Are there formal ways to bound the following quantity:
The expectation value is taken on an eigenstate of $S_x$.
$H$ is a dipolar Hamiltonian acting on $N$ distinguishable spin-3 particles. $S_x=\sum_i^NS_x^i$ is the sum of $N$ spin-3 operators acting on distinguishable particles. $H$ and $S_x$ do not commute.
I am looking for formal and general ways to bound these quantities, so I do not think the details of the operators or the system are needed, let me know if I am wrong.