Consider the Eigenvalue problem $\hat{H} \Psi = E \Psi$ on interval $[a,b]$.
Where \begin{align} \hat{H} = \begin{pmatrix} -i \partial_x &0 \\ 0 & i\partial_x \end{pmatrix} \end{align} And $\hat{H}$ acts on $\Psi(x) = \begin{pmatrix} \psi_1(x) \\ \psi_2(x) \end{pmatrix}$.
Assume that $\hat{H}$ is an self-adjoint operator, 1. i want to show the boundary condition as \begin{align} \frac{\psi_1(a)}{\psi_2(a)} = exp[i\theta_a], \qquad \frac{\psi_1(b)}{\psi_2(b)} = exp[i\theta_b] \end{align} for real $\theta_a$ and $\theta_b$.
- Further i want to find the eigenfunction and eigenvalue for this problem.
What i tried is following Set $\Psi_1= (\psi_{11}, \psi_{12})^T$, $\Psi_2=(\psi_{21}, \psi_{22})^T$, then
\begin{align} &\langle \Psi_1 |\hat{H} \Psi_2 \rangle = \int_a^b \left[ \psi_{11}^* ( -i \partial_x \psi_{21}) + \psi_{12}^* ( i \partial_x \psi_{22}) \right] dx \\ & \langle \hat{H} \Psi_1 | \Psi_2 \rangle = \int_{a}^b \left[ \psi_{21} i \partial_x \psi_{11}^* - \psi_{22} i\partial_x \psi_{12}^* \right] dx \\ & \langle \Psi_1 |\hat{H} \Psi_2 \rangle - \langle \hat{H} \Psi_1 | \Psi_2 \rangle = -i \left( \psi_{11}^* \psi_{21}\right) \big|_a^b + i (\psi_{12}^* \psi_{22} ) \big|_{a}^b = 0 \end{align} ...