  We study the double scaling limit for unitary invariant ensembles of random
matrices with non analytic potentials and find the asymptotic expansion for the
entries of the corresponding Jacobi matrix. Our approach is based on the
perturbation expansion for the string equations. The first order perturbation
terms of the Jacobi matrix coefficients are expressed through the
Hastings-McLeod solution of the Painleve II equation. The limiting reproducing
kernel is expressed in terms of solutions of the Dirac system of differential
equations with a potential defined by the first order terms of the expansion.
