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 c
ontrols a deformation of the boundary. The quantum eigenstates of the syste
m are \n(x)]. We describe how the parametric kernel P(n\m)=\[n(x)\m(x(0))]\
(2), also known as the local density of states, evolves as a function of de
ltax=x-x(0). We illuminate the nonunitary nature of this parametric evoluti
on, the emergence of nonperturbative features, the final nonuniversal satur
ation, and the limitations of random-wave considerations. The parametric ev
olution is demonstrated numerically for two distinct representative deforma
tion processes.