  We consider a classically chaotic system that is described by a Hamiltonian
H(Q,P;x), where (Q,P) describes a particle moving inside a cavity, and x
controls a deformation of the boundary. The quantum-eigenstates of the system
are |n(x)>. We describe how the parametric kernel P(n|m) = <n(x)|m(x0)>, also
known as the local density of states, evolves as a function of x-x0. We
illuminate the non-unitary nature of this parametric evolution, the emergence
of non-perturbative features, the final non-universal saturation, and the
limitations of random-wave considerations. The parametric evolution is
demonstrated numerically for two distinct representative deformation processes.
