(a) Show that if $f$ satisfies the equation

$$f^{\prime \prime}(x)-x^{2} f(x)=\mu f(x), \quad x \in \mathbb{R},$$

where $\mu$ is a constant, then its Fourier transform $\widehat{f}$ satisfies the same equation, i.e.

$$\widehat{f}^{\prime \prime}(\lambda)-\lambda^{2} \widehat{f}(\lambda)=\mu \widehat{f}(\lambda) .$$

(b) Prove that, for each $n \geq 0$, there is a polynomial $p_{n}(x)$ of degree $n$, unique up to multiplication by a constant, such that

$$f_{n}(x)=p_{n}(x) e^{-x^{2} / 2}$$

is a solution of $(*)$ for some $\mu=\mu_{n}$.

(c) Using the fact that $g(x)=e^{-x^{2} / 2}$ satisfies $\widehat{g}=c g$ for some constant $c$, show that the Fourier transform of $f_{n}$ has the form

$$\widehat{f_{n}}(\lambda)=q_{n}(\lambda) e^{-\lambda^{2} / 2}$$

where $q_{n}$ is also a polynomial of degree $n$.

(d) Deduce that the $f_{n}$ are eigenfunctions of the Fourier transform operator, i.e. $\widehat{f_{n}}(x)=c_{n} f_{n}(x)$ for some constants $c_{n} .$