Let $B=c A+s A^{\dagger}$, where $c \equiv \cosh \theta, s \equiv \sinh \theta$ with $\theta$ a real constant and $A, A^{\dagger}$ are the usual ladder operators. Show that $\left[B, B^{\dagger}\right]=1$. Consider the Hamiltonian
$$
H=\epsilon A^{\dagger} A+\frac{1}{2} \lambda\left(A^{\dagger} A^{\dagger}+A A\right)
$$
where $\epsilon$ and $\lambda$ are real and such that $\epsilon>\lambda>0 .$ Show that when
$$
\epsilon c-\lambda s=E c, \quad \lambda c-\epsilon s=E s
$$
with $E$ a constant, $[B, H]=E B .$ Hence determine the spectrum of $H$ in terms of $\epsilon$ and $\lambda$.