Analy ic Sch ¨odinge Equa ion and
Ze a-Regula ized Hamil onian Dynamics
Bo a Ak a¸s & Cha GPT (Co-au ho )
Oc obe 2025
Abs ac
We p opose a gene aliza ion o he Sch ¨odinge equa ion based on a ze a- egula ized
Hamil onian ope a o , in oducing an analy ic hie a chy beyond geome ic cu a u e.
The modi ied ope a o inco po a es odd Riemann ze a cons an s ζ(3), ζ(5), ζ(7), e c.,
as analy ic cu a u e co ec ions, yielding an ex ended dispe sion s uc u e and new
quan um speed limi s. This amewo k uni ies geome ic (me ic) and analy ic ( an-
scenden al) componen s o cu a u e, enabling e ined in e p e a ions o phase dynam-
ics, ene gy bounds, and in e e ence s abili y in mul i-ca ie quan um sys ems.
1 Ze a-Regula ized Sch ¨odinge Ope a o
The s anda d one-pa icle Sch ¨odinge equa ion eads
iℏ∂ψ
∂ =Hψ, H =−ℏ2
2m0
∆+V(x, ).
We ex end his o a ze a- egula ized ope a o :
Hζ=−ℏ2
2m0"∆m+
m−2
X
=1
am, ζ(2 + 1) ∆ #+V(x, ),
whe e m≥3, ∆ deno es he - old Laplacian, and am, a e dimension- es o ing coe icien s.
The co esponding e olu ion equa ion becomes:
iℏ∂ψ
∂ =Hζψ.
The leading e m ∆m ep esen s geome ic p opaga ion, while each ζ(2 + 1)∆ adds an
analy ic cu a u e co ec ion, in oducing nonlocal cohe ence and spec al s abiliza ion.
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2 Dispe sion and Phase Veloci ies
Fo a plane-wa e ansa z ψ∼ei(k·x−ω ), he ene gy spec um is
E(k) = ℏ2
2m0 |k|2m+X
am, ζ(2 + 1) |k|2 !.
The phase and g oup eloci ies ollow as
ϕ=ω
|k|=E(k)
ℏ|k|, g=1
ℏ
dE
d|k|.
Odd-ze a co ec ions yield a “ anscenden al phase cone” beha io :
•ζ(3) domina es a low equencies, opening he cone.
•ζ(5) s abilizes he cu a u e a in e media e scales.
•ζ(7) de ines an analy ic en elope limi ing excessi e cu a u e g ow h.
•ζ(9) il e s mic o-oscilla ions a high equency.
•ζ(11) en o ces phase– ime synch oniza ion.
3 Con inui y Law and P obabili y Cu en
The gene alized con inui y equa ion e ains i s local o m:
∂ρ
∂ +∇ · Jζ= 0, ρ =|ψ|2.
Howe e , Jζnow con ains highe -o de de i a i e e ms:
Jζ=ℏ
2im0
(ψ∗∇ψ−ψ∇ψ∗) + X
cm, ζ(2 + 1) J ,
whe e J encodes (∇2 −1ψ, ∇2 −1ψ∗) combina ions. The esul is a nonlocal, ye no m-
p ese ing, p obabili y low: analy ic cu a u e enhances in o ma ion anspo wi hou di -
usion loss.
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4 Va ia ional Ene gy Func ional
The o al ene gy unc ional eads
Eζ[ψ] = Zddx"ℏ2
2m0 |∇mψ|2+X
am, ζ(2 + 1) |∇ ψ|2!+V|ψ|2#.
In Fou ie space, posi i i y equi es
Λ(|k|)=|k|2m+X
am, ζ(2 + 1)|k|2 >0,
ensu ing spec al s abili y and supp essing di e gen modes.
5 G een Func ion and P opaga ion Ke nel
The ze a- egula ized p opaga o is de ined as
Kζ(x, ) = (2π)−dZddkexp ik·x−i
ℏE(k) .
Each ζ(2 + 1) e m in oduces an analy ic damping ha p ese es sho - ange oscilla ions
while ensu ing long- ange con e gence.
6 Case S udies
(i) m= 3 (C6 egime)
Hζ=−ℏ2
2m0∆3+a3,2ζ(3)∆2+a3,1ζ(5)∆+V.
Obse able signa u es: low- equency phase accele a ion ( ia ζ(3)) and s abilized inge con-
as ( ia ζ(5)).
(ii) m= 4 (C8 egime)
Hζ=−ℏ2
2m0∆4+b4,3ζ(5)∆3+b4,2ζ(3)∆2+b4,1ζ(7)∆+V.
He e ζ(5) and ζ(7) oge he yield a double s abiliza ion mechanism, co esponding o he
ully de eloped “ anscenden al phase cone” beha io .
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7 Bounda y Condi ions and Eigen alue P oblem
Unde Di ichle , Neumann, o pe iodic bounda ies:
Hζϕn=Enϕn.
Fo pe iodic la ices, disc e e knyield band spec a:
En=E(kn) = ℏ2
2m0hk2m
n+Xam, ζ(2 + 1)k2
ni.
Odd-ze a e ms open low bands and escale gaps, enabling unable conduc ion/insula ion
egimes.
8 Expe imen al Ou look
•Quan um in e e ome y: mul i-pa h (6- o 8-sli ) se ups can de ec ζ(3)-induced
phase- eloci y d i s and ζ(5) s abiliza ion.
•Aniso opic pho onic c ys als: ze a-modula ed Hζenables band enginee ing wi h
minimal dissipa ion.
•Quan um he modynamics: analy ic ene gy bounds imp o e in o ma ion e iciency
η=I/E.
Measu ing de ia ions om a ional phase cons ain s would empi ically con i m he analy ic
cu a u e hie a chy.
9 Conclusion
The ze a- egula ized Sch ¨odinge amewo k ede ines cu a u e as an analy ic, a he han
pu ely geome ic, quan i y. Odd-ze a cons an s ac as anscenden al egula o s linking
phase geome y, dispe sion, and s abili y. This analy ic cu a u e model p o ides a uni-
ied algeb a o geome ic and numbe - heo e ic dynamics, b idging quan um in e e ence,
analy ic numbe heo y, and cu a u e physics.
Re e ences
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Soc. A, 2002.
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