One-mode photon number operator and count PVM

Under the unitary $J_u$ of [F2], the one-dimensional $n$th sector of ${\mathcal{F}}_s(\mathbb{C}u)$ is spanned by the unit vector $e_n$. Therefore $Q_n$ are mutually orthogonal projections and their strong sum is the identity on ${\mathcal{F}}_s(\mathbb{C}u)$. Tensoring with the identity on ${\mathcal{F}}_s(K)$ and conjugating by $J_u$ shows that each $P_n$ is an orthogonal projection, $P_n{P}_m=0$ for $n\mathrm{\ne }m$, and ${\sum }_{n\ge 0}{P}_n\mathrm{\Psi }=\mathrm{\Psi }$ in norm for every $\mathrm{\Psi }\in {\mathcal{F}}_s(H)$.

Under $J_u$, write $\mathrm{\Psi }={⨁}_{n\ge 0}{e}_n\otimes {\varphi }_n$ with ${\varphi }_n\in {\mathcal{F}}_s(K)$, and put $m_n=n(n-1)$. The composition $N_u(N_u-I)$ acts diagonally by $m_n$ and has exactly the stated domain: ${\sum }_n{m}_n^2\mathrm{\parallel }{\varphi }_n{\mathrm{\parallel }}^2<\mathrm{\infty }$ implies $\mathrm{\Psi }\in D(N_u)$ because $n^2\le m_n^2$ for $n\ge 2$ and the $n=0,1$ block contributions are finite, and it gives $(N_u-I)\mathrm{\Psi }\in D(N_u)$; conversely, membership in the composition domain gives this sum. This domain contains ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$, dense by truncation of the square-summable Fock sum in [F1], and the operator is symmetric because each $m_n$ is real. If $\mathrm{\Phi }\in D(M_u^{\ast })$ and $\mathrm{\Xi }=M_u^{\ast }\mathrm{\Phi }$, testing the adjoint identity on vectors supported in one $n$th block forces ${\mathrm{\Xi }}_n=m_n{\mathrm{\Phi }}_n$; square summability gives $\mathrm{\Phi }\in D(M_u)$. Conversely, for $\mathrm{\Phi }\in D(M_u)$, Cauchy–Schwarz on the block sum gives the adjoint identity with value $M_u\mathrm{\Phi }$. Hence $M_u=M_u^{\ast }$ and is self-adjoint, therefore closed, by [F5].

For Borel $B$, the orthogonal sum defining $E_u(B)$ converges strongly because ${\sum }_n\mathrm{\parallel }{P}_n\mathrm{\Psi }{\mathrm{\parallel }}^2=\mathrm{\parallel }\mathrm{\Psi }{\mathrm{\parallel }}^2$. It is an orthogonal projection, $E_u(\mathrm{\varnothing })=0$, $E_u(\mathbb{R})=I$, and orthogonality gives $E_u(B)E_u(C)=E_u(B\cap C)$. For pairwise disjoint Borel sets $B_j$, their intersections with ${\mathbb{N}}_0$ partition $B\cap {\mathbb{N}}_0$, so the corresponding orthogonal series gives $E_u({\bigcup }_j{B}_j)\mathrm{\Psi }={\sum }_j{E}_u(B_j)\mathrm{\Psi }$ in norm. Thus $E_u$ is a PVM supported on ${\mathbb{N}}_0$, and $E_u(\{n\})=P_n$.

Under $J_u$, $N_u$ is the diagonal operator $({\mathrm{\Psi }}_n{)}_{n\ge 0}\mapsto (n{\mathrm{\Psi }}_n{)}_{n\ge 0}$ with domain ${\sum }_n{n}^2\mathrm{\parallel }{\mathrm{\Psi }}_n{\mathrm{\parallel }}^2<\mathrm{\infty }$. It is symmetric because every diagonal coefficient $n$ is real. If $\mathrm{\Phi }$ lies in its adjoint domain with adjoint value $\mathrm{\Xi }$, testing the adjoint identity on vectors supported in a single sector forces ${\mathrm{\Xi }}_n=n{\mathrm{\Phi }}_n$ for every $n$. Since $\mathrm{\Xi }$ is square-summable, ${\sum }_n{n}^2\mathrm{\parallel }{\mathrm{\Phi }}_n{\mathrm{\parallel }}^2<\mathrm{\infty }$, so $\mathrm{\Phi }\in D(N_u)$. Conversely, for $\mathrm{\Phi }\in D(N_u)$, Cauchy–Schwarz on the sector sum gives the adjoint identity with value $N_u\mathrm{\Phi }$. Hence $D(N_u^{\ast })=D(N_u)$ and $N_u^{\ast }=N_u$, and its projections are the constructed $E_u(\{n\})=P_n$.

By [F2], on a one-mode occupation vector $e_n\otimes \varphi$ one has $a(u)(e_n\otimes \varphi )=\sqrt{n}{e}_{n-1}\otimes \varphi$ and $a^{†}(u)(e_n\otimes \varphi )=\sqrt{n+1}{e}_{n+1}\otimes \varphi$; thus $a^{†}(u)a(u)=N_u$ on finite-particle vectors, including the $n=0$ sector. On the same invariant subspace, [F3] gives $a(u)a^{†}(u)=a^{†}(u)a(u)+I$. Consequently $N_u(N_u-I)=a^{†}(u)a(u)a^{†}(u)a(u)-a^{†}(u)a(u)=a^{†}(u{)}^{2}a(u{)}^2$ on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$, by substitution of the CCR and cancellation of the two $a^{†}(u)a(u)$ terms.

Truncate any $\mathrm{\Psi }\in D(M_u)$ to ${\mathrm{\Psi }}^{(M)}={\sum }_{n=0}^M{P}_n\mathrm{\Psi }$; its squared graph-norm tail is ${\sum }_{n>M}(1+m_n^2)\mathrm{\parallel }{P}_n\mathrm{\Psi }{\mathrm{\parallel }}^2\to 0$. For fixed $M$, write each $P_n\mathrm{\Psi }$ under $J_u^{-1}$ as $e_n\otimes {\varphi }_n$ and approximate each of the finitely many ${\varphi }_n$ by finite-particle vectors in ${\mathcal{F}}_s(K)$, which are dense by [F1]. The one-mode unitary preserves norm and the finitely many multipliers $m_n$ are bounded, so these approximations converge in graph norm and their images have finite total particle number. Thus ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ is a graph core for $M_u$; its restriction is $a^{†}(u{)}^{2}a(u{)}^2$ by step 3.1, and its closure is $M_u$ because $M_u$ is closed by step 2.1.