Consider a scattering problem that is described by the Hamiltonian \( H = \frac{p^2}{2M} - \frac{h}{2} \sigma_z + u \delta(x-(a/2)\sigma_x) \) . Note that \( \delta(x-(a/2)\sigma_x) \) can be expressed as sum of \( [(1+\sigma_x)/2]\delta(x-(a/2)) \) and \( [(1-\sigma_x)/2]\delta(x+(a/2)) \) . The target is prepared in the lower level, and hence looks like a double barrier. The kinetic energy of the incident particle is $E$. Write the matching equations at \( x=-a/2 \) and at $x=+a/2$. Explain in what limits the problem reduces to Fabry-Perrot. Find the explicit solution of the scattering problem for $h=0$ and for $|h|=\infty$.