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Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is $$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$ the adjoint operator is then $$(\Delta^* u)(n)=...