System of Schrodinger equations

Question

I know that Schrodinger equation for 1 dimension is given by: $-u''+qu=\lambda u$ and if I want to work on a system for say 2 dimensions, the system would be: \begin{equation} \left(\begin{array}{c} -u_1''\\ -u_2'' \end{array}\right) +\left(\begin{array}{cc} q_1 & q_2\\ q_3 & q_4\end{array}\right) \left(\begin{array}{c} u_1 \\ u_2 \end{array}\right) =\lambda \left(\begin{array}{c} u_1 \\ u_2 \end{array}\right) \end{equation} My question is: is this system right and does make sense? and if the q in 1 dimension is periodic can I make the matrix $Q$ in this system: \begin{equation} Q=\left(\begin{array}{cc} q_1 & q_2\\ q_3 & q_4\end{array}\right) \end{equation} to be periodic? If this system I made is correct, is there any examples for this case of system in quantum mechanics or other field of applications?