Proof of The Controlled von Neumann Hamiltonian at Bounded Positions: Evaluation, the Structure Condition and Uniform Continuity in the Momentum

Each result cited below is universally quantified over the data in its own statement.

Claim 1 (Evaluation). Since $s\Omega=X$, the pair $(X,P)$ of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations equals $(s\Omega,P)$, so $\pi=\mathrm{law}(X,P)$ satisfies $\pi=\mathrm{law}(s\Omega,P)$ and the first case of The Controlled von Neumann Hamiltonian on Phase-Space Noncommutative Laws §hamiltonian applies with the space $(H,M,\Omega)$, the tuple $s$ and the momentum $P$. By Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, $\mathcal{H}^{\mathrm{vN}}_{M}(X,P)=\mathcal{H}^{\mathrm{vN}}(\pi)$, hence

$$\mathcal{H}^{\mathrm{vN}}_{M}(X,P)=\frac{1}{2}\sum_{j=1}^{d}\lVert[P_{j},s_{j}]\rVert^{2}-f(\mathrm{pr}^{1}_{\#}\pi).$$

By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, $\mathrm{pr}^{1}_{\#}\pi=\mathrm{law}(\mathrm{pr}^{1}(X,P))$. The coordinate datum $\mathrm{pr}^{1}$ of Affine Data and Affine Substitutions of Noncommutative Polynomials §coordinate has zero shift and coefficient matrix with entries $1$ at the positions $(i,i)$, $i\in[d]$, and $0$ elsewhere, so the affine image of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations gives $(\mathrm{pr}^{1}(X,P))_{i}=X_{i}$ for $i\in[d]$, that is, $\mathrm{pr}^{1}(X,P)=X$. Hence $f(\mathrm{pr}^{1}_{\#}\pi)=f(\mathrm{law}(X))$, which is the claimed formula.

Claim 2 (Structure condition). Assume $f$ is uniformly continuous. Let $R>0$ and $\eta>0$ be real. By Uniformly Continuous Map Between Metric Spaces, with the metrics of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics, there is a real $r>0$ such that $|f(\mu)-f(\nu)|<\eta$ for all $\mu,\nu\in\Sigma^{2}_{d}$ with $\widehat{W}_{2}(\mu,\nu)<r$; we show that this $r$ has the property required in The Structure Condition at Bounded Positions for a Hamiltonian on Phase-Space Noncommutative Laws §structure. Let $(H,M,\Omega)$ be a tracial W*-probability space, let $X,Y$ be $L^{2}$ $d$-tuples with $\mathrm{law}(X),\mathrm{law}(Y)\in\kappa_{d}(\Sigma_{d,R})$, and let $\alpha>0$ be real with $\alpha\lVert X-Y\rVert_{2}^{2}+\lVert X-Y\rVert_{2}<r$. By Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §operator there are self-adjoint $d$-tuples $s$ and $t$ in $M$ with $s\Omega=X$ and $t\Omega=Y$. Put $P=\alpha(X-Y)$, the real multiple of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing; it is an $L^{2}$ $d$-tuple with $P_{j}=\alpha(X_{j}-Y_{j})$. For $j\in[d]$, $P_{j}=\alpha(s_{j}\Omega-t_{j}\Omega)=b_{j}\Omega$ with $b_{j}=\alpha(s_{j}-t_{j})\in M$. By The Commutator of a Square-Integrable Vector with a Bounded Self-Adjoint Operator: Linearity, the Norm Bound, Vacuum Vectors and Dependence on the Law Only §vacuum,

$$[P_{j},t_{j}]=(b_{j}t_{j}-t_{j}b_{j})\Omega=\alpha(s_{j}t_{j}-t_{j}s_{j})\Omega,\qquad[P_{j},s_{j}]=(b_{j}s_{j}-s_{j}b_{j})\Omega=\alpha(s_{j}t_{j}-t_{j}s_{j})\Omega,$$

since $t_{j}t_{j}$ and $s_{j}s_{j}$ cancel in the respective differences. Thus $[P_{j},t_{j}]=[P_{j},s_{j}]$ for every $j\in[d]$, and Claim 1, applied at $X$ with $s$ and at $Y$ with $t$, gives

$$\mathcal{H}^{\mathrm{vN}}_{M}(Y,P)-\mathcal{H}^{\mathrm{vN}}_{M}(X,P)=f(\mathrm{law}(X))-f(\mathrm{law}(Y)).$$

Since $\alpha\lVert X-Y\rVert_{2}^{2}\ge0$, we have $\lVert X-Y\rVert_{2}<r$, and $\widehat{W}_{2}(\mathrm{law}(X),\mathrm{law}(Y))\le\lVert X-Y\rVert_{2}$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz. Hence $f(\mathrm{law}(X))-f(\mathrm{law}(Y))<\eta$, which is the required inequality.

Claim 3 (Momentum continuity). Let $R>0$ and $\eta>0$ be real and put $r=\eta/(4dR^{3})$. Let $(H,M,\Omega)$ be a tracial W*-probability space, let $X$ be an $L^{2}$ $d$-tuple with $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,R})$, with $s$ as in Claim 1, so that $\lVert s_{j}\rVert_{\mathrm{op}}\le R$ by Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §bound, and let $P,P'$ be $L^{2}$ $d$-tuples with $\lVert P\rVert_{2}\le R$, $\lVert P'\rVert_{2}\le R$ and $\lVert P-P'\rVert_{2}<r$. By Claim 1 at $X$ for $P$ and for $P'$,

$$\mathcal{H}^{\mathrm{vN}}_{M}(X,P)-\mathcal{H}^{\mathrm{vN}}_{M}(X,P')=\frac{1}{2}\sum_{j=1}^{d}\Bigl(\lVert a_{j}\rVert^{2}-\lVert a'_{j}\rVert^{2}\Bigr),\qquad a_{j}=[P_{j},s_{j}],\quad a'_{j}=[P'_{j},s_{j}].$$

By The Commutator of a Square-Integrable Vector with a Bounded Self-Adjoint Operator: Linearity, the Norm Bound, Vacuum Vectors and Dependence on the Law Only §linear, $a_{j}-a'_{j}=[P_{j}-P'_{j},s_{j}]$, so The Commutator of a Square-Integrable Vector with a Bounded Self-Adjoint Operator: Linearity, the Norm Bound, Vacuum Vectors and Dependence on the Law Only §bound gives $\lVert a_{j}-a'_{j}\rVert\le2R\lVert P_{j}-P'_{j}\rVert$, $\lVert a_{j}\rVert\le2R\lVert P_{j}\rVert$ and $\lVert a'_{j}\rVert\le2R\lVert P'_{j}\rVert$. By the definition of the $L^{2}$ norm in Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples, $\lVert P_{j}\rVert\le\lVert P\rVert_{2}\le R$, $\lVert P'_{j}\rVert\le R$ and $\lVert P_{j}-P'_{j}\rVert\le\lVert P-P'\rVert_{2}$. The triangle inequality of The Induced Norm is a Norm, and Induces a Metric gives $\bigl|\lVert a_{j}\rVert-\lVert a'_{j}\rVert\bigr|\le\lVert a_{j}-a'_{j}\rVert$, hence

$$\bigl|\lVert a_{j}\rVert^{2}-\lVert a'_{j}\rVert^{2}\bigr|=\bigl|\lVert a_{j}\rVert-\lVert a'_{j}\rVert\bigr|\bigl(\lVert a_{j}\rVert+\lVert a'_{j}\rVert\bigr)\le2R\lVert P-P'\rVert_{2}\cdot4R^{2}=8R^{3}\lVert P-P'\rVert_{2}.$$

Summing over $j\in[d]$ and halving, $\bigl|\mathcal{H}^{\mathrm{vN}}_{M}(X,P)-\mathcal{H}^{\mathrm{vN}}_{M}(X,P')\bigr|\le4dR^{3}\lVert P-P'\rVert_{2}<4dR^{3}r=\eta$. This is the property of Hamiltonians on Phase-Space Noncommutative Laws that are Uniformly Continuous in the Momentum at Bounded Positions §momentum.