= Contact interaction between two box modes
{title2=$I_{1111}=I_{2222}=3/(2L),\quad I_{1122}=1/L$}
For distinct box modes $\varphi_1=\sqrt{2/L}\sin(p\pi x/L)$ and $\varphi_2=\sqrt{2/L}\sin(q\pi x/L)$, the displayed <contact interaction> overlaps produce self terms $3g\sum_jn_j(n_j-1)/(4L)$, a density term $2gn_1n_2/L$, and pair conversion $g(a_1^{\dagger2}a_2^2+a_2^{\dagger2}a_1^2)/(2L)$. Pair conversion preserves total <particle number> but prevents diagonalization by individual <occupation numbers>. The additional overlaps are $I_{1112}=-\delta_{q,3p}/(2L)$ and $I_{1222}=-\delta_{p,3q}/(2L)$, following from $\sin^3u=(3\sin u-\sin3u)/4$. Third-harmonic mode pairs therefore also permit single-particle transfer assisted by the other particles. For the first two box modes these additional terms vanish.
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