Semiclassical analytical solutions of the eigenstate thermalization hypothesis in a quantum billiard

We derive semiclassical analytical solutions for both the diagonal and off-diagonal functions in the eigenstate thermalization hypothesis (ETH) in a quarter-stadium quantum billiard. For a representative observable, we obtain an explicit expression and an asymptotic closed-form solution that naturally separate into a local contribution and a phase-space correlation term. These analytical results predict the band structure of the observable matrix, including its bandwidth and scaling behavior. We further demonstrate that our analytical formula is equivalent to the prediction of Berry's conjecture. Supported by numerical evidence, we show that Berry's conjecture captures the energetic long-wavelength behavior in the space of eigenstates, while our analytical solution describes the asymptotic behavior of the f function in the semiclassical limit. Finally, by revealing the connection between the bandwidth scaling and the underlying classical dynamics, our results suggest that the ETH carries important physical implications in single-particle and few-body systems, where "thermalization" manifests as the loss of information about initial conditions.