Unique properties of Hamiltonian

I think the Question is what is meant by "Hamiltonian", when not working in a context where the physical interpretation is immediate. However, you could ask under which circumstances a given system could be given a physical interpretation, and this will lead to two requirements, as far as i can see. Therefore assume $O$ is an operator on a Hilbert space $H$.

1. $O$ should be densely defined, self-adjoint. The reasoning behind this is that $O$ should generate a strongly continuous one-parameter group of unitaries $t mapsto U(t)$, and Stone's theorem singles out these operators. In physical terms, $U(t)$ is a time evolution of a system generated by $O$. In a physical system, a Hamiltonian is precisely that: the generator of time evolution, therefore an operator that does not generate a time evolution is no sensible Hamiltonian.

2. $O$ should be bounded below. Here the motivation is stability: a physical system should be be stable in the sense that one cannot extract infinitely much energy out of it.