Relation between HO and this hamiltonian

[Matiasss]

hi,
i have studied the annihilation and creation operators and number operator N in relation with the simple harmonic oscillator that is governed by:
H = hw(N+1/2)

i don't understand the relation between the harmonic oscillator and for example, this hamiltonian:

H = hw₁a⁺a+hw₂a⁺a⁺aa

that i have found it in an example in the lecture notes of a course. they calculate the energies of this system. They use the annihilation operator that is defined from the simple HO to solve that system.
what is physically this system? why i can use the SHO to calculate the energies?
i feel that i am confused with the a operator . i thought that it was defined from the hamiltonian of the simple harmonic oscillator ,,,, isn't it ?

thanks in advance

 

[Orodruin]

Formally, this new Hamiltonian is only an operator on the same Hilbert space as that used for the harmonic oscillator and expressed in terms of the operators $a$ and $a^\dagger$, which have a known action on the energy eigenstates of the harmonic oscillator. You can therefore simply treat it as a (hermitian) operator on that Hilbert space and compute its eigenvalues.