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## Homework Statement

I am trying to solve a problem of 1D electron system.

Given [tex]a,a^\dagger,b,b^\dagger[/tex] annihilation and creation operator which satisfy the fermion commutation relations diagonalize the following hamiltonian:

[tex]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})[/tex]

where [tex]v_F,\Delta[/tex] are c-numbers.

Prove that the spectrum is given by:

[tex]E=v_Fk_F\pm v_F(\Delta^2+k^2)^{1/2}[/tex]

**2. The attempt at a solution**

I try to define the following operators (that form an su(2) algebra):

[tex]J_3=\frac{1}{2}(a^\dagger_{k+k_F}a_{k+k_F}-b^\dagger_{k-k_F}b_{k-k_F})[/tex]

[tex]J_+=a^\dagger_{k+k_F}b_{k-k_F}[/tex]

[tex]J_-=b^\dagger_{k-k_F}a_{k+k_F}[/tex]

and to calculate the adjoint action but I don't know how to continue.

Please help me, thank you.