### abstract

- 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 $P(n\ensuremath{\mid}m)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}|〈n(x)\ensuremath{\mid}m({x}_{0})〉{|}^{2}$ evolves as a function of $\ensuremath{\delta}{x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}x\ensuremath{-}x}_{0}$. 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.