  Joint distribution function of N eigenvalues of U(N) invariant random-matrix
ensemble can be interpreted as a probability density to find N fictitious
non-interacting fermions to be confined in a one-dimensional space. Within this
picture a general formalism is developed to study the eigenvalue correlations
in non-Gaussian ensembles of large random matrices possessing non-monotonic,
log-singular level confinement. An effective one-particle Schroedinger equation
for wave-functions of fictitious fermions is derived. It is shown that
eigenvalue correlations are completely determined by the Dyson's density of
states and by the parameter of the logarithmic singularity. Closed analytical
expressions for the two-point kernel in the origin, bulk, and soft-edge scaling
limits are deduced in a unified way, and novel universal correlations are
predicted near the end point of the single spectrum support.
