# How can i write the matrix representation of the following Hatano - Nelson model Hamiltonian?

I have a $$1$$D and one band lattice model with hopping constants $$J_R$$ (to the right) and $$J_L$$ (to the left) and under open boundary condition. It has the following Hamiltonian :

$$H = \sum_{n} (J_R c^{\dagger}_{n+1} c_n +J_L c^{\dagger}_n c_{n+1} )$$ (1)

where $$J_L , J_R \in R$$ and $$c^{\dagger}_n , c_n$$ are creation and annihilation operators.

I want to write the matrix representation of this Hamiltonian. I've tried to rewrite this Hamiltonian without using second quantized operators. It looks like this:

$$H = \sum_n (J_R | n+1\rangle\langle n |+ J_L |n+1\rangle\langle n |)$$ (2)

I am not sure if this is correct. I've just used the n+1 and n's. How can i write the matrix representation of this Hamiltonian using the (1) Hamiltonian? Thanks in advance.

In the associated one-particle hamiltonian has the infinite matrix such that each row has 0 on the digaonal and $$J_L$$ in the entry to the left and $$J_R$$ in the entry to the right. Obviously it is not hermitian unless $$J_L=J_R^*$$. This one-particle Hamiltonian matrix is the one you write down in yoiur equation 2.