Let $a$ and $a^{\dagger}$ be the simple harmonic oscillator annihilation and creation operators, respectively. Write down the commutator $\left[a, a^{\dagger}\right]$.

Consider a new operator $b=c a+s a^{\dagger}$, where $c \equiv \cosh \theta, s \equiv \sinh \theta$ with $\theta$ a real constant. Show that

$$\left[b, b^{\dagger}\right]=1$$

Consider the Hamiltonian

$$H=\epsilon a^{\dagger} a+\frac{1}{2} \lambda\left(a^{\dagger^{2}}+a^{2}\right),$$

where $\epsilon$ and $\lambda$ are real and such that $\epsilon>\lambda>0$. Assuming that $\epsilon c-\lambda s=E c$ and $\lambda c-\epsilon s=E s$, with $E$ a real constant, show that

$$[b, H]=E b$$

Thus, calculate the energy of $b\left|E_{a}\right\rangle$ in terms of $E$ and $E_{a}$, where $E_{a}$ is an eigenvalue of $H$.

Assuming that $b\left|E_{\min }\right\rangle=0$, calculate $E_{\min }$ in terms of $\lambda, s$ and $c$. Find the possible values of $x=s / c$. Finally, show that the energy eigenvalues of the system are

$$E_{n}=-\frac{\epsilon}{2}+\left(n+\frac{1}{2}\right) \sqrt{\epsilon^{2}-\lambda^{2}}$$