High-Order SUSY-QM, the Quantum XP Model and zeroes of the Riemann Zeta function

TL;DR

This paper uses supersymmetric quantum mechanics to construct Hamiltonians whose zero-energy states correspond to the nontrivial zeros of the Riemann Zeta function, revealing new potential models related to these zeros.

Contribution

It introduces a novel SUSY-QM approach using wave functions |x|^{-S} to locate Riemann zeros on the critical line, expanding the class of partner potentials to inverse squared distance types.

Findings

Partner Hamiltonians have zero modes at Riemann zeros.

Constructed potentials are inverse squared distance with complex couplings.

Method links quantum mechanics to the distribution of Riemann zeros.

Abstract

Making use of the first- and second-order algorithms of supersymmetric quantum mechanics (SUSY-QM), we construct quantum mechanical Hamiltonians whose spectra are related to the zeroes of the Riemann Zeta function $\zeta(s)$. Inspired by the model of Das and Kalauni (DK), which corresponds to this function in the strip $0<Re[s]<1$, and taking the factorization energy equal to zero, we use the wave function $|x|^{-S}$, $S\in\mathbb{C}$, as a seed solution for our algorithms, obtaining XP-like operators. Thus, we construct SUSY-QM partner Hamiltonians whose zero energy mode locates exactly the nontrivial zeroes of $\zeta(s)$ along the critical line $Re[s]=1/2$ in the complex plane. We further find that unlike the DK case, where the SUSY-QM partner potentials correspond to free particles, our partner potentials belong to the family of inverse squared distance potentials with complex couplings.