Semiclassical study of diagonal and offdiagonal functions in the eigenstate thermalization hypothesis

The so-called eigenstate thermalization hypothesis (ETH), which has been tested in various manybody models by numerical simulations, supplies a way of understanding eventual thermalization and is believed to be important for understanding processes of thermalization. Two functions play important roles in the application of ETH, one for averaged diagonal elements and the other for the variance of offdiagonal elements of an observable addressed by ETH on the energy basis. For the former function, a semiclassical expression is known of the zeroth order of hbar, while, little is known analytically for the latter. In this paper, a semiclassical expression is derived for the former function, which includes higher-order contributions of hbar. And, a semiclassical approximation is derived for the latter function, under the assumption of negligible correlations among energy eigenfuntions on an action basis. Relevance of the analytical predictions are tested numerically in the Lipkin-Meshkov-Glick model.