Eigenvalue-by-Eigenvalue Comparison of a Sierra--Rodríguez-Laguna-Type Spectrum with the Riemann Zeros
Abstract
We study the self-adjoint extension of the Sierra--Rodríguez-Laguna (SR) H=xp-type Hamiltonian ĤSR. Its discrete spectrum is fixed by the equation Re [e−iθ/2K1/2+iE/2(2π)]=0. We solve this equation numerically for θ=1.417π and obtain the first 606 eigenvalues En. We compare them, one by one, with the ordinates γn of the first 606 nontrivial zeros of the Riemann zeta function. Using the steepest-descent (saddle-point) method for the modified Bessel function, together with the Riemann--von Mangoldt formula, we derive a closed-form prediction for En−γn in terms of the Lambert-W function: En−γn≃(θ−π/4)/[1+W(n/e)], which decays only like 1/ln n. Here the n-th numerical root corresponds to the (n−2)-th root of the asymptotic quantization condition. The leading-order phase of K1/2+iE/2(2π) agrees with a known rigorous asymptotic formula for modified Bessel functions of large imaginary order. The prediction is checked against the first 105 tabulated zeta zeros (A.~Odlyzko): the measured coefficient of 1/[1+W(n/e)] is $3.666$ (mean) and $3.667$ (median) for n≥103, in agreement with θ−π/4=3.666. The scatter of En−γn around this smooth curve comes from the fluctuating term S(T) of the zero-counting function, which the spectrum En does not contain.
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BibTeX
@misc{hwang2026eigenvaluebyeigenvalue,
title = {{Eigenvalue-by-Eigenvalue Comparison of a Sierra--Rodríguez-Laguna-Type Spectrum with the Riemann Zeros}},
author = {Mi-Ra Hwang and Eylee Jung and MuSeong Kim and DaeKil Park},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23353},
eprint = {2609.23353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23353},
note = {CC0},
}
RIS
TY - UNPB
TI - Eigenvalue-by-Eigenvalue Comparison of a Sierra--Rodríguez-Laguna-Type Spectrum with the Riemann Zeros
AU - Mi-Ra Hwang
AU - Eylee Jung
AU - MuSeong Kim
AU - DaeKil Park
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2609.23353
UR - https://arxiv.org/abs/2609.23353
AB - We study the self-adjoint extension of the Sierra--Rodríguez-Laguna (SR) H=xp-type Hamiltonian Ĥ_SR. Its discrete spectrum is fixed by the equation Re [e^−iθ/2K_1/2+iE/2(2π)]=0. We solve this equation numerically for θ=1.417π and obtain the first 606 eigenvalues E_n. We compare them, one by one, with the ordinates γ_n of the first 606 nontrivial zeros of the Riemann zeta function. Using the steepest-descent (saddle-point) method for the modified Bessel function, together with the Riemann--von Mangoldt formula, we derive a closed-form prediction for E_n−γ_n in terms of the Lambert-W function: E_n−γ_n≃(θ−π/4)/[1+W(n/e)], which decays only like 1/ln n. Here the n-th numerical root corresponds to the (n−2)-th root of the asymptotic quantization condition. The leading-order phase of K_1/2+iE/2(2π) agrees with a known rigorous asymptotic formula for modified Bessel functions of large imaginary order. The prediction is checked against the first 10^5 tabulated zeta zeros (A.~Odlyzko): the measured coefficient of 1/[1+W(n/e)] is $3.666$ (mean) and $3.667$ (median) for n≥10^3, in agreement with θ−π/4=3.666. The scatter of E_n−γ_n around this smooth curve comes from the fluctuating term S(T) of the zero-counting function, which the spectrum E_n does not contain.
ER -