a) This is just an eigenvalue problem for a $2\times 2$ matrix. The eigenkets and for the two eigenvalues are
\[\ket{\Delta}=\frac{\ket R+\ket L}{\sqrt 2},\quad
\ket{-\Delta}=\frac{\ket R-\ket L}{\sqrt 2}.
\]
b) The time-evolution operator is
\[U(t)=U(0,t)=\exp (-\ii Ht/\hbar).\]
We can expand $\ket {\alpha(0)}$ in energy eigenstates as
\[\ket{\alpha(0)}=\ip{\Delta}{\alpha(0)}\ket\Delta+\ip{-\Delta}{\alpha(0)}\ket{-\Delta},\]
so the evolution equation
\[\ket{\alpha(t)}=\exp (-\ii Ht/\hbar)\ket{\alpha(0)}\]