If I find a Hamiltonian $H = \sum_{k} \varepsilon_k a_k^{\dagger} a_k + \sum_k V_k a_k^{\dagger} a_k$ then I was wondering:
As far as I know this is many body theory and so these operators act on symmetrized or antisymmetrized states respectively, but I am not sure which quantity would determine in this case what the state that we symmetrize or antisymmetrize are?
So which states do these operators $a_k$ actually create and annihilate? I guess that the answer is eigenstates of the Hamiltonian, but actually the spectrum of the Hamiltonian does not have to be discrete, so I don't know this.
EDIT: It was suggested that these are the eigenstates of the kinetic part of the Hamiltonian, but then we have the problem, that the spectrum is not necessarily discrete (think of unrestricted motion in $x$ direction)
Best Answer
Usually in many body theory these operators create and annihilate particles. There are different annihilation and creation operators for fermions and bosons (they obey different commutation relations).
The states they act upon and the states "created" by them respect the required symmetries (antisymmetric for fermions, symmetric for bosons). The operators in your hamiltonian could act for instance on a fock state (note that the operator pairs in your Hamiltonian represent the number operator). You might want to take a look at http://en.wikipedia.org/wiki/Fock_state , especially take a look at the two sections of "Action on some specific Fock states".
NB: If you came across your Hamiltonian while dealing with harmonic oscillators, you rather want to think of $a_k^{\dagger}$ and $a_k$ in terms of ladder operators.