Phase De i a i e, Spec al Densi y
and Windowed Readou :
Uni ied Measu emen F amewo k
Au ic
Ve sion 2.1
No embe 24, 2025
Abs ac
Es ablish uni ied amewo k connec ing sca e ing phase de i a i e, spec al densi y,
and windowed eadou . Co e o mula holding a.e.:
φ′(E)
π=ρ el(E) = 1
2π Q(E)
whe e φsca e ing phase, ρ el ela i e spec al densi y om Bi man–K e˘ın de S=
e−2πiξ wi h ρ el =−ξ′,Q=−iS†∂ESWigne –Smi h delay ma ix.
Windowed eadou Rw=Rw(E)[h∗ρ el](E)dE wi h NPE h ee- e m e o decom-
posi ion. Applica ions: quan um me ology, sca e ing heo y, condensed ma e .
1 Co e De ini ions
De ini ion 1.1 (Rela i e Spec al Densi y).Fo sca e ing pai (H, H0) wi h S(E) sca e ing
ma ix:
ρ el(E) = −ξ′(E) = 1
2πi (S†∂ES) = 1
2π Q(E)
whe e ξspec al shi unc ion, QWigne –Smi h delay.
De ini ion 1.2 (Windowed Readou ).Fo window w, ke nel h:
Rw[ρ el] = ZR
w(E) [h∗ρ el](E)dE
2 Main Theo ems
Theo em 2.1 (Phase–Densi y Uni ica ion).On absolu ely con inuous spec um a.e., single-
channel S=e2iφ gi es
φ′(E) = π ρ el(E) = 1
2 Q(E).
Mul i-channel: 1
2π Q(E) = ρ el(E) = −ξ′(E).
P oo . F om Bi man–K e˘ın de S=e−2πiξ ge ∂EA g de S=−2πξ′. Jacobi o mula ∂Elog de S=
(S−1∂ES) wi h uni a i y S−1=S†gi es (S†∂ES) = −2πiξ′. De ini ion Q=−iS†∂ES
yields Q= 2πξ′, hus ρ el =−ξ′= (2π)−1 Q.
1
Theo em 2.2 (Windowed Readou NPE Decomposi ion).Fo disc e e app oxima ion wi h
s ep ∆, unca ion N:
|Rw−b
Rw| ≤ |εalias|+|εEM|+|ε ail|
wi h εalias = 0 when bandlimi ed + Nyquis ∆≤π/Ω.
Theo em 2.3 (Bo n P obabili y as I-P ojec ion).Unde alignmen condi ion, Bo n p oba-
bili y pi=⟨ψ, Eiψ⟩equals I-p ojec ion minimizing DKL(p∥q)o e cons ain se .
Theo em 2.4 (Poin e Basis as Ky Fan Minimum).Poin e basis {ek}minimizes Pk⟨ek, Wwek⟩
o window ope a o Ww=Rw(E)dEA(E)(Ky Fan minimum sum).
3 Applica ions
3.1 Quan um Me ology
Phase de i a i e measu emen ia windowed eadou p o ides op imal ene gy es ima ion
wi hin bandwid h cons ain s.
3.2 Sca e ing Theo y
Wigne –Smi h delay di ec ly measu able ia phase–ene gy co ela ion, connec ion o F iedel
sum ule.
3.3 Condensed Ma e
Local densi y o s a es (LDOS) in mesoscopic sys ems, quan um poin con ac s, esonan
unneling.
4 Discussion and Ou look
Uni ied amewo k es ablished connec ing:
Phase de i a i e φ′(obse able)
Spec al densi y ρ el ( heo e ical)
Delay ace Q(dynamical)
ia Bi man–K e˘ın–Wigne –Smi h chain, wi h windowed eadou p o iding expe imen al
b idge.
Key achie emen s:
1. Rigo ous scale iden i y o mula
2. NPE non-asymp o ic e o closu e
3. In o ma ion-geome ic Bo n p obabili y
4. Ky Fan poin e basis cha ac e iza ion
Fu u e di ec ions:
Ex ension o ime-dependen sca e ing
2
Open quan um sys ems and decohe ence
Expe imen al implemen a ions in quan um op ics
Connec ions o quan um ield heo y and g a i y
3
