Selection rules for the triple–sine overlap integral.

Consider

$\displaystyle I_{klm} = \int_{0}^{L} \sin\!\left(\frac{k\pi x}{L}\right) \sin\!... ...ight) \sin\!\left(\frac{m\pi x}{L}\right)\,dx, \qquad k,l,m\in\mathbb{Z}_{>0}.$

Parity rule. The integral vanishes unless the sum of the three mode numbers is odd:

$\displaystyle I_{klm}=0$ if $\displaystyle k+l+m$ is even $\displaystyle .$
Equivalently, the interaction is nonzero only if an odd number of the indices are odd.
Triad (index–sum) rule. A nonzero contribution arises only from combinations satisfying

$\displaystyle \pm k \pm l \pm m =$ odd integer $\displaystyle ,$
which correspond to near-resonant conditions $m \approx k+l$ or $m \approx \vert k-l\vert$ .
Symmetry rule. The overlap integral is invariant under any permutation of the indices:

$\displaystyle I_{klm}=I_{lmk}=I_{mkl}.$
Strength hierarchy. Among the admissible triads, the coupling coefficient is largest when one mode number is close to the sum or the absolute difference of the other two; triads far from these conditions are progressively weaker.

$\begin{indisplay}\boxed{ \begin{aligned} I_{klmn} \;\equiv\; &\int_{0}^{L} \sin\... ...\delta_{k-l+m+n,\,2r} -\delta_{k+l+m+n,\,2r} \Big) \end{aligned}}\end{indisplay}$

$\displaystyle \boxed{ I_{klmn} = \frac{L}{8} \sum_{\text{all }\pm} (\pm)(\pm)(\pm)\; \delta_{\,k \pm l \pm m \pm n,\, \text{even}} }$

Another version

Examples - SUM part:

$\displaystyle (1,2,3),\ (2,3,5),\ (3,4,7).$

Example - DIFFERENCE:

$\displaystyle (3,1,2),(5,3,2),(7,4,3).$