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arXiv · 2609.23353

Eigenvalue-by-Eigenvalue Comparison of a Sierra--Rodríguez-Laguna-Type Spectrum with the Riemann Zeros

Mi-Ra Hwang·Eylee Jung·MuSeong Kim·DaeKil Park
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Abstract

We study the self-adjoint extension of the Sierra--Rodríguez-Laguna (SR) $H=xp$-type Hamiltonian $\widehat H_{SR}$. Its discrete spectrum is fixed by the equation $\mathrm{Re}\!\left[e^{-iθ/2}K_{1/2+iE/2}(2π)\right]=0$. 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: $E_n-γ_n\simeq(θ-π/4)/[1+W(n/e)]$, 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 $n\ge10^3$, 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.

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Mi-Ra Hwang, Eylee Jung, MuSeong Kim, DaeKil Park. 2026-10-25. Eigenvalue-by-Eigenvalue Comparison of a Sierra--Rodríguez-Laguna-Type Spectrum with the Riemann Zeros. https://arxiv.org/abs/2609.23353

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