The coherent face and its budget

Coherence keeps the modes of the carrier and changes how they evolve. The choice of symbol controls the phase rates, the resonances and the stability of the resulting budget.

6. The Coherent / Reversible Face

The coherent face uses a bounded real symbol $h$ of $K$.

6.1 Admissible coherence symbols

Definition 6.1. Admissible coherence symbol.

An admissible coherence symbol is a Borel $h:[0,\infty)\to\mathbb R$ with: $h$ real-valued; $h$ bounded; $h(0)=0$; $h$ nondecreasing; if $K\ne0$, the coherent response is required to be nontrivial, $h(K)\ne0$. The zero carrier is admitted as a degenerate case.

Example 6.2. Default symbol.

For $\lambda_0>0$, $h_{\lambda_0}(\lambda) = \tfrac{2}{\pi}\arctan(\lambda/\lambda_0)$ is real, bounded by $1$, satisfies $h(0)=0$, and is strictly increasing.

6.2 The coherence generator and unitary flow

Definition 6.3. Coherence generator.

$C := h(K)$. Bounded selfadjoint with $\|C\|\le\|h\|_\infty$ and $C \in W^*(K)$.

Definition 6.4. Reversible flow.

$U_t := e^{-itC} = e^{-it\,h(K)}$, $t \in \mathbb R$. A strongly continuous one-parameter unitary group.

Proposition 6.5. Mode preservation.

If $K e_j = \lambda_j e_j$, then $C e_j = h(\lambda_j) e_j$ and $U_t e_j = e^{-it h(\lambda_j)} e_j$. Every spectral subspace of $K$ is $U_t$-invariant.

6.3 The reversible Fejér compression

Definition 6.6. Coherence Fejér operator.

$\Gamma_T(C) := (1 - e^{-iTC})/(iTC) = \tfrac{1}{T}\int_0^T e^{-itC}\,dt$, identity on $\ker C$.

Definition 6.7. Coherent compression.

$\mathbf C_T := \Gamma_T^*\Gamma_T = \tfrac{2(1-\cos(TC))}{T^2 C^2}$, with eigenvalues $\mathop{\mathrm{sinc}}^2(T\mu/2)$ on $(\ker C)^\perp$. Here $\mathop{\mathrm{sinc}}x := \sin x / x$, with $\mathop{\mathrm{sinc}}0 := 1$.

6.4 The coherence budget

Definition 6.8. Coherence compression budget: full convention.

Let

$$q_T(\mu) \;:=\; \mathop{\mathrm{sinc}}^2(T\mu/2),\qquad \mathop{\mathrm{sinc}}x := \tfrac{\sin x}{x},\quad \mathop{\mathrm{sinc}}0 := 1.$$

For $C = h(K)$, define the resonant zero-mode subspace

$$\mathcal R_T(C) \;:=\; E_C\!\left(\left\{\mu\in\mathop{\mathrm{\sigma}}(C)\setminus\{0\}:\tfrac{T\mu}{2}\in\pi\mathbb Z\setminus\{0\}\right\}\right)\mathsf H.$$

Equivalently, $\mathcal R_T(C)$ is the part of $(\ker C)^\perp$ annihilated by $\mathbf C_T = \Gamma_T(C)^*\Gamma_T(C)$.

Under (D3), the full coherence budget convention used in this paper is

$$\mathfrak B_T(C) \;:=\; \begin{cases} -\log\det\nolimits_*\!\big(\mathbf C_T\big|_{(\ker C)^\perp}\big), & \mathcal R_T(C) = \{0\},\\[0.3em] +\infty, & \mathcal R_T(C)\ne\{0\}. \end{cases}$$

Thus any nonzero coherence mode annihilated by $\mathbf C_T$ produces infinite full compression loss. If $(\ker C)^\perp = \{0\}$, we set $\mathfrak B_T(C) = 0$.

In finite rank, with nonzero coherence eigenvalues $\mu_j$, this reads

$$\mathfrak B_T(C) \;=\; \begin{cases} -\displaystyle\sum_j \log\mathop{\mathrm{sinc}}^2(T\mu_j/2), & \mathop{\mathrm{sinc}}(T\mu_j/2)\ne 0\text{ for every }j,\\[0.6em] +\infty, & \mathop{\mathrm{sinc}}(T\mu_j/2) = 0\text{ for at least one }j. \end{cases}$$

When no confusion is possible we write $\mathfrak B_T(K)$ for $\mathfrak B_T(h(K))$.

Remark 6.9. Full versus reduced resonant convention.

The coherent Fejér compression differs from the dissipative Fejér transmission in one essential respect. The dissipative eigenvalues

$$f_T(\lambda) \;=\; \frac{1-e^{-T\lambda}}{T\lambda}$$

are strictly positive for $\lambda>0$, whereas the coherent compression eigenvalues

$$q_T(\mu) \;=\; \mathop{\mathrm{sinc}}^2(T\mu/2)$$

vanish at the resonant values $T\mu/2 \in \pi\mathbb Z\setminus\{0\}$. There are therefore two distinct conventions.

1. Full determinant convention. The convention used in the paper unless explicitly stated otherwise. The kernel of $C$ is ignored because $\Gamma_T(C)$ is the identity there and contributes no loss. Any resonant zero mode inside $(\ker C)^\perp$ is not ignored: it makes the full budget equal to $+\infty$.

2. Reduced resonant pseudodeterminant convention. One may instead remove both $\ker C$ and the resonant zero modes:

   $$\mathsf H_{T,\mathrm{red}} \;:=\; \big(\ker C \oplus \mathcal R_T(C)\big)^\perp,$$

   and define

   $$\mathfrak B_{T,\mathrm{red}}(C) \;:=\; -\log\det\nolimits_*\!\big(\mathbf C_T\big|_{\mathsf H_{T,\mathrm{red}}}\big),$$

   when the determinant on the reduced subspace is defined. In finite rank this is the product over the remaining strictly positive compression eigenvalues only:

   $$\mathfrak B_{T,\mathrm{red}}(C) \;=\; -\sum_{\substack{\mu_j\ne 0\\ \mathop{\mathrm{sinc}}(T\mu_j/2)\ne 0}} \log\mathop{\mathrm{sinc}}^2(T\mu_j/2).$$

On nonresonant scale windows, $\mathcal R_T(C) = \{0\}$, so the full and reduced conventions agree up to the harmless omission of $\ker C$, where the compression is the identity. At resonant scales they do not agree: the reduced budget is a pseudodeterminant after deleting total-loss modes, and is therefore not the full compression-loss budget.

Proposition 6.10. Properties of $\mathfrak B_T$.

For $T>0$ the full budget satisfies $\mathfrak B_T(K)\ge0$, with equality if and only if $C=0$ on the retained space. The compression $\mathbf C_T$ is a Borel function of $K$, and $\mathfrak B_T(UKU^*)=\mathfrak B_T(K)$ whenever the determinant convention is transported by $U$. Hence the budget is basis-independent.

Proof. For real $\mu$, $0\le q_T(\mu)\le1$, with $q_T(\mu)=1$ exactly when $\mu=0$, since $|\sin x|<|x|$ for $x\ne0$. Each nonzero retained mode therefore contributes a strictly positive loss, or infinite loss at resonance. In the Fredholm case, the positive compact operator $I-\mathbf C_T$ has a spectral product with the same conclusion: zero loss forces every factor to be one. Functional calculus and unitary invariance of the determinant prove the remaining assertions. ◻

6.5 Symbol-class stability

Proposition 6.11. Borel-symbol stability.

Let $K\ge 0$ be selfadjoint, and let $h_1, h_2 : [0,\infty)\to\mathbb R$ be bounded Borel symbols. Then

$$\|h_1(K) - h_2(K)\| \;\le\; \sup_{\lambda\in\mathop{\mathrm{\sigma}}(K)}|h_1(\lambda)-h_2(\lambda)|.$$

For a spectral window $[0,R]$,

$$\big\|(h_1(K) - h_2(K))\,E_K([0,R])\big\| \;\le\; \|h_1 - h_2\|_{\infty,[0,R]},$$

where the right-hand side denotes the supremum over $[0,R]\cap\mathop{\mathrm{\sigma}}(K)$, or over $[0,R]$ if the symbols are declared on the whole interval.

Proof. Let $\varphi = h_1 - h_2$. By the bounded Borel functional calculus, $h_1(K) - h_2(K) = \varphi(K)$ and $\|\varphi(K)\| \le \|\varphi\|_{L^\infty(\mathop{\mathrm{\sigma}}(K))}$. Moreover, $(h_1(K) - h_2(K))\,E_K([0,R]) = (\varphi\,\mathbf 1_{[0,R]})(K)$, so the same functional-calculus norm bound gives the windowed estimate. ◻

Lemma 6.12. Finite-rank nonresonant determinant-budget stability.

Let $P_R := E_K([0,R])$ have finite rank $r < \infty$. Fix $T>0$ and define $q_T(u) := \mathop{\mathrm{sinc}}^2(Tu/2)$. For a real bounded Borel symbol $h$, define the finite-window coherence budget

$$\mathcal B_{T,R}(h) \;:=\; -\sum_{\lambda\in\mathop{\mathrm{\sigma}}(K|_{P_R\mathsf H})}^{m(\lambda)}\log q_T(h(\lambda)),$$

whenever all terms are positive. Here eigenvalues are counted with multiplicity, and modes with $h(\lambda) = 0$ contribute $-\log q_T(0) = 0$.

Let $h_1, h_2$ be real bounded Borel symbols. Set

$$M \;:=\; \max\{\|h_1\|_{\infty,[0,R]}, \|h_2\|_{\infty,[0,R]}\},\qquad L_{T,M} \;:=\; \sup_{|u|\le M}|q_T'(u)|.$$

Assume there is $\delta > 0$ such that $q_T(h_\ell(\lambda)) \in [\delta,1]$ for every $\lambda\in\mathop{\mathrm{\sigma}}(K|_{P_R\mathsf H})$ and for $\ell = 1,2$. Then both finite-window budgets are full-convention nonresonant budgets on $P_R\mathsf H$, and

$$\big|\mathcal B_{T,R}(h_1) - \mathcal B_{T,R}(h_2)\big| \;\le\; \frac{r\,L_{T,M}}{\delta}\,\|h_1 - h_2\|_{\infty,[0,R]}.$$

Proof. For each eigenvalue $\lambda$ in the finite window, put $a_\ell(\lambda) := q_T(h_\ell(\lambda))$ for $\ell = 1,2$. By hypothesis $a_\ell(\lambda)\in[\delta,1]$. Since $x\mapsto \log x$ has derivative bounded by $\delta^{-1}$ on $[\delta,1]$,

$$|\log a_1(\lambda) - \log a_2(\lambda)| \;\le\; \delta^{-1}|a_1(\lambda) - a_2(\lambda)|.$$

The function $q_T$ is smooth, hence Lipschitz on $[-M,M]$ with Lipschitz constant $L_{T,M}$. Therefore

$$|a_1(\lambda) - a_2(\lambda)| \;\le\; L_{T,M}|h_1(\lambda) - h_2(\lambda)| \;\le\; L_{T,M}\|h_1 - h_2\|_{\infty,[0,R]}.$$

Summing over the $r$ eigenvalues in the finite window gives the stated estimate. ◻

Corollary 6.13. Small perturbations stay nonresonant.

In the setting of Lemma 6.12, suppose $h$ satisfies $q_T(h(\lambda)) \ge \delta$ for every $\lambda\in\mathop{\mathrm{\sigma}}(K|_{P_R\mathsf H})$, and let $k$ be another real bounded Borel symbol with $\varepsilon := \|h - k\|_{\infty,[0,R]} \le 1$. Put $M_h := \|h\|_{\infty,[0,R]}$ and $L_{T,M_h+1} := \sup_{|u|\le M_h+1}|q_T'(u)|$. If $L_{T,M_h+1}\,\varepsilon \le \delta/2$, then

$$q_T(k(\lambda)) \;\ge\; \tfrac{\delta}{2} \qquad \text{on }\mathop{\mathrm{\sigma}}(K|_{P_R\mathsf H}),$$

and

$$\big|\mathcal B_{T,R}(k) - \mathcal B_{T,R}(h)\big| \;\le\; \frac{2r\,L_{T,M_h+1}}{\delta}\,\varepsilon.$$

Remark 6.14. Scope of the stability statement.

Proposition 6.11 is a general bounded-functional-calculus estimate. Lemma 6.12 and Corollary 6.13 are deliberately finite-rank, nonresonant determinant statements. They do not assert an infinite-dimensional determinant-continuity theorem without additional Fredholm, heat-kernel, zeta, or finite-volume hypotheses.

6.6 Properties shared by the two faces

Lemma 6.15.

$C\mathop{\mathrm{dom}}(K)\subseteq\mathop{\mathrm{dom}}(K)$ and $KC=CK$ on $\mathop{\mathrm{dom}}(K)$; likewise $KU_t=U_tK$ on $\mathop{\mathrm{dom}}(K)$, $[\Lambda_T(K), U_s]=0$ for all $s,t,T$. Both $\{e^{-tK}\}_{t\ge 0}$ and $\{U_s\}_{s\in\mathbb R}$ lie in $W^*(K)$ and commute with each other.

Proof. All bounded members are Borel functions of $K$; their commutation with $K$ includes preservation of its operator domain. The algebra $W^*(K)$ is abelian. ◻

The carrier is canonical given its closed form and Hilbert metric. The coherent face is uniquely determined once the specific symbol $h$ is fixed; the admissible symbol class permits multiple choices.