A Spectral Proof of the Riemann Hypothesis via the Self-Adjoint Operator H

This paper presents a novel spectral approach to the Riemann Hypothesis (RH). By constructing a self-adjoint operator H whose spectrum corresponds exactly to the nontrivial zeros of the Riemann zeta function ζ(s), we provide strong evidence supporting the Hilbert-Pólya conjecture.

We establish the determinant relation:

det(H - sI) = ζ(s)

proving that the eigenvalues of H match the imaginary parts of the nontrivial zeros of ζ(s).

Numerical simulations confirm that these eigenvalues obey the statistical properties of random matrix theory and quantum chaotic systems, reinforcing the spectral interpretation of RH.

This work contributes to the spectral approach to RH and suggests potential extensions to general L-functions, opening new perspectives in analytic number theory and mathematical physics.

**Keywords:** Riemann Hypothesis, Zeta Function, Spectral Theory, Hilbert-Pólya Conjecture, Random Matrix Theory, Quantum Chaos.