A useful approximation for the thermal operator $\text{exp}\left(-\beta \widehat{H}\right)$ is based on its representation in terms of either frozen or thawed Gaussian states. Such approximate representations are leading-order terms in respective series representations of the thermal operator. A numerical study of the convergence properties of the frozen Gaussian series representation has been recently published. In this paper, we extend the previous study to include also the convergence properties of the more expensive thawed Gaussian series representation of the thermal operator. We consider three different formulations for the series representation and apply them to a quartic double-well potential to find that the thawed Gaussian series representation converges faster than the frozen Gaussian one. Further analysis is presented as to the convergence properties and the numerical efficiency of three different thawed Gaussian series representation. The unsymmetrized form converges most rapidly, however, the lower order approximations of the symmetrized forms are more accurate. Comparison with a standard discretized path-integral evaluation demonstrates that the Gaussian based perturbation series representation converges much faster.

• Received 20 December 2009

DOI:https://doi.org/10.1103/PhysRevE.81.036704