Sequence-space models of test-function spaces

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Problem 556 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.2.1 "Characters of an Abelian Group"

Let $S(\mathbb {Z})$ be the space of two-sided sequences $\left\{ c_n\right\}$ having the property that $c_n=O(n^{-k})$ for all $k$ . The topology in $S(\mathbb {Z})$ is defined by the family of norms

p_k(\left\{ c_n\right\} )=\sup _n\left|n^k c_n\right|,\qquad k=0,1,\ldots .

Prove that the Fourier transformation establishes an isomorphism between the linear topological spaces $\mathcal{E}(\mathrm{T})$ and $S(\mathbb {Z})$ .

Problem 580 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.2.3 "The Fourier Integral"

Let $D_k=\partial /\partial x_k$ , and $M_k$ the operator of multiplication by $x_k$ . Define the operators $A_k=iD_k+M_k$ , $A_k^*=iD_k-M_k$ , $k=1,2,\ldots ,n$ (the so-called creation and annihilation operators in quantum field theory) in the space $S(\mathbb {R}^{n})$ .

(a)

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

(b)

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 $S(\mathbb {R}^{n})$ .

(c)

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.

(d)

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 $k\in \mathbb {N}$ .

(e)

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

References

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.