Shift Semicontinuity of a Hamiltonian on Phase-Space Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$, with lifts $\mathcal{H}_{M}$ as in Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, and let $\sigma>0$ be real. Sums and real multiples of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing and the $L^{2}$ norm $\lVert\cdot\rVert_{2}$ is that of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples. For a tracial W*-probability space $(H,M,\Omega)$, $L^{2}$ $d$-tuples $X,P,Q$ of it and real $\delta>0$, write

$$G^{+}_{\delta}(X,P,Q)=\mathcal{H}_{M}(X,P+\delta Q)+\frac{\sigma^{2}\delta}{2}\lVert Q\rVert_{2}^{2},\qquad G^{-}_{\delta}(X,P,Q)=\mathcal{H}_{M}(X,P-\delta Q)-\frac{\sigma^{2}\delta}{2}\lVert Q\rVert_{2}^{2}.$$

Shift-convergent sequences are those of Shift-Convergent Sequences of Positions, Momenta and Shifts in a Tracial W*-Probability Space §sequence.

1. (Lower shift semicontinuity)¶ $\mathcal{H}$ is lower shift-semicontinuous at noise level $\sigma$ if for every real $r>0$ there is a real $\delta_{0}>0$ such that, for every tracial W*-probability space $(H,M,\Omega)$, every shift-convergent sequence $(X_{n},P_{n},Q_{n})\to(X,P,Q)$ at radius $r$ in it, every real $\delta$ with $0<\delta\le\delta_{0}$ and every real $\varepsilon>0$, there is $N\in\mathbb{N}$ with

$$G^{+}_{\delta}(X,P,Q)\le G^{+}_{\delta}(X_{n},P_{n},Q_{n})+\varepsilon\qquad\text{for every }n\ge N.$$

2. (Upper shift semicontinuity)¶ $\mathcal{H}$ is upper shift-semicontinuous at noise level $\sigma$ if for every real $r>0$ there is a real $\delta_{0}>0$ such that, for every tracial W*-probability space $(H,M,\Omega)$, every shift-convergent sequence $(X_{n},P_{n},Q_{n})\to(X,P,Q)$ at radius $r$ in it, every real $\delta$ with $0<\delta\le\delta_{0}$ and every real $\varepsilon>0$, there is $N\in\mathbb{N}$ with

$$G^{-}_{\delta}(X_{n},P_{n},Q_{n})\le G^{-}_{\delta}(X,P,Q)+\varepsilon\qquad\text{for every }n\ge N.$$

3. (Shift semicontinuity)¶ $\mathcal{H}$ is shift-semicontinuous at noise level $\sigma$ if it is both lower and upper shift-semicontinuous at noise level $\sigma$.