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

We study a classically chaotic system which is described by a Hamiltonian $H(Q,P;x)$ where $(Q,P)$ are the canonical coordinates of a particle in a 2D 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 $x-x0$. This kernel, regarded as a function of $n-m$, characterizes the shape of the wavefunctions, and it also can be interpreted as the local density of states (LDOS). The kernel $P(n|m)$ has a well defined classical limit, and the study addresses the issue of quantum-classical correspondence (QCC). We distinguish between restricted QCC and detailed QCC. Both the perturbative and the non-perturbative regimes are explored. The limitations of the random-matrix-theory (RMT) approach are demonstrated.