The Gaussian Weyl average is a rank-one projection

By [F7], $\mathrm{\parallel }{\mathrm{\Omega }}_a{\mathrm{\parallel }}_2^2=(a\mathrm{/}\pi {)}^{1\mathrm{/}2}\int e^{-a{x}^2}dx=1$. For fixed $f\in L^2(\mathbb{R})$, the map $(q,p)\mapsto e^{-a{q}^2\mathrm{/}4-p^2\mathrm{/}(4a)}W(q,p)f$ is strongly measurable by [F1] and has integrable norm. Therefore [F5] defines a bounded operator $A_a$ by the displayed weak integral.

Suppose $g\in L^2(\mathbb{R})$ is orthogonal to $W(0,p){\mathrm{\Omega }}_a$ for every $p$. Then $h(x)={\mathrm{\Omega }}_a(x)\overline{g(x)}$ belongs to $L^1$ by [F8], and $\widehat{h}(\xi )=0$ for every $\xi$, because $\lambda =1$ in the modulation and $p$ ranges over all real numbers. By [F6], $h=0$ almost everywhere; ${\mathrm{\Omega }}_a(x)>0$ everywhere, so $g=0$. Hence the modulation orbit, and therefore the full Weyl orbit, has dense linear span.

Take $f,g\in C_c(\mathbb{R})$. The bounded functional $h\mapsto ⟨h,g⟩$ commutes with the Bochner integral by [F5], and the resulting integral over $x,q,p$ is absolutely integrable: $f,g$ are bounded with compact support, while the $q,p$ weights are Gaussian. Thus [F4] permits integrating first in $p$. By [F2] with $t=1\mathrm{/}(4\pi a)$,${\int }_{\mathbb{R}}{e}^{-p^2\mathrm{/}(4a)}{e}^{ip(x-q\mathrm{/}2)}dp=2\sqrt{\pi a}{e}^{-a(x-q\mathrm{/}2{)}^2}\mathrm{.}$It follows that$⟨A_{a}f,g⟩=\sqrt{a\mathrm{/}\pi }{\int }_{{\mathbb{R}}^2}{e}^{-a{q}^2\mathrm{/}4-a(x-q\mathrm{/}2{)}^2}f(x-q)\overline{g(x)}dqdx$.

Set $y=x-q$. Then $q^2\mathrm{/}4+(x-q\mathrm{/}2{)}^2=(x^2+y^2)\mathrm{/}2$, so the matrix coefficient in step 2.1 becomes$\sqrt{a\mathrm{/}\pi }({\int }_{\mathbb{R}}{e}^{-a{y}^2\mathrm{/}2}f(y)dy)\left({\int }_{\mathbb{R}}{e}^{-a{x}^2\mathrm{/}2}\overline{g(x)}dx\right)=⟨f,{\mathrm{\Omega }}_a⟩⟨{\mathrm{\Omega }}_a,g⟩=⟨⟨f,{\mathrm{\Omega }}_a⟩{\mathrm{\Omega }}_a,g⟩\mathrm{.}$Thus $A_a$ and the rank-one operator agree on matrix coefficients for $f,g\in C_c(\mathbb{R})$. Both are bounded, and [F3] makes $C_c(\mathbb{R})$ dense in $L^2$, so they agree on all of $L^2$. Since $\mathrm{\parallel }{\mathrm{\Omega }}_a\mathrm{\parallel }=1$, $A_a$ is the orthogonal projection onto $\mathbb{C}{\mathrm{\Omega }}_a$.

For any regular one-mode system, the Weyl relation gives $W(q,p)=e^{iqp\mathrm{/}2}W(q,0)W(0,p)$. Regularity and the unitary norm bound therefore make each vector orbit jointly continuous. On expanding bounded squares, sampling that orbit on finite dyadic grids gives measurable simple functions converging pointwise to it, proving strong measurability. The Gaussian weight has integrable norm, so [F5] defines its average as in step 1.1. If bounded $T$ intertwines two such systems, $TW(q,p)=W^{\mathrm{\prime }}(q,p)T$ and [F5] gives $T{A}_a=A_a^{\mathrm{\prime }}T$. In an incomplete tensor product, each coordinate system is regular: strong continuity holds on finite products of one-coordinate vectors, whose span is dense, and extends by the unitary bound. On these vectors its average is the one-mode average in that coordinate and the identity elsewhere; boundedness extends this identity to the completion.

Steps 1.2, 3.1, and 4.1 prove, respectively, totality of the Gaussian Weyl orbit, the rank-one projection, and its intertwiner and tensor-factor properties.