\D \o_n$ in (\[toy\]) agrees with the order $g_2^2$ contribution to the scale dimension of single-trace BMN operators in (\[eig1\]).

The generalized model includes three types of states: i) the discrete “single-trace” $|n \>$, ii)  continuous “double-trace” $|\a_2 \>$, and iii)  continuous “triple-trace” $|\a_3 \>$. The free Hamiltonian $H_0$ is diagonal with energy eigenvalues $E_n, E_{\a_2}$, and $
E_{\a_3}$, respectively. The interaction matrix and time-dependent state vector which generalize those of the model above are: \[int\] V&=&(

[ccc]{} 0 & V\^i\_[\_2]{} & 0\
V\^[\_2]{}\_i & 0 & V\^[\_2]{}\_[\_3]{}\
0 & V\^[\_3]{}\_[\_2]{} & 0

),\
|(t) &=&(

[c]{} a\^i(t)e\^[-iE\_it]{}\
(\_2,t)e\^[-iE\_[a\_2]{}t]{}\
(\_3,t)e\^[-iE\_[a\_3]{}t]{}

). Again we need the solution of the Schrödinger equation $ i d / dt
|\Psi(t) \ra  = (H_0+V) |\Psi(t) \ra $ with initial conditions: $a^m(0) = \d^m_n,~\phi^{\a_2}(0)=0, \s^{\a_3}(0)=0$. The equations linking the components of $|\Psi(t) \ra$ are i [d dt]{} a\^n(t) &=& \_2 V\^n\_[\_2]{} e\^[iE\_[n\_2]{}t]{}  \^[\_2]{}(t)\[one\]\
i \^[\_2]{}(t) &=& \_m V\^[\_2]{}\_m e\^[iE\_[\_2m]{}t]{}  a\^m(t) + \_3 V\^[\_2]{}\_[\_3]{} e\^[ iE\_[\_2\_3]{}t]{}  \^[\_3]{}(t)\[two\]\
i \^[\_3]{}(t) &=& \_2 V\^[\_3]{}\_[\_2]{} e\^[ iE\_[\_3\_2]{}t]{}  \^[\_2]{}(t) \[three\] We use the notation $E_{n\a_2} = E_n - E_{\a_2}$, etc. for differences of energy.

We now wish to process the information in (\[one\]-\[three\]) to obtain a relation describing the coupling of 