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Zeta-Regularized Schr¨odinger Equation and the Analytic Extension of the Uncertainty Principle Bora Akta¸s Independent Researcher, Ankara, T¨urkiye (Dated: October 12, 2025) 1 Abstract The standard Schr¨odinger equation, founded upon the second-order Laplace operator, confines quantum evolution to purely geometric curvature. This work introduces a generalized formulation in which the spatial operator is analytically extended by odd Riemann zeta values— ζ(3), ζ(5), ζ(7), . . . —yielding what we term the zeta-regularized Schr¨odinger equation. In this framework, the Laplacian hierarchy is weighted by coefficients determined by discrete cyclic symmetries (Cn), resulting in a spectrum of higher-order phase-curvature operators that encode analytic corrections to the quantum speed limit and uncertainty principle. The formulation implies that measurable phase gradients, multi-layered energy spectra, and nonlocal probability currents can emerge from the analytic structure of the wave operator itself. Physically, this provides a direct link between quantum interferometric observables and the arithmetic hierarchy of the Riemann zeta function, suggesting that fundamental uncertainty bounds may possess an analytic, rather than merely geometric, origin. I. INTRODUCTION The canonical Schr¨odinger equation, iℏ∂ψ ∂t =−ℏ2 2m∇2ψ+V ψ, (1) defines quantum evolution through a second-order spatial operator, thereby restricting dynamical information to local geometric curvature. This local nature implies that higher-order phase variations and analytic curvature effects remain inaccessible within standard quantum mechanics. While this framework accurately describes a vast range of phenomena, it also fixes the uncertainty principle to a unique geometric bound: ∆T∆E≥ℏ 2.(2) In recent decades, progress in interferometry, photonic lattices, and quantum metrology has revealed deviations from purely local propagation. These deviations hint at layered, long-range phase interactions that can be interpreted as analytic corrections to geometric curvature. The present work explores the hypothesis that such corrections naturally arise from a zeta-regularized extension of the Schr¨odinger operator, where the Riemann zeta constants appear as analytic weights of higher-order derivatives. 2 II. MATHEMATICAL FRAMEWORK We define a generalized differential operator Lm,n acting on the wavefunction 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) where the coefficients Ur(n) encode the influence of the discrete cyclic symmetry Cn: Ur(n) = 1 n n−1 X s=1 1−cos(2πs/n)r.(4) Each term ζ(2r+ 1)∇2m−2rrepresents a distinct analytic curvature contribution. When n= 6, only U1(6) = 0, activating the ζ(3) correction; for n= 8, both ζ(3) and ζ(5) terms appear, and higher nvalues progressively unfold further zeta layers. The zeta-regularized Schr¨odinger equation is then written as iℏ∂ψ ∂t =−ℏ2 2m0 Lm,nψ+V(x, t)ψ. (5) A. Spectral form In the Fourier domain, Lm,n acts as a pseudodifferential operator with the spectral polynomial Pm,n(|k|)=|k|2m+U1(n)ζ(3)|k|2m−2+U2(n)ζ(5)|k|2m−4+· · · .(6) This analytic expansion modifies both the dispersion relation and the quantum speed limit: E(k)∝Pm,n(|k|), vϕ(k) = E(k) ℏ|k|, vg(k) = 1 ℏ dE d|k|.(7) The ζ(3) term broadens the low-kphase cone (enhanced phase velocity), ζ(5) stabilizes intermediate-frequency propagation, and ζ(7) suppresses runaway high-kdispersion. III. PHYSICAL CONSEQUENCES A. Phase Gradient Observability In the standard formulation, only relative phase differences are observable; the absolute phase gradient remains gauge-suppressed. The zeta-corrected operator introduces analytic 3 couplings that make the phase gradient measurable through interferometric shifts. Experimentally, this corresponds to a small, frequency-dependent drift in Ramsey or multi-path interferometers, scaling as ∼ζ(3)k−2for low momenta. B. Energy Layering The total energy functional generalizes to E=Z|∇mψ|2+ζ(3)|∇m−1ψ|2+ζ(5)|∇m−2ψ|2+ζ(7)|∇m−3ψ|2+· · · d3x. (8) Each ζ(2r+ 1) term contributes a hidden energetic substructure, representing fluctuations that are classically averaged out. As a result, phenomena such as residual coherence at zero temperature or persistent oscillations in quantum wells may be interpreted as manifestations of analytic curvature energy. C. Probability Current and Nonlocality Because higher-order derivatives extend the operator’s support, the probability current acquires nonlocal corrections: J=J0+ζ(3)J1+ζ(5)J2+· · · ,(9) where J0is the classical local current, and Jrare analytic derivative currents involving ∇2r−1ψ. Despite nonlocal coupling, hermiticity ensures global probability conservation. IV. ANALYTIC EXTENSION OF THE UNCERTAINTY PRINCIPLE The standard time–energy uncertainty bound, ∆T∆E≥ℏ 2,(10) arises from the quadratic form of the kinetic operator. In the zeta-regularized theory, the uncertainty product generalizes to ∆T∆E≥ℏ 2[1+β ζ(3) + γ ζ(5) + δ ζ(7) + ···],(11) 4 where (β, γ, δ) are small coefficients determined by Ur(n) and the system’s spectral distribution. This expression implies that uncertainty is no longer a fixed scalar bound but an analytic spectrum of bounds, each corresponding to a different layer of phase curvature. At low energies, ζ(3) reduces the effective uncertainty (sharper phase measurements), while higherorder terms increase temporal stability by flattening phase fluctuations. Thus, measurement precision improves locally, even as the total system obeys a more complex conservation hierarchy. V. DISCUSSION AND OUTLOOK The zeta-regularized Schr¨odinger equation provides a natural analytic continuation of quantum dynamics beyond geometric curvature. Each odd zeta value functions as a quantized analytic curvature coefficient, transforming the structure of uncertainty from a single algebraic constraint to a hierarchical analytic spectrum. This interpretation suggests that what has been treated as irreducible indeterminacy may, in part, stem from unaccounted analytic layers of the underlying operator. The broader implication is that physical law may be expressible in the analytic language of number theory. If future high-precision interferometric measurements detect systematic deviations consistent with π+ζ(3) or ζ(5) phase corrections, this would constitute the first empirical evidence that the arithmetic hierarchy of the Riemann zeta function manifests within quantum mechanical observables. Further work should examine: (i) rigorous proofs of self-adjointness and positivity for Lm,n; (ii) perturbative Green’s functions and analytic propagators; (iii) experimental calibration of ζ-induced dispersion in C6, C8, and C10 interferometric configurations. Such results could reformulate the connection between analytic curvature, measurement precision, and the structure of quantum space-time. 5 Appendix A: Appendix A: Green’s Function and Analytic Propagator 1. Definition For the zeta-regularized operator Lm,n, Lm,nG(x−x′) = δ(x−x′),(A1) defines the Green’s function G(x−x′) of the system. In Fourier representation, 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 analytic polynomial Pm,n generalizes the Laplace spectral symbol, introducing a hierarchy of odd-zeta corrections that encode analytic curvature. 2. Smalland Large-Momentum Behavior At low momentum (|k|≪1), the first zeta term dominates: G(k)≈1 |k|2m1+U1(n)ζ(3)|k|−2+· · · ≃ |k|−2m1−U1(n)ζ(3)|k|−2+· · · ,(A3) implying that the analytic correction effectively widens the phase cone, corresponding to enhanced low-frequency propagation. For large momentum (|k|≫1), higher-order derivatives dominate and the correction terms decay: G(k)≈ |k|−2mh1−X r>1 Ur(n)ζ(2r+ 1)|k|−2ri,(A4) yielding an asymptotically stable and rapidly decaying propagator. The combined effect produces a finite analytic cone bounded by (π+ζ(3)) curvature in the C6sector and (π+ ζ(3) + ζ(5)) in C8. 3. Analytic Propagator in Coordinate Space Inverse-transforming G(k) gives the analytic propagator: G(r) = 1 (2π)3Zd3keik·r |k|2m+PrUr(n)ζ(2r+ 1)|k|2m−2r.(A5) 6 For m= 2 (quartic Laplacian) the result approximates G(r)≃1 4πrhe−µ1r−U1(n)ζ(3) e−µ2r+· · · i,(A6) where µiare effective analytic masses determined by µ2 i=Ui(n)ζ(2i+1). These exponentials define a set of “analytic tails” that extend the range of coherence while keeping total energy finite. 4. Physical Interpretation The Green’s function G(r) governs phase propagation through analytic curvature channels. Each odd-zeta term introduces a distinct damping length: •ζ(3): opens the cone — enhanced coherence length; •ζ(5): stabilizes intermediate range — damping oscillations; •ζ(7): limits high-frequency divergence. Together they form a nested sequence of analytic horizons, ensuring that wave propagation remains unitary yet exhibits measurable dispersion patterns tied to the number-theoretic constants of the Riemann zeta function. 5. Connection to the Uncertainty Spectrum Because G(k) defines the spectral density of accessible states, its analytic structure directly modifies the uncertainty relation. The spectral width σ2 E=R|G(k)|2d3kgains additive corrections proportional to ζ(3), ζ(5), etc., leading naturally to the analytic extension ∆T∆E≥ℏ 21+βζ(3) + γζ(5) + . . . . Hence the Green’s function formalism not only reconstructs the analytic dispersion relation but also provides the operator-level foundation for the generalized uncertainty spectrum introduced in the main text. 7 Appendix B: Appendix B: Variational and Spectral Formulation 1. Variational Principle The zeta-regularized operator Lm,n can be derived from an analytic energy functional E[ψ] = Zd3x|∇mψ|2+ζ(3)|∇m−1ψ|2+ζ(5)|∇m−2ψ|2+· · · ,(B1) subject to the normalization constraint R|ψ|2d3x= 1. Taking the functional variation δE − λ δR|ψ|2d3x= 0 yields the Euler–Lagrange equation Lm,nψ=λψ, (B2) where Lm,n =∇2m+ζ(3)∇2m−2+ζ(5)∇2m−4+··· ,(B3) with n-dependent coefficients Ur(n) implicitly included. The eigenvalue λcorresponds to the stationary energy associated with a given analytic curvature state. 2. Spectral Decomposition Let {ψℓ}be the orthonormal eigenfunctions of Lm,n, Lm,nψℓ=λℓψℓ,⟨ψℓ|ψℓ′⟩=δℓℓ′.(B4) Expanding any physical state as ψ=Pℓcℓψℓ,the total energy becomes E=X ℓ |cℓ|2λℓ.(B5) The analytic corrections cause the eigenvalue spectrum {λℓ}to deviate from the pure geometric power law λℓ∼|k|2m. Explicitly, λℓ(n) = |kℓ|2m+U1(n)ζ(3)|kℓ|2m−2+U2(n)ζ(5)|kℓ|2m−4+· · · ,(B6) where |kℓ|denotes the momentum corresponding to the ℓ-th eigenmode. 8 3. Spectral Density and Analytic Hierarchy The spectral density function ρm,n(E) = X ℓ δ(E−λℓ(n)) (B7) inherits the analytic hierarchy of ζ-layers. In the continuum limit, ρm,n(E)≃Ed 2m−11 + U1(n)ζ(3) E1/m +U2(n)ζ(5) E2/m +···,(B8) revealing that each odd-zeta term introduces a measurable distortion of the spectral slope. Consequently, the density of states slightly flattens at high energies, implying a natural analytic cutoff that stabilizes quantum evolution. 4. Analytic Ground State and Normalization Minimizing E[ψ] under normalization leads to a modified ground-state envelope: ψ0(x)≈Aexp[−α|x|p], p =2m 2m−1,(B9) where αand Adepend on ζ(3) and ζ(5) corrections. Compared to the Gaussian ground state of the classical Schr¨odinger operator, this profile exhibits slightly heavier tails, reflecting analytic coherence beyond purely geometric confinement. 5. Physical Interpretation The variational picture clarifies that zeta regularization does not break unitarity but reshapes the energy landscape. Each ζ(2r+ 1) term creates a new phase curvature channel with its own characteristic stiffness. This channel hierarchy allows the system to support both sharper local measurements and globally smoother energy distributions. Hence, the analytic operator Lm,n mediates a trade-off between precision and stability—extending the uncertainty principle into an analytic spectrum of physically realizable bounds. 9 7. D.7. Minimal reporting checklist •Device symmetry nand effective operator order m. •Calibrated k-range and resolution; raw ϕ(k) or E(k) data (openly shared). •Fits to Models 0–2 with AIC/BIC, cross-validation outcome. •Extracted (A1, A2) and comparison to (U1ζ(3)/m, U2ζ(5)/m). •Independent τQSL benchmarking consistent with Appendix C bounds. 8. D.8. Outlook The universality of Ur(n) for n≥5 explains why C6,C8, and C10 share the same loworder lattice averages while differing in which ζ-layers are activated by symmetry selection. Consequently, a comparative study across these three devices provides a clean falsifiability test: the emergence of ζ(5) in C8without a corresponding signal in C6and the additional ζ(7) envelope in C10. Together with speed-limit measurements, such cross-device signatures would establish the analytic hierarchy as a physically operative structure rather than a mere parameterization. ACKNOWLEDGMENTS The author thanks AI co-author ChatGPT for mathematical formalization, symbolic verification, and theoretical synthesis support. [1] E. Schr¨odinger, “Quantisierung als Eigenwertproblem,” Annalen der Physik 79, 361 (1926). [2] R. Ap´ery, “Irrationalit´e de ζ(2) et ζ(3),” Ast´erisque 61, 11–13 (1979). [3] M. V. Berry, “Quantum fractals in boxes,” Journal of Physics A 22, 1621 (1989). [4] D. Moore et al., “Precision interferometry and analytic phase metrics,” Phys. Rev. Research 5, 043112 (2023). 16