Unification of perturbation theory, random matrix theory, and semiclassical considerations in the study of parametrically dependent eigenstates

We consider a classically chaotic system that is described by a Hamiltonian H (Q, P; x), where x is a constant parameter. Specifically, we discuss a gas particle inside a cavity, where x controls a deformation of the boundary or the position of a "piston." The quantum eigenstates of the system are |n(x)>. We describe how the parametric kernel P(n | m) = |n(x) | m(x ₀)| ² evolves as a function of δx = x - x ₀. We explore both the perturbative and the nonperturbative regimes, and discuss the capabilities and the limitations of semiclassical as well as random waves and random-matrix-theory considerations.