Dresden 2026 – wissenschaftliches Programm
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DY: Fachverband Dynamik und Statistische Physik
DY 42: Poster: Quantum Dynamics and Many-body Systems (joint session DY/TT)
DY 42.13: Poster
Mittwoch, 11. März 2026, 15:00–18:00, P5
Periodic orbit theory approach for a non-Hermitian Riemann operator — •Sebastian Hörhold, Andreas Hötzinger, Juan Diego Urbina, and Klaus Richter — Institut für Theoretische Physik, Universität Regensburg, Germany
The Riemann Hypothesis (RH) is one of the most important open problems in mathematics. Among the various approaches toward its proof is the Hilbert-Pólya (HP) conjecture, stating that there should exist a Hermitian operator whose eigenvalues tn are given by the zeros of the Riemann zeta function ζ(1/2+itn). The RH would then follow from the reality of these eigenvalues.
In a recent contribution toward a proof of the RH, a non-Hermitian Hamiltonian has been introduced, referred to as a Riemann operator, whose spectrum contains the tn, and from which one can construct an HP operator [1]. Our work focuses on a similar Hamiltonian, and we intend to make use of semiclassical tools to support earlier work by Berry and Keating, who obtained a formal asymptotic expression for the counting function of the nontrivial Riemann zeros [2]. Their results suggest a strong connection between the spectral statistics of these zeros and classically chaotic systems.
In this poster contribution, we show the emergence of the Riemann zeros within the spectrum of our non-Hermitian Hamiltonian and discuss how periodic orbit theory can be applied.
[1] E. Yakaboylu, arXiv:2408.15135
[2] M. V. Berry and J. P. Keating, SIAM Review 41.2 pp. 236-266
Keywords: non-Hermitian quantum mechanics; semiclassical methods; Riemann Hypothesis; Hilbert-Pólya conjecture