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In a previous paper (hep-th/0509071), it was shown that quantum 1/Jcorrections to the BMN spectrum in an effective Landau-Lifshitz (LL) modelmatch with the results from the one-loop gauge theory, provided one chooses anappropriate regularization. In this paper we continue this study for theconjectured Bethe ansatz for the long range spin chain representingperturbative planar N=4 Super Yang-Mills in the SU(2) sector, and the ``quantumstring" Bethe ansatz for its string dual. The comparison is carried out forcorrections to BMN energies up to 3rd order in the effective expansionparameter $\tl=\lambda/J^2$. After determining the ``gauge-theory'' LL actionto order $\tl^3$, which is accomplished indirectly by fixing the coefficientsin the LL action so that the energies of circular strings match with theenergies found using the Bethe ansatz, we find perfect agreement. We interpretthis as further support for an underlying integrability of the system. We thenconsider the ``string-theory'' LL action which is a limit of the classicalstring action representing fast string motion on an S^3 subspace of S^5 andcompare the resulting $\tl^3/J^2$ corrections to the prediction of the``string'' Bethe ansatz. As in the gauge case, we find precise matching. Thisindicates that the LL Hamiltonian supplemented with a normal orderingprescription and zeta-function regularization reproduces the full superstringresult for the $1/J^2$ corrections, and also signifies that the string Betheansatz does describe the quantum BMN string spectrum to order $1/J^2$. We alsocomment on using the quantum LL approach to determine the non-analyticcontributions in $\lambda$ that are behind the strong to weak couplinginterpolation between the string and gauge results.Comment: 36 pages, v2: new Appendix C on non-analytic terms in the circular  solution example added, v3: typos correcte

1/J2corrections to BMN energies from the quantum long range Landau-Lifshitz model

1/J²corrections to BMN energies from the quantum long range Landau-Lifshitz model