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

We study a classically chaotic system that is described by a Hamiltonian H(Q,P;x), where (Q,P) 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 \|n(x)>. We analyze numerically how the parametric kernel P(n\|m)=\|<n(x)\|m(x₀)>\|² evolves as a function of δx≡(x-x₀). This kernel, regarded as a function of n-m, characterizes the shape of the wave functions, and it also can be interpreted as the local density of states. The kernel P(n\|m) 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.