I have an eigenvalue problem $$ \mathbf{L} \Psi=\lambda \Psi, $$ where $$ \mathbf{L}=i\left(\begin{array}{cc} G_0 & \nabla^2+G_1 \\ \nabla^2+G_2 & -G_0 \end{array}\right), \quad \Psi=\left(\begin{array}{c} v \\ w \end{array}\right), $$ $\nabla$ means $\frac{d}{dx}$. I expand the eigenfunction $\Psi$ as well as the functions $G_0, G_1$ and $G_2$ into Fourier series. So, the eigenvalue problem turns into a matrix eigenvalue problem for the Fourier coefficients of the eigenfunction $\Psi$ on the interval $x\in[-L/2,L/2]$. Let's assume Fourier series are \begin{gathered} v(x)=\sum_n a_n e^{i n k_0 x}, \quad w(x)=\sum_n b_n e^{i n k_0 x}, \\ G_p=\sum_n c_n^{(p)} e^{i n k_0 x}, \quad p=0,1,2, \end{gathered} where $k_0=2\pi/L$. When we substitute these expansions into the eigenvalue problem we get the following equations \begin{aligned} \sum_n c_n^{(0)} a_{j-n}-\left(k_0 j\right)^2 b_j+\sum_n c_n^{(1)} b_{j-n} & =-i \lambda a_j, \\ -\left(k_0 j\right)^2 a_j+\sum_n c_n^{(2)} a_{j-n}-\sum_n c_n^{(0)} b_{j-n} & =-i \lambda b_j, \end{aligned} where $j,n \in [-N,N]$.
(Yang, Jianke, Nonlinear waves in integrable and nonintegrable systems., Mathematical Modeling and Computation 16. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) p. (2010).)
My question: What if we had $G_0 \nabla$ in $L(1,1)$ and $L(2,2)$ instead $G_0$? I mean, if we had the eigenvalue problem: $$ \mathbf{L}=i\left(\begin{array}{cc} G_0 \nabla & \nabla^2+G_1 \\ \nabla^2+G_2 & -G_0 \nabla \end{array}\right), \quad \Psi=\left(\begin{array}{c} v \\ w \end{array}\right), $$ then, would the equations above be like \begin{aligned} &\sum_n c_n^{(0)}. i(j-n)k_0 .a_{j-n}-\left(k_0 j\right)^2 b_j+\sum_n c_n^{(1)} b_{j-n} =-i \lambda a_j, \\ &-\left(k_0 j\right)^2 a_j+\sum_n c_n^{(2)} a_{j-n}-\sum_n c_n^{(0)}.i(j-n). b_{j-n} =-i \lambda b_j \end{aligned} ?