Can a Hamiltonian be unbounded ?

[zetafunction]

the idea is can a Hamiltonian in 1-D of the form $$H=p^2 + V(x)$$ for a certain function V(x) be unbounded and have NEGATIVE energies , for example a Hamiltonian whose spectra may be $$E_{n} = ...,-3,-2,-1,1,2,3,...$$ and so on, so we have an UNBOUNDED Hamiltonian with positive and negative energies with the property

$$E_{-n}=-E_{n}$$

 

[tom.stoer]

I think a Hamiltonian must always be bounded from below, b/c otherwiese it would be possible (e.g. due to perturbations) that there is a non-zero possibility for a state |n> to decay to |n-k>, where n is not bounded from below. So the whole system is unstable and decays into a singular state plus infinitly many photons, phonons or whatever.

 

[dextercioby]

In QFT, boundedness from below is a postulated for an acceptable Hamiltonian. In nonrelativistic physics, it is a derived result for each analyzed system.