  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.
