# Diagonalize the hamiltonian!

## 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.
$$E=v_Fk_F\pm v_F(\Delta^2+k^2)^{1/2}$$
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
$$E=\frac{1}{2} (v_F k_F \pm (v_F^2 k^2 + 4\Delta^2)^{1/2})$$