In some cases, such as finite and infinite square wells, the Hamiltonian has energy eigenstates which correspond to physical wavefunctions.
In other cases, such as a one dimensional universe with constant potential, it doesn't. It has the plane wave, but that's not normalizable. (Also, for all Hamiltonians, the constant zero wavefunction is an eigenstate, but that's equally non normalizable.)
Is there any physical meaning to whether the Hamiltonian has energy eigenstates? Under what sort of situations does it have energy eigenstates?