Selection rules for the four–sine overlap integral.

Consider

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

1. Parity rule. The integral vanishes unless the sum of the four mode numbers is even:
   $$I_{klmn}=0$$   if $$k+l+m+n$$    is odd$$.$$
   Equivalently, the interaction is nonzero only if an even number of the indices $k,l,m,n$ are odd.

2. Quartet (index–sum) rule. Nonzero contributions occur only for combinations satisfying
   $$\pm k \pm l \pm m \pm n = 0$$   or, more generally, an even integer$$.$$
   In practice this corresponds to conditions such as
   $$k+l \approx m+n, \qquad k \approx l+m+n,$$
   and permutations thereof.

3. Symmetry rule. The overlap integral is invariant under any permutation of the indices:
   $$I_{klmn}$$    is symmetric in $$(k,l,m,n).$$

4. Strength hierarchy. Among the admissible quartets, the coupling is strongest when two mode numbers nearly balance the other two (e.g. $k+l \approx m+n$); couplings decay as the mismatch between these sums increases.