Parametric evolution for a deformed cavity

We consider a classically chaotic system that is described by a Hamiltonian (Formula presented) where (Formula presented) describes a particle moving inside a cavity, and x controls a deformation of the boundary. The quantum eigenstates of the system are (Formula presented) We describe how the parametric kernel (Formula presented) also known as the local density of states, evolves as a function of (Formula presented) 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.