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H = xp model revisited and the Riemann zeros.

Germán Sierra1, Javier Rodríguez-Laguna

  • 1Instituto de Física Teórica, CSIC-UAM, Madrid, Spain.

Physical Review Letters
|June 15, 2011
PubMed
Summary
This summary is machine-generated.

Researchers found a new Hamiltonian model that exhibits closed orbits, aligning its energy spectrum with Riemann zeta function zeros. This advances understanding of the connection between classical mechanics and number theory.

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Area of Science:

  • Mathematical Physics
  • Quantum Mechanics
  • Number Theory

Background:

  • The Berry-Keating conjecture proposes a link between the classical Hamiltonian H = xp and Riemann zeta function zeros.
  • Previous models showed average semiclassical energies matching Riemann zeros but lacked closed classical trajectories, indicating an incomplete model.

Purpose of the Study:

  • To investigate a modified Hamiltonian that incorporates closed periodic orbits.
  • To demonstrate the spectral coincidence of this new Hamiltonian with average Riemann zeros.
  • To generalize findings to Dirichlet L functions and explore experimental connections.

Main Methods:

  • Introduced a new Hamiltonian: H = x(p + ℓ(p)²/p).
  • Analyzed the properties of classical trajectories for the modified Hamiltonian.
  • Examined the energy spectrum of the Hamiltonian and its relation to Riemann zeros.
  • Utilized self-adjoint extensions for generalization to Dirichlet L functions.

Main Results:

  • The modified Hamiltonian H = x(p + ℓ(p)²/p) possesses closed periodic orbits.
  • The energy spectrum of this Hamiltonian aligns with the average Riemann zeros.
  • The results were successfully generalized to Dirichlet L functions.

Conclusions:

  • The study provides a more complete model connecting classical mechanics and Riemann zeros through a Hamiltonian with closed orbits.
  • The findings offer a potential pathway for experimental verification using the Landau model.
  • The work deepens the understanding of the interplay between quantum mechanics, classical dynamics, and number theory.