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 $|n\left(x\right)〉$. We describe how the parametric kernel $P\left(n\mid m\right)=|〈n\left(x\right)\mid m\left(x_0\right)〉{|}^2$ evolves as a function of $\delta {x=x-x}_0$. 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.