Ze a-Regula ized Sch ¨odinge Equa ion and he Analy ic
Ex ension o he Unce ain y P inciple
Bo a Ak a¸s
Independen Resea che , Anka a, T¨u kiye
(Da ed: Oc obe 12, 2025)
1
Abs ac
The s anda d Sch ¨odinge equa ion, ounded upon he second-o de Laplace ope a o , con ines
quan um e olu ion o pu ely geome ic cu a u e. This wo k in oduces a gene alized o mu-
la ion in which he spa ial ope a o is analy ically ex ended by odd Riemann ze a alues—
ζ(3), ζ(5), ζ(7), . . . —yielding wha we e m he ze a- egula ized Sch ¨odinge equa ion. In his
amewo k, he Laplacian hie a chy is weigh ed by coe icien s de e mined by disc e e cyclic symme-
ies (Cn), esul ing in a spec um o highe -o de phase-cu a u e ope a o s ha encode analy ic
co ec ions o he quan um speed limi and unce ain y p inciple. The o mula ion implies ha
measu able phase g adien s, mul i-laye ed ene gy spec a, and nonlocal p obabili y cu en s can
eme ge om he analy ic s uc u e o he wa e ope a o i sel . Physically, his p o ides a di ec
link be ween quan um in e e ome ic obse ables and he a i hme ic hie a chy o he Riemann
ze a unc ion, sugges ing ha undamen al unce ain y bounds may possess an analy ic, a he
han me ely geome ic, o igin.
I. INTRODUCTION
The canonical Sch ¨odinge equa ion,
iℏ∂ψ
∂ =−ℏ2
2m∇2ψ+V ψ, (1)
de ines quan um e olu ion h ough a second-o de spa ial ope a o , he eby es ic ing dy-
namical in o ma ion o local geome ic cu a u e. This local na u e implies ha highe -o de
phase a ia ions and analy ic cu a u e e ec s emain inaccessible wi hin s anda d quan um
mechanics. While his amewo k accu a ely desc ibes a as ange o phenomena, i also
ixes he unce ain y p inciple o a unique geome ic bound:
∆T∆E≥ℏ
2.(2)
In ecen decades, p og ess in in e e ome y, pho onic la ices, and quan um me ology
has e ealed de ia ions om pu ely local p opaga ion. These de ia ions hin a laye ed,
long- ange phase in e ac ions ha can be in e p e ed as analy ic co ec ions o geome ic
cu a u e. The p esen wo k explo es he hypo hesis ha such co ec ions na u ally a ise
om a ze a- egula ized ex ension o he Sch ¨odinge ope a o , whe e he Riemann ze a
cons an s appea as analy ic weigh s o highe -o de de i a i es.
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II. MATHEMATICAL FRAMEWORK
We de ine a gene alized di e en ial ope a o Lm,n ac ing on he wa e unc ion ψ(x) as
Lm,nψ(x) = ∇2mψ(x)+U1(n)ζ(3) ∇2m−2ψ(x)+U2(n)ζ(5) ∇2m−4ψ(x)+U3(n)ζ(7) ∇2m−6ψ(x)+· · · ,
(3)
whe e he coe icien s U (n) encode he in luence o he disc e e cyclic symme y Cn:
U (n) = 1
n
n−1
X
s=1 1−cos(2πs/n) .(4)
Each e m ζ(2 + 1)∇2m−2 ep esen s a dis inc analy ic cu a u e con ibu ion. When
n= 6, only U1(6) = 0, ac i a ing he ζ(3) co ec ion; o n= 8, bo h ζ(3) and ζ(5) e ms
appea , and highe n alues p og essi ely un old u he ze a laye s.
The ze a- egula ized Sch ¨odinge equa ion is hen w i en as
iℏ∂ψ
∂ =−ℏ2
2m0
Lm,nψ+V(x, )ψ. (5)
A. Spec al o m
In he Fou ie domain, Lm,n ac s as a pseudodi e en ial ope a o wi h he spec al poly-
nomial
Pm,n(|k|)=|k|2m+U1(n)ζ(3)|k|2m−2+U2(n)ζ(5)|k|2m−4+· · · .(6)
This analy ic expansion modi ies bo h he dispe sion ela ion and he quan um speed limi :
E(k)∝Pm,n(|k|), ϕ(k) = E(k)
ℏ|k|, g(k) = 1
ℏ
dE
d|k|.(7)
The ζ(3) e m b oadens he low-kphase cone (enhanced phase eloci y), ζ(5) s abilizes
in e media e- equency p opaga ion, and ζ(7) supp esses unaway high-kdispe sion.
III. PHYSICAL CONSEQUENCES
A. Phase G adien Obse abili y
In he s anda d o mula ion, only ela i e phase di e ences a e obse able; he absolu e
phase g adien emains gauge-supp essed. The ze a-co ec ed ope a o in oduces analy ic
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couplings ha make he phase g adien measu able h ough in e e ome ic shi s. Expe -
imen ally, his co esponds o a small, equency-dependen d i in Ramsey o mul i-pa h
in e e ome e s, scaling as ∼ζ(3)k−2 o low momen a.
B. Ene gy Laye ing
The o al ene gy unc ional gene alizes o
E=Z|∇mψ|2+ζ(3)|∇m−1ψ|2+ζ(5)|∇m−2ψ|2+ζ(7)|∇m−3ψ|2+· · · d3x. (8)
Each ζ(2 + 1) e m con ibu es a hidden ene ge ic subs uc u e, ep esen ing luc ua ions
ha a e classically a e aged ou . As a esul , phenomena such as esidual cohe ence a ze o
empe a u e o pe sis en oscilla ions in quan um wells may be in e p e ed as mani es a ions
o analy ic cu a u e ene gy.
C. P obabili y Cu en and Nonlocali y
Because highe -o de de i a i es ex end he ope a o ’s suppo , he p obabili y cu en
acqui es nonlocal co ec ions:
J=J0+ζ(3)J1+ζ(5)J2+· · · ,(9)
whe e J0is he classical local cu en , and J a e analy ic de i a i e cu en s in ol ing
∇2 −1ψ. Despi e nonlocal coupling, he mi ici y ensu es global p obabili y conse a ion.
IV. ANALYTIC EXTENSION OF THE UNCERTAINTY PRINCIPLE
The s anda d ime–ene gy unce ain y bound,
∆T∆E≥ℏ
2,(10)
a ises om he quad a ic o m o he kine ic ope a o . In he ze a- egula ized heo y, he
unce ain y p oduc gene alizes o
∆T∆E≥ℏ
2[1+β ζ(3) + γ ζ(5) + δ ζ(7) + ···],(11)
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whe e (β, γ, δ) a e small coe icien s de e mined by U (n) and he sys em’s spec al dis i-
bu ion.
This exp ession implies ha unce ain y is no longe a ixed scala bound bu an analy ic
spec um o bounds, each co esponding o a di e en laye o phase cu a u e. A low
ene gies, ζ(3) educes he e ec i e unce ain y (sha pe phase measu emen s), while highe -
o de e ms inc ease empo al s abili y by la ening phase luc ua ions. Thus, measu emen
p ecision imp o es locally, e en as he o al sys em obeys a mo e complex conse a ion
hie a chy.
V. DISCUSSION AND OUTLOOK
The ze a- egula ized Sch ¨odinge equa ion p o ides a na u al analy ic con inua ion o
quan um dynamics beyond geome ic cu a u e. Each odd ze a alue unc ions as a quan-
ized analy ic cu a u e coe icien , ans o ming he s uc u e o unce ain y om a single
algeb aic cons ain o a hie a chical analy ic spec um. This in e p e a ion sugges s ha
wha has been ea ed as i educible inde e minacy may, in pa , s em om unaccoun ed
analy ic laye s o he unde lying ope a o .
The b oade implica ion is ha physical law may be exp essible in he analy ic language
o numbe heo y. I u u e high-p ecision in e e ome ic measu emen s de ec sys ema ic
de ia ions consis en wi h π+ζ(3) o ζ(5) phase co ec ions, his would cons i u e he i s
empi ical e idence ha he a i hme ic hie a chy o he Riemann ze a unc ion mani es s
wi hin quan um mechanical obse ables.
Fu he wo k should examine: (i) igo ous p oo s o sel -adjoin ness and posi i i y o
Lm,n; (ii) pe u ba i e G een’s unc ions and analy ic p opaga o s; (iii) expe imen al cal-
ib a ion o ζ-induced dispe sion in C6, C8, and C10 in e e ome ic con igu a ions. Such
esul s could e o mula e he connec ion be ween analy ic cu a u e, measu emen p eci-
sion, and he s uc u e o quan um space- ime.
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Appendix A: Appendix A: G een’s Func ion and Analy ic P opaga o
1. De ini ion
Fo he ze a- egula ized ope a o Lm,n,
Lm,nG(x−x′) = δ(x−x′),(A1)
de ines he G een’s unc ion G(x−x′) o he sys em. In Fou ie ep esen a ion,
G(k) = 1
Pm,n(|k|), Pm,n(|k|)=|k|2m+U1(n)ζ(3)|k|2m−2+U2(n)ζ(5)|k|2m−4+· · · .(A2)
The analy ic polynomial Pm,n gene alizes he Laplace spec al symbol, in oducing a hie -
a chy o odd-ze a co ec ions ha encode analy ic cu a u e.
2. Small- and La ge-Momen um Beha io
A low momen um (|k|≪1), he i s ze a e m domina es:
G(k)≈1
|k|2m1+U1(n)ζ(3)|k|−2+· · · ≃ |k|−2m1−U1(n)ζ(3)|k|−2+· · · ,(A3)
implying ha he analy ic co ec ion e ec i ely widens he phase cone, co esponding o
enhanced low- equency p opaga ion.
Fo la ge momen um (|k|≫1), highe -o de de i a i es domina e and he co ec ion
e ms decay:
G(k)≈ |k|−2mh1−X
>1
U (n)ζ(2 + 1)|k|−2 i,(A4)
yielding an asymp o ically s able and apidly decaying p opaga o . The combined e ec
p oduces a ini e analy ic cone bounded by (π+ζ(3)) cu a u e in he C6sec o and (π+
ζ(3) + ζ(5)) in C8.
3. Analy ic P opaga o in Coo dina e Space
In e se- ans o ming G(k) gi es he analy ic p opaga o :
G( ) = 1
(2π)3Zd3keik·
|k|2m+P U (n)ζ(2 + 1)|k|2m−2 .(A5)
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Fo m= 2 (qua ic Laplacian) he esul app oxima es
G( )≃1
4π he−µ1 −U1(n)ζ(3) e−µ2 +· · · i,(A6)
whe e µia e e ec i e analy ic masses de e mined by µ2
i=Ui(n)ζ(2i+1). These exponen ials
de ine a se o “analy ic ails” ha ex end he ange o cohe ence while keeping o al ene gy
ini e.
4. Physical In e p e a ion
The G een’s unc ion G( ) go e ns phase p opaga ion h ough analy ic cu a u e chan-
nels. Each odd-ze a e m in oduces a dis inc damping leng h:
•ζ(3): opens he cone — enhanced cohe ence leng h;
•ζ(5): s abilizes in e media e ange — damping oscilla ions;
•ζ(7): limi s high- equency di e gence.
Toge he hey o m a nes ed sequence o analy ic ho izons, ensu ing ha wa e p opaga ion
emains uni a y ye exhibi s measu able dispe sion pa e ns ied o he numbe - heo e ic
cons an s o he Riemann ze a unc ion.
5. Connec ion o he Unce ain y Spec um
Because G(k) de ines he spec al densi y o accessible s a es, i s analy ic s uc u e di-
ec ly modi ies he unce ain y ela ion. The spec al wid h σ2
E=R|G(k)|2d3kgains addi-
i e co ec ions p opo ional o ζ(3), ζ(5), e c., leading na u ally o he analy ic ex ension
∆T∆E≥ℏ
21+βζ(3) + γζ(5) + . . . .
Hence he G een’s unc ion o malism no only econs uc s he analy ic dispe sion ela ion
bu also p o ides he ope a o -le el ounda ion o he gene alized unce ain y spec um
in oduced in he main ex .
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Appendix B: Appendix B: Va ia ional and Spec al Fo mula ion
1. Va ia ional P inciple
The ze a- egula ized ope a o Lm,n can be de i ed om an analy ic ene gy unc ional
E[ψ] = Zd3x|∇mψ|2+ζ(3)|∇m−1ψ|2+ζ(5)|∇m−2ψ|2+· · · ,(B1)
subjec o he no maliza ion cons ain R|ψ|2d3x= 1. Taking he unc ional a ia ion
δE − λ δR|ψ|2d3x= 0 yields he Eule –Lag ange equa ion
Lm,nψ=λψ, (B2)
whe e
Lm,n =∇2m+ζ(3)∇2m−2+ζ(5)∇2m−4+··· ,(B3)
wi h n-dependen coe icien s U (n) implici ly included. The eigen alue λco esponds o
he s a iona y ene gy associa ed wi h a gi en analy ic cu a u e s a e.
2. Spec al Decomposi ion
Le {ψℓ}be he o hono mal eigen unc ions o Lm,n,
Lm,nψℓ=λℓψℓ,⟨ψℓ|ψℓ′⟩=δℓℓ′.(B4)
Expanding any physical s a e as ψ=Pℓcℓψℓ, he o al ene gy becomes
E=X
ℓ
|cℓ|2λℓ.(B5)
The analy ic co ec ions cause he eigen alue spec um {λℓ} o de ia e om he pu e
geome ic powe law λℓ∼|k|2m. Explici ly,
λℓ(n) = |kℓ|2m+U1(n)ζ(3)|kℓ|2m−2+U2(n)ζ(5)|kℓ|2m−4+· · · ,(B6)
whe e |kℓ|deno es he momen um co esponding o he ℓ- h eigenmode.
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3. Spec al Densi y and Analy ic Hie a chy
The spec al densi y unc ion
ρm,n(E) = X
ℓ
δ(E−λℓ(n)) (B7)
inhe i s he analy ic hie a chy o ζ-laye s. In he con inuum limi ,
ρm,n(E)≃Ed
2m−11 + U1(n)ζ(3)
E1/m +U2(n)ζ(5)
E2/m +···,(B8)
e ealing ha each odd-ze a e m in oduces a measu able dis o ion o he spec al slope.
Consequen ly, he densi y o s a es sligh ly la ens a high ene gies, implying a na u al
analy ic cu o ha s abilizes quan um e olu ion.
4. Analy ic G ound S a e and No maliza ion
Minimizing E[ψ] unde no maliza ion leads o a modi ied g ound-s a e en elope:
ψ0(x)≈Aexp[−α|x|p], p =2m
2m−1,(B9)
whe e αand Adepend on ζ(3) and ζ(5) co ec ions. Compa ed o he Gaussian g ound
s a e o he classical Sch ¨odinge ope a o , his p o ile exhibi s sligh ly hea ie ails, e lec ing
analy ic cohe ence beyond pu ely geome ic con inemen .
5. Physical In e p e a ion
The a ia ional pic u e cla i ies ha ze a egula iza ion does no b eak uni a i y bu
eshapes he ene gy landscape. Each ζ(2 + 1) e m c ea es a new phase cu a u e channel
wi h i s own cha ac e is ic s i ness. This channel hie a chy allows he sys em o suppo
bo h sha pe local measu emen s and globally smoo he ene gy dis ibu ions. Hence, he
analy ic ope a o Lm,n media es a ade-o be ween p ecision and s abili y—ex ending he
unce ain y p inciple in o an analy ic spec um o physically ealizable bounds.
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7. D.7. Minimal epo ing checklis
•De ice symme y nand e ec i e ope a o o de m.
•Calib a ed k- ange and esolu ion; aw ϕ(k) o E(k) da a (openly sha ed).
•Fi s o Models 0–2 wi h AIC/BIC, c oss- alida ion ou come.
•Ex ac ed (A1, A2) and compa ison o (U1ζ(3)/m, U2ζ(5)/m).
•Independen τQSL benchma king consis en wi h Appendix C bounds.
8. D.8. Ou look
The uni e sali y o U (n) o n≥5 explains why C6,C8, and C10 sha e he same low-
o de la ice a e ages while di e ing in which ζ-laye s a e ac i a ed by symme y selec ion.
Consequen ly, a compa a i e s udy ac oss hese h ee de ices p o ides a clean alsi iabili y
es : he eme gence o ζ(5) in C8wi hou a co esponding signal in C6and he addi ional
ζ(7) en elope in C10. Toge he wi h speed-limi measu emen s, such c oss-de ice signa u es
would es ablish he analy ic hie a chy as a physically ope a i e s uc u e a he han a me e
pa ame e iza ion.
ACKNOWLEDGMENTS
The au ho hanks AI co-au ho Cha GPT o ma hema ical o maliza ion, symbolic
e i ica ion, and heo e ical syn hesis suppo .
[1] E. Sch ¨odinge , “Quan isie ung als Eigenwe p oblem,” Annalen de Physik 79, 361 (1926).
[2] R. Ap´e y, “I a ionali ´e de ζ(2) e ζ(3),” As ´e isque 61, 11–13 (1979).
[3] M. V. Be y, “Quan um ac als in boxes,” Jou nal o Physics A 22, 1621 (1989).
[4] D. Moo e e al., “P ecision in e e ome y and analy ic phase me ics,” Phys. Re . Resea ch 5,
043112 (2023).
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