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 ${\hat{H}}_{SR}$. Its discrete spectrum is fixed by the equation $\mathrm{Re}\left[e^{-i\theta /2}{K}_{1/2+iE/2}(2\pi )\right]=0$. We solve this equation numerically for $\theta =1.417\pi$ and obtain the first 606 eigenvalues $E_n$. We compare them, one by one, with the ordinates ${\gamma }_n$ 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 $E_n-{\gamma }_n$ in terms of the Lambert-$W$ function. We show that our leading-order phase for $K_{1/2+iE/2}(2\pi )$ matches exactly a known, rigorous asymptotic formula for modified Bessel functions of large imaginary order. This rules out the Bessel function as the source of a numerical mismatch we found earlier. We then trace that mismatch to how the counting function $N(T)$ must be treated exactly at $T={\gamma }_n$: the usual midpoint convention for the fluctuating term $S(T)$ shifts the effective quantum number by $1/2$. With this fix, ${\gamma }_n\sim g(n-11/8)$ instead of $g(n-7/8)$, and the new prediction for $E_n-{\gamma }_n$ matches, to within $0.03\%$, the value we measure directly from the first $10^5$ tabulated zeta zeros (A.~Odlyzko).

10 pages, 3 figures included