Canonical commutation relations of position and momentum on the Schwartz core

[Schwartz invariance and domains] Let $f\in \mathcal{S}$. By [L2], $x_{j}f$ and ${\mathrm{\partial }}_{k}f$ are Schwartz and in $L^2$; by [L1], $f\in D(Q_j)\cap D(P_k)$, $Q_{j}f=x_{j}f\in \mathcal{S}$, and $P_{k}f=-i\mathrm{\hslash }{\mathrm{\partial }}_{k}f\in \mathcal{S}$. This proves (a) and ensures both products in (b) are defined.

[The position operators have Schwartz cores] Fix $j$ and $f\in D(Q_j)$, and define $f_n={\chi }_{n}f$ using [L3]. Since ${\chi }_n(x)\to 1$ for each fixed $x$ and $\mathrm{\mid }{\chi }_n\mathrm{\mid }\le 1$, dominated convergence applied to $\mathrm{\mid }{f}_n-f{\mathrm{\mid }}^2$ and $\mathrm{\mid }{x}_j(f_n-f){\mathrm{\mid }}^2$ gives $\mathrm{\parallel }{f}_n-f{\mathrm{\parallel }}_2+\mathrm{\parallel }{x}_j{f}_n-x_{j}f{\mathrm{\parallel }}_2\to 0$. Each $f_n$ is supported in $\mathrm{\mid }x\mathrm{\mid }\le 2n$. For fixed $n$, let $g_{n,ϵ}={\rho }_{ϵ}\ast f_n$; by [L4] it is smooth, compactly supported in $\mathrm{\mid }x\mathrm{\mid }\le 2n+2ϵ$, and $\mathrm{\parallel }{g}_{n,ϵ}-f_n{\mathrm{\parallel }}_2\to 0$. For $0<ϵ\le 1$, $\mathrm{\mid }{x}_j\mathrm{\mid }\le 2n+2$ on both supports, so $\mathrm{\parallel }{x}_j(g_{n,ϵ}-f_n){\mathrm{\parallel }}_2\le (2n+2)\mathrm{\parallel }{g}_{n,ϵ}-f_n{\mathrm{\parallel }}_2$. Choose ${ϵ}_n\le 1$ so the latter $L^2$ error is below $n^{-2}$. Then $g_{n,{ϵ}_n}\in C_c^{\mathrm{\infty }}\subseteq \mathcal{S}$ and converges to $f$ in the graph norm of $Q_j$. Thus $\mathcal{S}$ is a core for $Q_j$.

[Classical product rule and commutator] Since $x_{j}f\in \mathcal{S}$, its classical derivative is its weak derivative by [L1]. By [L5], $D_k(x_{j}f)={\mathrm{\partial }}_k(x_{j}f)={\delta }_{jk}f+x_j{\mathrm{\partial }}_{k}f$. Therefore $P_k{Q}_{j}f=-i\mathrm{\hslash }({\delta }_{jk}f+x_j{\mathrm{\partial }}_{k}f)$, while $Q_j{P}_{k}f=-i\mathrm{\hslash }{x}_j{\mathrm{\partial }}_{k}f$. Subtraction yields $(Q_j{P}_k-P_k{Q}_j)f=i\mathrm{\hslash }{\delta }_{jk}f$, proving (b) directly on Schwartz representatives.

[The momentum operators have Schwartz cores] Fix $k$ and $f\in D(P_k)$; put $f_n={\chi }_{n}f$. For every $\phi \in C_c^{\mathrm{\infty }}$, the weak derivative identity for $f$ applied to ${\chi }_n\phi$ gives $\int f_n{\mathrm{\partial }}_k\phi =-\int ({\chi }_n{D}_{k}f+({\mathrm{\partial }}_k{\chi }_n)f)\phi$, so $D_k{f}_n={\chi }_n{D}_{k}f+({\mathrm{\partial }}_k{\chi }_n)f\in L^2$. Dominated convergence gives $f_n\to f$ and ${\chi }_n{D}_{k}f\to D_{k}f$ in $L^2$, while [L3] gives $\mathrm{\parallel }({\mathrm{\partial }}_k{\chi }_n)f{\mathrm{\parallel }}_2\le n^{-1}\mathrm{\parallel }{\mathrm{\partial }}_k\chi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }f{\mathrm{\parallel }}_2\to 0$; hence $f_n\to f$ in the graph norm of $P_k$. For fixed $n$, let $g_{n,ϵ}={\rho }_{ϵ}\ast f_n$. For a test function $\phi$, change variables and apply the weak-derivative identity inside the convolution integral: $\int g_{n,ϵ}(x){\mathrm{\partial }}_k\phi (x)dx=\int {\rho }_{ϵ}(y)\int f_n(z){\mathrm{\partial }}_k\phi (z+y)dzdy=-\int ({\rho }_{ϵ}\ast D_k{f}_n)(x)\phi (x)dx$. The interchange is valid by [L6], so ${\mathrm{\partial }}_k{g}_{n,ϵ}={\rho }_{ϵ}\ast D_k{f}_n$ as a weak derivative; since $g_{n,ϵ}$ is smooth, it is also its classical derivative. Both $g_{n,ϵ}\to f_n$ and ${\mathrm{\partial }}_k{g}_{n,ϵ}\to D_k{f}_n$ in $L^2$ by [L4]. Choose ${ϵ}_n>0$ so both errors are below $1\mathrm{/}n$. Then $g_{n,{ϵ}_n}\in C_c^{\mathrm{\infty }}\subseteq \mathcal{S}$ and converges to $f$ in the graph norm of $P_k=-i\mathrm{\hslash }{D}_k$. Thus $\mathcal{S}$ is a core for $P_k$. Since $j,k$ were arbitrary, the proof establishes the common-core assertion in the Statement and completes (a) and (b).