and an order $g_2^2$ contact term. For large finite $J$ the matrix elements are V\_[\_k]{}\^[’\_[k+1]{}]{}&=& () \_[ss’]{}\[Vcoef\]\
V\_[\_[k+1]{}]{}\^[’\_[k]{}]{}&=& ()\_[s’s]{}\[Vcoef2\] where $\Delta_{ss'}$ is a product of delta functions, \[delta\] \_[ss’]{}= \_[PS\_[k+1]{}]{}\_[s\_1,s’\_[P(1)]{}]{} \_[s\_[k]{},s’\_[P(k)]{}]{}\_[s\_0,s’\_0+s’\_[P(k+1)]{}]{}.

It is in this form that we will use these equations. In the BMN limit, matrix elements we calculate agree with those of the continuum formulation of ([@golm1]).

We have also obtained a version of (\[heff\]) valid at any finite $J$ in the $\phi \leftrightarrow \psi$ symmetric channel. Eigenoperators of $H_0$ are superpositions of those of fixed spacing, namely \^[J\_0,J\_1…J\_k]{}\_n = \_[l=0]{}\^[J\_0]{} ([n (2l+1) J\_0+1]{}) \^[J\_0,J\_1…J\_k]{}\_l These operators have eigenvalues of $H_0$ given by ${g_{YM}^2 N
\over \pi^2} \sin^2({\pi n \over J_0 +1}) $ which approaches the eigenvalue in (\[heff\]) as $J_0 \to \infty$. These results agree with those of [@beisert; @minahan]. The interaction terms still describe order $g_2$ splitting/joining of traces, but they are more complicated than those of (\[heff\]). For example when $H_+$ acts on the the single-trace $\co^J_n$ one obtains a superposition of double-trace operators $\co^{J_0,J_1}_m$ where $J_0=J-J_1$ with expansion coefficients ([m J\_0+1]{}) A(n,m,J\_1) A(n,m,J\_1) = ([ n J\_1 J+1]{}){\[\] - \[ m -m\] } One notes that these coefficients exactly reduce to (\[Vcoef\]) in the 