Unification of perturbation theory, RMT and semiclassical considerations in the study of parametrically-dependent eigenstates

We consider a classically chaotic system that is described by an Hamiltonian $H(Q,P;x)$ where x is a constant parameter. Our main interest is in the case of 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 $\mathrm{\mid }n(x)>$. We describe how the parametric kernel $P(n\mathrm{\mid }m)=\mathrm{\mid }<n(x)\mathrm{\mid }m(x_0)>{\mathrm{\mid }}^2$ evolves as a function of $\delta x=(x-x_0)$. We explore both the perturbative and the non-perturbative regimes, and discuss the capabilities and the limitations of semiclassical as well as of random-waves and random-matrix-theory (RMT) considerations.