High-order SUSY-QM, the quantum XP model and zeroes of the Riemann Zeta function

Abstract

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 ζ(s). Inspired by the model of Das and Kalauni (DK) A Das and P Kalauni (2019 Physics Letters B 791, 265-269) 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 ∈ 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 ζ(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.