  Unitary random matrix ensembles Z_{n,N}^{-1} (\det M)^alpha exp(-N Tr V(M))
dM defined on positive definite matrices M, where alpha > -1 and V is real
analytic, have a hard edge at 0. The equilibrium measure associated with V
typically vanishes like a square root at soft edges of the spectrum. For the
case that the equilibrium measure vanishes like a square root at 0, we
determine the scaling limits of the eigenvalue correlation kernel near 0 in the
limit when n, N tend to infinity such that n/N - 1 = O(n^{-2/3}). For each
value of alpha > -1 we find a one-parameter family of limiting kernels that we
describe in terms of the Hastings-McLeod solution of the Painleve II equation
with parameter alpha + 1/2.
