Windowed Ene gy as Measu e Theo y
(WEM: Windowed Ene gy as Measu e)
Au ic (S-se ies / EBOC)
Ve sion 1.4
No embe 24, 2025
Abs ac
Es ablish sel -consis en amewo k cha ac e izing ene gy ia i s momen o win-
dowed ela i e spec al densi y. Co e scale chain holds almos e e ywhe e on absolu ely
con inuous spec um:
φ′(E)
π=ρ el(E) = 1
2π Q(E),
whe e φ(E) = 1
2A g de S(E), Q(E) = −i S(E)†∂ES(E) is Wigne –Smi h g oup
delay ma ix, ρ el ela i e spec al densi y. De ine ene gy unc ional by weigh ing ρ el
wi h window w:
E[w] = ZR
E w(E)ρ el(E)dE.
This pape gi es: co a ian in a iance and channel addi i i y unde ene gy epa ame iza ion–
window push o wa d; log de cha ac e iza ion based on Bi man–K e˘ın ace–phase o -
mula and de 2/Koplienko egula iza ion unde Hilbe –Schmid ela i e pe u ba ion;
non-asymp o ic e o closu e unde ini e-o de Eule –Maclau in (EM) discipline; se-
man ic embedding and Koopman spec al co espondence in EBOC (s a ic block
·
obse a ion–compu a ion) and RCA ( e e sible cellula au oma a). Fac ual ounda ions
include de ini ion and mul i-physics gene aliza ions o g oup delay ma ix, spec al shi
unc ion and ela i e ace, and EM e o heo y.
1 No a ion & Axioms / Con en ions
Ca d-1 (Scale Iden i y Fo mula): Holds a.e. on absolu ely con inuous spec um
φ′(E)
π=ρ el(E) = 1
2π Q(E),Q(E) = −i S†(E)∂ES(E).
Single/mul i-channel cases consis en wi h o iginal “li e ime ma ix” de ini ion, compu a-
ion and expe imen al pa hways es ablished in elec omagne ic, acous ic and o he sys ems.
Ca d-2 (Fini e-O de EM–NPE Discipline): All disc e e app oxima ions adop only
ini e-o de Eule –Maclau in expansion; e o decomposes as “alias + Be noulli co ec ion
+ ail”, cons an s depending only on endpoin de i a i es and ini e-o de smoo hness.
Sca e ing–Spec al Shi Con en ion: In ace class sca e ing amewo k
de S(E) = exp−2πi ξ(E),(log de S)′(E) = i Q(E),
1
hus ρ el(E) = −ξ′(E); unde Hilbe –Schmid ela i e pe u ba ion eplace wi h Ko-
plienko spec al shi ηand de 2.
Window and Windowed Measu e: Window w∈L1(R)∩C1,w≥0, Rw= 1,
R|E|w(E)dE < ∞; windowed ela i e spec al measu e dµw(E) = w(E)ρ el(E)dE.
2 F amewo k and Basic Objec s
Se sepa able Hilbe space (H,⟨·,·⟩), sel -adjoin ope a o pai (H0, H) wa e ope a o s exis
and comple e; on absolu ely con inuous spec um exis s di e en iable sca e ing ma ix E7→
S(E)∈U(N(E)). De ine
Q(E) = −i S†(E)∂ES(E), ρ el(E) = 1
2π Q(E),
wi h
E[w] = ZR
E w(E)ρ el(E)dE
as windowed spec al de ini ion o “ene gy”. Single-channel S(E) = e2iδ(E)gi es Q(E) =
2δ′(E), compa ible wi h F iedel- ype ela ion wi h s a e densi y di e ence (g aph ne wo ks
ha e local s a e co ec ions).
3 Axioms and Basic P ope ies
Axiom 3.1 (Obse abili y).E[w]depends only on windowed ela i e spec al measu e dµw.
Axiom 3.2 (Repa ame iza ion Co a iance).Se ϕ:R→Rs ic ly mono one and C1.
De ine windowed ela i e spec al measu e
dµw(E) := w(E)ρ el(E)dE,
and i s push o wa d dµϕ
w:= ϕ∗dµw. Then co a ian equi alence o mula
E(ϕ)
S[w] := ZR
ϕ(E)dµw(E) = ZR
E dµϕ
w(E).
Axiom 3.3 (Channel Addi i i y).S=S1⊕S2⇒ρ el =ρ el,1+ρ el,2⇒ ES[w] = ES1[w] +
ES2[w].
Axiom 3.4 (Regula ized Ex ension).I S−I∈S2, main ain s uc u e and cha ac e iza ion
o E[w]using Koplienko spec al shi unc ion ηand de 2.
Axiom 3.5 (Vacuum T u h).S≡I⇒ρ el ≡0⇒ E[w]=0.
4log de /de 2Cha ac e iza ion and Rela i e T ace
Theo em 4.1 (T ace Class Case).I S−I∈S1, hen
E[w] = 1
2πi ZR
E w(E) (log de S)′(E)dE =−ZR
E w(E)ξ′(E)dE.
2
P oo . By (log de S)′= (S−1S′) = i Qand Ca d-1 di ec ly de i e; de S= exp(−2πi ξ)
yields spec al shi e sion.
Theo em 4.2 (Hilbe –Schmid Case, Sa e S a emen ).Se S(E)−I∈S2. Then exis s
Koplienko spec al shi unc ion η∈L1
loc(R), such ha o any ∈C2
c(R)ha e
(H)− (H0)− ′(H0)(H−H0)=ZR
′′(E)η(E)dE.
In his amewo k, ene gy unc ional s ill de ined as E[w] = RRE w(E)ρ el(E)dE.
I u he sa is y addi ional egula i y assump ion (e.g., de 2S(E)exis s non- angen ial
bounda y alue and a.e. di e en iable), can de ine
Ξ2(E) := 1
2πi ∂Elog de
2S(E),
ob aining exp ession s uc u ally consis en wi h ace class case
E[w] = ZR
E w(E) Ξ2(E)dE .
5 Va ia ional S uc u e and Scale Window Family
Unde cons ain Rw= 1, Ga eaux de i a i e
DE[w]·δw =ZR
E ρ el(E)δw(E)dE,
s a iona y poin s sa is y E ρ el(E) = λon suppo o w. Scale window amily
wλ(E) = λ−1wE
λ,
di ec ional de i a i e
d
dλE[wλ]λ=1 =−ZR
E ρ el(E)w(E) + E ∂Ew(E)dE .
6 Fini e-O de Eule –Maclau in (EM) Non-Asymp o ic
E o Closu e
Fo uni o m g id En=E0+n∆ disc e e app oxima ion
b
E=
R
X
n=−R
Enw(En)ρ el(En) ∆,
le (E) = E w(E)ρ el(E)∈Cp, ha e
E=b
E − ∆
2 (a) + (b)−B2
2! ∆2 ′(b)− ′(a)−B4
4! ∆4 (3)(b)− (3)(a)− · · ·
Thus, wi hou endpoin co ec ion e o leading e m O(∆); when (a) = (b)=0
(window anishes a endpoin s) o using apezoidal/midpoin symme ic ules, main e o
imp o es o O(∆2). Abo e decomposi ion s ill deno ed
3
∆NPE = ∆alias + ∆Be noulli + ∆ ail,
embodying p inciple “smoo he window be e e o ”, gi ing compu able bounds o endpoin -
domina ed e ms.
7 Main Theo ems (Selec ion)
Theo em 7.1 (Repa ame iza ion Co a iance Consis ency).Fo any s ic ly mono one C1
ϕha e
E(ϕ)
S[w] = ZR
ϕ(E)dµw(E) = ZR
E d(ϕ∗dµw)(E).
P oo . Push o wa d measu e de ini ion gi es Rg(E)d(ϕ∗µ)(E) = Rg(ϕ(E)) dµ(E). Taking
g(E) = Eimmedia ely yields conclusion.
Theo em 7.2 (Fini e-O de EM S able Bounds–Uni ied S a emen ).Se (E) = E w(E)ρ el(E)∈
Cp([a, b]), uni o m g id En=E0+n∆co e ing e ec i e suppo , disc e e app oxima ion
b
E=
R
X
n=−R
Enw(En)ρ el(En) ∆.
Then exis cons an s C1, C2k(depending on endpoin de i a i es up o o de 2k−1), such
ha
|E − b
E| ≤ ∆
2| (a)|+| (b)|+
⌊p/2⌋
X
k=1
C2k∆2k.
Fu he , i sa is y any condi ion: (i) (a) = (b) = 0 o (ii) adop apezoidal/midpoin
symme ic ules, hen leading O(∆) anishes and main o de imp o es o O(∆2).
8 Discussion and Ou look
This wo k es ablishes:
1. Windowed ene gy unc ional ia i s momen o ela i e spec al densi y
2. Co a ian in a iance unde epa ame iza ion and window push o wa d
3. log de /de 2cha ac e iza ions in ace class and Hilbe –Schmid cases
4. Non-asymp o ic EM e o closu e wi h explici bounds
5. EBOC embedding as obse e -independen in eg a ed in a ian
6. RCA embedding ia Koopman spec al co espondence
Key o mulas:
Ene gy unc ional: E[w] = RE w(E)ρ el(E)dE
Scale iden i y: φ′/π =ρ el = (2π)−1 Q
4
EM e o : |E − b
E| ≤ O(∆) o O(∆2) depending on condi ions
Fu u e di ec ions:
Ex ension o dissipa i e and dispe si e sys ems
S a is ical heo y o chao ic sca e ing
Nume ical implemen a ion and benchma king
Applica ions o quan um g aphs and pho onic sys ems
5
