Harmonic oscillator and Hermite functions

Prove that the system of equations $A_kf=0$, $1\leq k\leq n$, has a one-dimensional space of solutions in $S(\mathbb {R}^{n})$.

Let $f_0\in S(\mathbb {R}^{n})$ be a basis vector in the solution space of the system $A_kf=0$, $1\leq k\leq n$ (the so-called vacuum vector). Prove that the system of functions $f_m=(A_1^*)^{m_1}\cdots (A_n^*)^{m_n}f_0$, $m\in \mathbb {N}^{n}$, is dense in $S(\mathbb {R}^{n})$.

Let $N_k=(1/4\pi )A_k^*A_k$, $N=\sum_{k=1}^nN_k$ (the so-called occupation number operator and particle number operator). Prove that the functions $f_m$, $m\in \mathbb {Z}^{n}$, are eigenfunctions for the operators $N_k$ and $N$, and compute the corresponding eigenvalues.

Construct an isomorphism between the space $S(\mathbb {R}^{n})$ and the space of $n$-fold sequences $\left\{ c_m\right\}$, $m\in \mathbb {N}^{n}$, with the property $\left|c_m\right|=o(\left|m\right|^{-k})$ for all $k\in \mathbb {N}$.

Compute the Fourier transforms of the functions $f_m$, $m\in \mathbb {N}^{n}$.