Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$. Lifts are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, sums and real multiples of $L^{2}$ $d$-tuples those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing, and $0$ denotes the $L^{2}$ $d$-tuple $(0,\dots,0)$, which is an $L^{2}$ $d$-tuple by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded applied to the zero $d$-tuple of $M$, whose entries are self-adjoint.

$\mathcal{H}$ is quadratic with a convex Lipschitz remainder if there are a function $\mathcal{H}_{0}:\Sigma^{2}_{2d}\to\mathbb{R}$, the remainder, and reals $L\ge0$ and $C$ with the following four properties.

1. (Decomposition)¶ $\mathcal{H}(\pi)=\tfrac{1}{2}\widehat{M}(\mathrm{pr}^{2}_{\#}\pi)+\mathcal{H}_{0}(\pi)$ for every $\pi\in\Sigma^{2}_{2d}$, with the second moment $\widehat{M}$ and the push-forward under the marginal datum $\mathrm{pr}^{2}$.

2. (Lipschitz remainder)¶ $\mathcal{H}_{0}$ is Lipschitz in the momentum with linear growth with constant $L$.

3. (Convex remainder)¶ For every tracial W*-probability space $(H,M,\Omega)$, all $L^{2}$ $d$-tuples $X,P,P'$ of $(H,M,\Omega)$ and every real $t$ with $0\le t\le1$,

4. (Bound at zero momentum)¶ $\mathcal{H}_{0,M}(X,0)\le C$ for every tracial W*-probability space $(H,M,\Omega)$ and every $L^{2}$ $d$-tuple $X$ of $(H,M,\Omega)$.