Confusion on Hamiltonian unbounded from below and Ostrogradsky Instability

Question

This might be a silly question but I failed to get it. In Ostrogradsky Instability, we deduced that Lagrangian of higher-order derivatives leads to Hamiltonian linear to canonical momenta, and thus, Hamiltonian is unbounded from below. However, a particle in potential: $V(r)=-\frac{1}{r}$ also has a Hamiltonian unbounded from below as $r$ approaches to zero, so why doesn’t it lead to an instability similar to Ostrogradsky instability? What makes them different?

Answer 1

What is important is that the spectrum of the Hamiltonian operator $\hat{H}$ is bounded from below, and that it has a ground state; not that the classical Hamiltonian $H$ is bounded from below in the classical phase space. E.g. quantum mechanically the hydrogen atom has a ground state despite the Coulomb potential is unbounded from below.