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arXiv · 2301.05360

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

Juan D García-Muñoz·A Raya·Y Concha-S
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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.

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Juan D García-Muñoz, A Raya, Y Concha-S. 2023-01-13. High-Order SUSY-QM, the Quantum XP Model and zeroes of the Riemann Zeta function. https://arxiv.org/abs/2301.05360

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