How to Diagonalize a Hamiltonian with Fermion Operators?

[Enialis]

Homework Statement

I am trying to solve a problem of 1D electron system.
Given a,a^\dagger,b,b^\dagger annihilation and creation operator which satisfy the fermion commutation relations diagonalize the following hamiltonian:

H=v_F\sum_{k>0}k(a^\dagger_ka_k-b^\dagger_kb_k)+\Delta\sum_k(b^\dagger_{k-k_F}a_{k+k_F}+a^\dagger_{k+k_F}b_{k-k_F})

where v_F,\Delta are c-numbers.

Prove that the spectrum is given by:
E=v_Fk_F\pm v_F(\Delta^2+k^2)^{1/2}

2. The attempt at a solution
I try to define the following operators (that form an su(2) algebra):
J_3=\frac{1}{2}(a^\dagger_{k+k_F}a_{k+k_F}-b^\dagger_{k-k_F}b_{k-k_F})
J_+=a^\dagger_{k+k_F}b_{k-k_F}
J_-=b^\dagger_{k-k_F}a_{k+k_F}
and to calculate the adjoint action but I don't know how to continue.
Please help me, thank you.

 

[halucka]

The spectrum cannot be given by
E=v_Fk_F\pm v_F(\Delta^2+k^2)^{1/2}
because the units of \Delta and k are not the same. I tried rewriting the first term (linear kinetic energy) using operators shifted by k_F and then diagonalizing it, and I got
E=\frac{1}{2} (v_F k_F \pm (v_F^2 k^2 + 4\Delta^2)^{1/2})
Still may not be correct, but at least the units match.