Unification of perturbation theory, RMT and semiclassical considerations in the study of parametrically-dependent eigenstates
Abstract
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
|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 non-perturbative regimes, and discuss the capabilities and the limitations of semiclassical as well as of random-waves and random-matrix-theory (RMT) considerations.