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(nm)0ex0ex=0ex0ex|〈n(x)m(x₀)〉|² evolves as a function of δx0ex0ex=0ex0exx-x₀. 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.