Consider the Bose Hamiltonian


\begin{displaymath}
H = {\frac{{1}}{{2}}}(a^{\dag } a)\left( {{\begin{array}{*{2...
... {a} \hfill \\
{a^{\dag }} \hfill \\
\end{array}}} \right),
\end{displaymath}

where$ c, b$ reals.

Find the canonical transformation that reduces it to diagonal form and the spectrum when $c>b$.