Parametric dependent Hamiltonians, wave functions, random matrix theory, and quantal-classical correspondence

We study a classically chaotic system that is described by a Hamiltonian (Formula presented) where (Formula presented) are the canonical coordinates of a particle in a two-dimensional well, and x is a parameter. By changing x we can deform the “shape” of the well. The quantum eigenstates of the system are (Formula presented) We analyze numerically how the parametric kernel (Formula presented) evolves as a function of (Formula presented) This kernel, regarded as a function of (Formula presented) characterizes the shape of the wave functions, and it also can be interpreted as the local density of states. The kernel (Formula presented) has a well-defined classical limit, and the study addresses the issue of quantum-classical correspondence. Both the perturbative and the nonperturbative regimes are explored. The limitations of the random matrix theory approach are demonstrated.