Parametric evolution for a deformed cavity

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(x(0))]\(2), also known as the local density of states, evolves as a function of deltax=x-x(0). We illuminate the nonunitary nature of this parametric evolution, the emergence of nonperturbative features, the final nonuniversal saturation, and the limitations of random-wave considerations. The parametric evolution is demonstrated numerically for two distinct representative deformation processes.