The Quadratic Control Hamiltonian: Legendre Formula, Bounds, Differences in the Law and Admissibility

In the setting of Unitary Laws with Free Unitary Noise: Standing Data, with the conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces, let $[d]$ be the initial segment determined by $d$, let $H_{Q}$ be the quadratic control Hamiltonian, $\kappa_{d}$ the constant of Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency §constant, $\Xi^{i}_{p,n}$ the truncated cyclic gradients, and $\lambda(X)$ the evaluation of a word polynomial. For $\lambda\in\mathcal{L}_{d}$, $p\in E_{d}$ and $n\in\mathbb{N}$ the numbers $\lambda\bigl(\Xi^{i}_{p,n}\Xi^{i}_{p,n}\bigr)$, $i\in[d]$, are real by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §positive, $\Xi^{i}_{p,n}$ being self-adjoint by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra. Then the following hold.

1. (Legendre formula)¶ For every $\lambda\in\mathcal{L}_{d}$ and $p\in E_{d}$, the sequence $n\mapsto\frac12\sum_{i\in[d]}\lambda\bigl(\Xi^{i}_{p,n}\Xi^{i}_{p,n}\bigr)$ converges in $(\mathbb{R},d_{\mathbb{R}})$, and its limit is $H_{Q}(\lambda,p)$.

2. (Bounds)¶ For every $\lambda\in\mathcal{L}_{d}$ and $p\in E_{d}$, $0\le H_{Q}(\lambda,p)\le\frac12\kappa_{d}^{2}\lVert p\rVert_{d}^{2}$.

3. (Differences in the law)¶ For all $\mu,\nu\in\mathcal{L}_{d}$ and $p\in E_{d}$,

$$H_{Q}(\nu,p)-H_{Q}(\mu,p)=-\frac12\lim_{n\to\infty}\sum_{i\in[d]}\Bigl(\mu\bigl(\Xi^{i}_{p,n}\Xi^{i}_{p,n}\bigr)-\nu\bigl(\Xi^{i}_{p,n}\Xi^{i}_{p,n}\bigr)\Bigr),$$

the limit existing in $(\mathbb{R},d_{\mathbb{R}})$.

4. (Admissibility)¶ $H_{Q}$ is an admissible Hamiltonian.