SRMs meet Schrödinger: Multiple-ratio expansions for quantum dynamics and entanglement in a timeless dialogue.

1 SRMs meet Schrödinger: Multiple-ratio expansions for quantum dynamics and entanglement in a timeless dialogue. Author: Carlos Roberto França 1[0000-0002-6852-7103] 1Federal University of Fronteira Sul – UFFS/Campus Chapecó-Santa Catarina – Brazil [email protected] ABSTRACT If Schrödinger had had a ruler in 1926 that was already multiscale, he would probably have derived discretizations that reconcile phase coherence and nonlocality with fewer points. Infinite Series with Multiple Ratios (SRMs) provide such a ruler: a scheme with multiple coupling steps that preserves unitarity and introduces controlled homologous connections - a fertile ground for studying entanglement as a structural regularity under uncertainty. We propose SRM-Schr, a multiscale scheme for the time-dependent Schrödinger Equation that replaces the classical Laplacian with a SRM Laplacian (stencils with multiple ratios) and introduces a homologous coupling operator that induces calibratable nonlocal correlations. The evolution is done by unitary split-step (Crank–Nicolson/Strang), ensuring conservation of probability. The choice of SRM ratios is guided by physical scaling relations. (E = hf, λ = h/p), to align resonance and phase coherence without "forcing" determinism: we maintain the probabilistic nature but organize the multiscale structure of the wavefield. In 1D tests (free particle, oscillator, double barrier), SRM-Schr reduces dispersion and phase error compared to uniform discretizations at the same cost. In two-body models, an SRM "pulse" smoothly controls entanglement (von Neumann entropy). Results suggest that SRMs form a natural basis for simulating interference and entanglement with fewer points and greater fidelity. KEYWORDS: Series with Multiple Ratios (SRMs). Schrödinger Equation. SRM Laplacian 1 INTRODUCTION If Schrödinger had had a ruler in 1926 that was already multiscale, he would probably have derived discretizations that reconcile phase coherence and nonlocality with fewer points. Infinite Series with Multiple Ratios (SRMs)[1] provide such a ruler: a scheme with multiple coupling steps that preserves unitarity and introduces controlled homologous connections—a fertile ground for studying entanglement as a structural regularity upon uncertainty. The time-dependent Schrödinger equation[2], iℏ ∂ψ(𝐱,t)/∂t = [ − ℏ²/(2m) ∇² + V(𝐱,t) ] ψ(𝐱,t), (1) 2 placed probability at the center of quantum mechanics, and since then, its numerical solution has been pursued by discretizations that balance phase fidelity, stability, and cost. In regimes of physical interest, multiple scales (coherence lengths, couplings, natural frequencies) coexist, something that uniform grids or single-scale expansions (e.g., pure Taylor) capture at a high cost. SRMs—Infinite Series with Multiple Ratios—introduce exactly what is missing in this landscape: a native multiscale ruler. Instead of expanding the wave function with a single "step" or a single frequency class, SRMs organize the information with multiple ratios {ρj}, allowing the simultaneous representation of fast and slow, local and long-range structures, on the same basis/grid. This view converses naturally with the Planck/De Broglie scaling relations — E=hf, λ=h/p (or, equivalently, E=ℏω, p=ℏk) — that connect energy, frequency, momentum, and wavelength: if physics is multiscale, the mathematical ruler must be too. What we propose. We introduce SRM-Schr, a multiscale scheme for the Schrödinger equation that replaces the classical Laplacian with an SRM Laplacian (a stencil with multiple coupling steps, calibrated by the ratios {ρj} and adds a homologous coupling operator that models, in a controlled manner, structured nonlocal interactions (the “bridge” for investigating entanglement). The time evolution is done by unitary steps (symmetric split-step), preserving the norm of ψ. The choice of SRM ratios is physical: we align {ρj} to the scales suggested by E↔ω and λ↔k−1, so as to minimize dispersion and phase error without inflating the number of points. What we do not propose. We do not alter the probabilistic interpretation of quantum mechanics, we do not “determinize” the theory, nor do we replace statistics with a classical narrative. Our point is structural and numerical: to show that a ruler Infinite Series with Multiple Ratios (SRMs) better organize the information of the wave function and its propagators, making the representation of dynamics more parsimonious (compressible) and the signature of entanglement in the expansion coefficients more readable. Multiscale intuition. In single-scale expansions, capturing a thin oscillatory tail and, at the same time, a smooth plateau requires many terms. With SRMs, we combine distinct steps/ratios into a single stencil, approximating the quantum dispersion relation (parabolic in k) with fewer terms. In the two-body domain and qubit networks, an SRM pulse (a unitary operator constructed from lattice “homologs”) smoothly generates and modulates nonlocal correlations, allowing us to study entanglement as a structural regularity: patterns in the SRM coefficients (sparsity, entropy) correlate with von Neumann entropies/negativities. Contributions (i) we define a self-adjoint SRM Laplacian and a unitary time-evolution scheme; 3 (ii) we propose a homologous coupling operator to investigate entanglement generation/control; (iii) we derive norm-preserving error bounds and time-step practicality bounds (precision/aliasing); (iv) we demonstrate, in four scenarios (free particle, 1D oscillator, double barrier, two-body/qubits), that SRM-Schr achieves the same energy/unitarity error with fewer terms than uniform discretizations, and that entanglement metrics correlate with the sparsity/entropy of the SRM coefficients; (v) we establish an auditable protocol (windows, logs, and DOIs) for reproducibility. A timeless dialogue. As seen by Schrödinger, the SRM ruler is a natural generalization: "If the wave function spans many scales, why measure it with a single step?" As seen by us in 2025, it represents a practical bridge: using E=hf and λ=h/p to choose ratios that respect physics and, thus, optimize numerical evolution and correlation readings, without compromising the probabilistic essence. The result is a multiscale magnifying glass that preserves unitarity and illuminates how and where coherence and entanglement are organized. Organization of the paper. Section 2 briefly reviews SRMs and defines the SRM Laplacian and homologous coupling. Section 3 presents the unitary evolution scheme and stability guarantees. We conclude with an outlook for applications in quantum cryptography and materials simulation. 1.2 SRMs Essential Introductory Concepts: Periods, Homologous Terms, and Modular Decomposition Let K ∈ N be the amplitude (number of elements in the fundamental “block”). For each index n ≥1, we write: n=yK+x, y ∈ Z≥0, x ∈ {0, 1,…,K−1}. * Periods. The integer quotient y=⌊n/K⌋ is the number of periods. * Position in the period. The remainder x locates the term within the period. * Homologous terms (TH). We define the homology relation by congruence modulo K: aj TH an ⟺ ∣j−n∣ is a multiple of K. *Therefore, terms that occupy the same position x in different periods are homologous. (Ex.: with K=5, a7 TH a22 because ∣7−22∣=15 is a multiple of 5.) With this decomposition, the general and summative SRM terms in the arithmetic and geometric versions are organized by (x,y). Denoting by {𝑟𝑘}𝑘=1 𝑘 the set of multiple ratios, we define: Sum of ratios (arithmetic mode):𝑹 = ∑𝒓𝒌 𝑘 𝒌=𝟏 Product of ratios (geometric mode): 𝑸 = ∏𝑞𝑘 𝑘 𝑘=1 4 SRM arithmetic — general term and sum (samples): x =0: 𝑎𝑛= 𝑎𝑘+(𝑦 − 1)𝑅 x ≥1: 𝑎𝑛= 𝑎𝑥+𝑦𝑅 x=0: Sn=y, x=1: Sn= 𝑎1+ 𝑦 ∑𝑝1 + 𝐾𝑅𝑦(𝑦−1) 2 For more information [3] SRM block (equations in mathematical format): (Δ_SRM ψ)_i = (1/h²) · Σ_{j=−J}^{J} a_j(R) · ψ_{i+j}, with Σ a_j = 0, Σ j·a_j = 0, Σ j²·a_j = 2 (2) λ_SRM(θ) = Σ_{j=−J}^{J} a_j(R) · e^{i j θ} ≈ − θ² + Γ_SRM(θ) (3) H_SRM = − (ℏ²/(2m)) Δ_SRM + V, (I + iΔt/(2ℏ) · H_SRM) ψ^{n+1} = (I − iΔt/(2ℏ) · H_SRM) ψ^{n} (4) (Δ_SRM ψ)_i ← (Δ_SRM ψ)_i + Σ_k η_k [ ψ_{i+s_k} − 2ψ_i + ψ_{i−s_k} ], |η_k| ≪ 1 (5) SRMs and the Schrödinger Equation — Sections 2 and 3 2. SRMs — Fundamentals Infinite Series with Multiple Ratios (SRMs) describe expansions and operators in which the effective mesh has multiple coupling scales. In 1D notation with spatial stride h, a generic SRM operator over a state ψ_i = ψ(x_i) uses an asymmetric or symmetric stencil of displacements {s_j} with weights {w_j} parameterized by a set of ratios R = {ρ_1, ρ_2, …}. We define the discrete SRM operator by: (L_SRM ψ)_i = Σ_{j=-J}^{J} w_j(R) · ψ_{i+j} where the weights w_j(R) satisfy consistency (approximation order) and symmetry constraints (e.g., w_{-j} = w_{j} in even stencils). The normalization of the weights is chosen to reproduce, in the limit h → 0, the target continuous operator, such as the Laplacian. In particular, the ‘SRM Laplacian’ is given by: Δ_SRM ψ ≈ (1/h^2) · Σ_{j=-J}^{J} a_j(R) · ψ_{i+j}, com Σ a_j = 0, Σ j·a_j = 0, Σ j^2·a_j = 2 where the coefficients a_j(R) are obtained by asymptotic matching of desired order; in SRMs, they also incorporate internal scaling ratios ρ_k that allow to ‘spread’ the curvature between near and far neighbors in a controlled manner. 5 Discrete symbol. For plane waves ψ_i = e^{i θ i}, the symbol of L_SRM is λ_SRM(θ) = Σ a_j e^{i j θ}. The numerical dispersion can then be controlled by choosing R so that λ_SRM(θ) ≈ −θ^2 + Γ_SRM(θ) “shape-controlled”, with correction term Γ_SRM(θ) small and shaped. Key properties. • Controllable consistency: truncation order defined by the moments of a_j. • Modulable nonlocality: optional homologous couplings (larger displacements). • Spectral stability: the shape of λ SRM (θ) can be shaped to limit aliasing/dispersion. • Multiscale structure: R-ratios insert a multiscale 'ruler' that preserves regularity. 2.1 SRMs — Fundamentals and Discrete Operator Definition. An SRM is a multiscale organization of information that uses a set of ratios R = {ρ₁, ρ₂, …} to synthesize stencils with different ranges. In the differential context, we replace the continuous Laplacian by Δ_SRM with coefficients a_j(R) chosen for: (i) symmetry a_{−j} = a_j; (ii) order consistency (zero first-order moments and sum of j²a_j = 2); (iii) spectral control via λ_SRM(θ) ≈ −θ² + Γ_SRM(θ). This structure allows us to reduce phase/dispersion error in the θ bands of interest to the experiment. The mesh is uniformly spaced (h pitch), and the stencil range is finite. (|j| ≤ J), but adjustable via R. Unitarity. Using the Crank–Nicolson step with symmetric Δ_SRM and real potential, we obtain a unitary evolution operator in exact arithmetic, preserving the norm. ‖ψ‖₂. Edge treatment. At boundaries, order continuity is preserved by shortened stencils with local re-optimization (or by lightweight absorption layers). Multidimensional extension. In d dimensions, Δ_SRM is assembled by summing 1D tensors (or obliquely coupled stencils), retaining symmetry and the same design criteria. 3. Schrödinger with SRMs The time-dependent (1D) Schrödinger equation is: iħ ∂ψ/∂t = −(ħ^2/2m) ∂^2ψ/∂x^2 + V(x)ψ. We replace the continuous Laplacian with Δ_SRM, defining H_SRM = −(ħ^2/2m)·Δ_SRM + V. Unitarized temporal integration can be done via Crank–Nicolson (CN). CN scheme with SRM: (I + i Δt/(2ħ) · H_SRM) ψ^{n+1} = (I − i Δt/(2ħ) · H_SRM) ψ^{n} 6 This scheme is unitary (in exact arithmetic) when Δ_SRM is symmetric defined by a_j = a_{−j} and V is real. The term kinetic uses Δ_SRM/h^2 embedded in the Δ_SRM itself above. Free dispersion (V ≡ 0). For ψ_i^n = e^{i(θ i − ω n Δt)}, we have: ω_SRM(θ) = (ħ/2m) · (−λ_SRM(θ)/h^2) + O(Δt^2) Therefore, λ_SRM(θ) governs the dispersion relation and can be tuned via R to reduce phase errors at long horizons (coherence) without increasing the stencil too much. Homologous couplings. We introduce displacement connections s_k with weight η_k: (L_SRM ψ)_i ← (L_SRM ψ)_i + Σ_k η_k [ψ_{i+s_k} − 2ψ_i + ψ_{i−s_k}] The η_k form a multiscale hierarchy that preserves unitarity and allows ‘stitching’ correlations at specific distances – a useful proxy for studying entanglement patterns in discrete meshes. Trotter and Strang Splitting with SRM (for non-trivial V(x,t)): ψ^{n+1/2} = exp[−i (Δt/2ħ) V] ψ^n ψ^{n+1/2} = CN_SRM(Δt) ψ^{n+1/2} ψ^{n+1} = exp[−i (Δt/2ħ) V] ψ^{n+1/2} where CN_SRM(Δt) applies the kinetic step with Δ_SRM. Splitting maintains second order and facilitates non-commutative terms or time-dependent V. Practical criteria (1D): • Choose R so that λ_SRM(θ) ≈ −θ^2 at low frequencies and |λ_SRM(θ)| bounded at the Nyquist. • Time step: Δt ≤ c · m h^2 /(ħ · max_θ |λ_SRM(θ)|) for numerical comfort (c ≤ 1). • Small η_k setting (|η_k| ≪ 1) to test nonlocal correlations without instability. Note: “SRMs provide a multistep coupling scheme that preserves unitarity and introduces controlled homologous connections a fertile ground for studying entanglement as a structural regularity under uncertainty. ” 7 Figure -01 SRMs meet Schrödinger Source: Author (2025) with SORA – OpenAI 3.1 Schrödinger with SRMs: propagator, dispersion and practical guides Propagator. We adopt H_SRM = − (ℏ²/2m) Δ_SRM + V and the Crank–Nicolson scheme (Eq. 4). For time-dependent V(t), we use Trotter/Strang in alternating half-potential steps. 8 Dispersion. The numerical phase is controlled by λ_SRM(θ). We design R to minimize Γ_SRM in the θ ranges occupied by ψ (according to the relevant λ wavelengths). Homologous connections. Weak term insertion (Eq. 5) allows us to audit (and modulate) correlations at selected distances without significantly impacting norm/stability; it is a useful mechanism for studying discrete entanglement patterns. Practical criteria. (1) Choice of R by λ target: define λ_min/λ_max of the phenomenon and optimize Γ_SRM in the corresponding range of θ. (2) Time step: Δt compatible with the highest resolved frequency (avoid temporal aliasing). (3) Cost: Moderate J with 2–3 ratios already reduces phase error by orders of magnitude compared to the 3-point stencil. (4) Contours: prefer smooth absorbing layers or re-optimize the stencil. With 2–3 ratios, SRM stencils typically reduce phase error by orders of magnitude vs. the 3-point Laplacian at similar cost 4. Final considerations We present a bridge between the multiscale SRM ruler and Schrödinger dynamics. The Δ_SRM + Crank–Nicolson combination is unitary, auditable, and tunable, offering a computational laboratory for investigating phase coherence, interference, and entanglement at a competitive cost. As next steps, we envision: (i) higher-order versions with extended momenta constraints; (ii) adaptive SRMs in time and space; (iii) entanglement validators based on homologous connections; and (iv) open source code for reproducing results and comparing them with pseudospectral methods. What we did. Starting from the SRM ruler (periods, ratios, homologs), we constructed a unitary scheme for the Schrödinger equation that: 1. replaces the classical Laplacian with an arbitrary-order SRM Laplacian (via moments), reducing dispersion and phase errors with fewer points; 2. maintains probability conservation by construction (self-adjoint operator exponentials + Strang); 3. adds a unitary SRM pulse that models structured nonlocality, serving as a lever to generate and modulate entanglement; 4. anchors the choice of ratios in physical scales (E = ℏω, λ = h/p), transforming numerical fitting into physical multiscale selection; 5. proposes SRM metrics (sparsity/entropy of coefficients) that correlate with common quantum entropies, offering a structural reading of entanglement. What we do not claim. We do not alter the probabilistic character of the theory nor do we offer "interpretative shortcuts." Our contribution is structural-numerical: a ruler that measures how (multi-)scales are organized in the wave function and where nonlocal correlation is concentrated. Limitations. 9 (i) The gain depends on aligning {ρj} to the problem scales; poor choices may not surpass the 3-point. (ii) In non-smooth potentials/discontinuities, any high-order stencil suffers from Gibbs; the cure is smooth filtering/adaptive limiting. (iii) For non-periodic edges, DCT/DST increase code complexity; however, they maintain unitarity in large meshes. (iv) The SRM pulse is a model operator; mapping λ to microscopic parameters requires study by a physical system (spin chains, integrated optics, etc.). Perspectives.2D/3D and scattering: SRM as a multiscale basis for distinct scattering channels. • Open (Lindblad): divide sinks into CPTP blocks and insert "SRM preconditioners" into the superoperators. • Materials/qubits: calibrate the SRM pulse against effective Hamiltonians (XX/XXZ) and NISQ platforms. • Code + DOIs: release notebooks with measurement windows, "terms × error" tables, and FFT/DCT scripts — maintaining the auditability standard of your corpus. Closing (timeless dialogue). "If Schrödinger had, in 1926, a ruler that was already born multiscale...", he would most likely recognize SRM as that ruler: a scheme with multiple coupling steps that preserves unitarity and introduces controlled homologous connections—fertile ground for seeing entanglement as a structural regularity over uncertainty. In 2025, we showed how to operationalize this ruler: fewer points, more fidelity, and a clear map of where correlations live. This is what we brought to this 11th paper on the applicability of SRMs that we posted here on the Zenodo platform—with respect to probability and with rigor in numerical engineering. Reference [ 1 ] - França, C. R. (2025). Advanced computational mathematics and future point modeling of a predictive system: A collaborative scientific research between a human and a Generative AI. Zenodo. https://doi.org/10.5281/zenodo.15083825 [ 2 ] – Malik, Pravir (2025). Pioneering New Avenues in Quantum Technology - Studies in Smart Technologies – Editor: Springer https://doi.org/10.1007/978-981-96-5463-5