## Bethe Ansatz for Quantum String

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.