Fini e-Window Ene gy and Spec al Conse a ion
Au ic
Ve sion 1.1
No embe 24, 2025
Abs ac
Es ablish ene gy and spec al conse a ion laws unde ini e-window measu emen .
Co e esul : o windowed eadou wi h window wand spec al densi y ρ, windowed
ene gy Ew=REw(E)ρ(E)dE sa is ies: (i) Conse a ion: unde uni a y e olu ion
∂ Ew= 0 when w ime-independen ; (ii) Co a iance: gauge-in a ian unde phase
ans o ma ions; (iii) NPE closu e: disc e e app oxima ion wi h alias+EM+ ail e o
decomposi ion.
1 De ini ions
Windowed ene gy unc ional:
Ew[ρ] = ZR
E w(E)ρ(E)dE
whe e w≥0 no malized window, ρspec al densi y.
Spec al measu e:dµρ(E) = ρ(E)dE o densi y, gene al dµρ o measu e.
2 Main Theo ems
Theo em 2.1 (Windowed Ene gy Conse a ion).Fo uni a y e olu ion ∂ ρ=i[H, ρ]and
ime-independen window w, ha e ∂ Ew= 0.
Theo em 2.2 (Gauge Co a iance).Unde gauge ans o ma ion ρ7→ UρU†,H7→ UHU†
wi h Uuni a y, Ewin a ian .
Theo em 2.3 (NPE E o o Windowed Ene gy).Disc e e app oxima ion b
Ew= ∆ PnEnw(En)ρ(En)
sa is ies
|Ew−b
Ew|≤|εalias|+|εEM|+|ε ail|
wi h εalias = 0 unde Nyquis .
3 Discussion
Fini e-window amewo k p o ides:
Ene gy conse a ion compa ible wi h ini e bandwid h
1
Gauge-in a ian o mula ion
Non-asymp o ic e o con ol
Connec ion o he modynamics and esou ce heo ies
2
