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April 17, 20260 citationsOpen Access

A Hilbert–Polya Operator derived from a Prime Oscillator System

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GGGuillermo Garcia

Key Points

  • The aim is to construct an explicit self-adjoint operator arising from a dynamical system of prime oscillators.
  • Constructed the operator L = A + V on ℓ2(N, 1/n).
  • Defined the free Hamiltonian A as log n to generate dilations.
  • Encoded interactions via V = Re(TΛ) using Dirichlet convolution with the von Mangoldt function.
  • The operator successfully encodes prime-power firing events.
  • Demonstrates a novel connection between dynamical systems and number theory.

Abstract

We construct an explicit self-adjoint operator L= A+ V on ℓ2(N,1/n) arising from a dynamical system of prime oscillators. The free Hamiltonian A= log n generates dilations; the interaction V = Re(TΛ) encodes prime-power firing events via Dirichlet convolution with the von Mangoldt function.

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Cite This Study

Guillermo Garcia (2026) studied this question.

synapsesocial.com/papers/69e1d0165cdc762e9d85931chttps://doi.org/10.5281/zenodo.19587054
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Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

  1. 1A Candidate for the Hilbert-Pólya Operator: Rigorous Construction of an Explicit Hermitian Operator from Symmetrized Prime Factorization2026
  2. 2QUADRATIC VARIATIONAL STRUCTURES AND THE OPERATOR L*L: SPECTRAL CONTROL OF LOCAL STABILITY, DYNAMICS, AND FLUCTUATIONS2026
  3. 3An Explicit Hilbert-Pólya Operator from Symmetrized Prime Factorization: Construction, Proof of Hermiticity, and Numerical Evidence2026
  4. 4Twin Prime Asymptotics from a Finite–Range Positive Operator: A Structural and Spectral Approach2025
  5. 5Twin Prime Asymptotics from a Finite–Range Positive Operator: A Structural and Spectral Approach2025