  We propose Bethe equations for the diagonalization of the Hamiltonian of
quantum strings on AdS_5 x S^5 at large string tension and restricted to
certain large charge states from a closed su(2) subsector. The ansatz differs
from the recently proposed all-loop gauge theory asymptotic Bethe ansatz by
additional factorized scattering terms for the local excitations. We also show
that our ansatz quantitatively reproduces everything that is currently known
about the string spectrum of these states. Firstly, by construction, we recover
the integral Bethe equations describing semiclassical spinning strings.
Secondly, we explain how to derive the 1/J energy corrections of arbitrary
M-impurity BMN states, provide explicit, general formulae for both distinct and
confluent mode numbers, and compare to asymptotic gauge theory. In the special
cases M=2,3 we reproduce the results of direct quantization of Callan et al.
Lastly, at large string tension and relatively small charge we recover the
famous 2 (n^2 lambda)^(1/4) asymptotics of massive string modes at level n.
Remarkably, this behavior is entirely determined by the novel scattering terms.
This is qualitatively consistent with the conjecture that these terms occur due
to wrapping effects in gauge theory. Our finding does not in itself cure the
disagreements between gauge and string theory, but leads us to speculate about
the structure of an interpolating Bethe ansatz for the AdS/CFT system at finite
coupling and charge.
