu Zhou, P. F. Meier +2 more

Jay D. Mancini, William J. Massano, Janice D. Prie +1 more

J. M. Noriega, Luis Pesquera, M. A. Rodrguez

The steady-state intensity correlation functions C(t) of the single-mode laser are analyzed with the generalized-moment-expansion method proposed by Nadler and Schulten [Z. Phys. B 59, 53 (1985); J. Chem. Phys. 82, 151 (1985)]. This method approximates C(t) by a superposition of N exponentials that exactly reproduces n derivatives of C(t) at t=0 and m relaxation moments (n+m+1=2N). In this way, the short- and long-time regimes of C(t) are well described even near threshold. With the use of relaxation moments we develop a method to obtain the eigenvalues of the Fokker-Planck operator. This method is applied to the single-mode laser. We show that the knowledge of only two relaxation moments is enough to obtain good results for the two first eigenvalues. The method gives excellent results when the eigenvalues are well separated (below threshold).

Daniel M. Greenberger, Klaus Hentschel, Friedel Weinert

Vassilios Fessatidis, Jay D. Mancini, Samuel P. Bowen +1 more

S. M. Blinder

Abstract A wave function which is other than an exact eigenfunction, if it obeys appropriate analytical conditions, can be considered to represent the initial configuration of a nonstationary state. In the course of its subsequent time development the quantum system exhibits implicitly its entire eigenvalue spectrum. A method based, in principle, on Fourier analysis of the evolving quantum system is applied to the direct calculation of the energy eigenvalue spectrum. The spectral function is expressed as a moment expansion, in terms of expectation values of powers of the Hamiltonian. When the expansion is truncated, as it must be in any practical application, the corresponding spectral function represents a smeared‐out eigenvalue spectrum. An alternative approximation leads to the quantum‐mechanical method of moments. As the number of terms is increased, the computed spectrum becomes sharper and more accurate. In certain cases the moment expansion can be circumvented, if the action of the evolution operator can be evaluated in closed form. This is equivalent to finding some solution of the time‐dependent Schrödinger equation. The various methods of eigenvalue spectroscopy are applied to the harmonic oscillator.