  We find explicit expressions for two first finite size corrections to the
distribution of Bethe roots, the asymptotics of energy and high conserved
charges in the sl(2) quantum Heisenberg spin chain of length J in the
thermodynamical limit J->\infty for low energies E\sim 1/J. This limit was
recently studied in the context of integrability in perturbative N=4
super-Yang-Mills theory. We applied the double scaling technique to Baxter
equation, similarly to the one used for large random matrices near the edge of
the eigenvalue distribution. The positions of Bethe roots are described near
the edge by zeros of Airy function. Our method can be generalized to any order
in 1/J. It should also work for other quantum integrable models.
