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May 20, 2026Open Access

Self-Adjointness of a First-Order Operator with Dense Mangoldt Potential

Key Points

To prove the essential self-adjointness of the first-order differential operator with a Mangoldt potential in L²(ℝ).
Constructed a strongly continuous one-parameter unitary group.
Applied Stone's theorem for analysis of the operator.
Relied on the unitarity of jump conditions and the Prime Number Theorem.
Confirmed essential self-adjointness of the operator A = −i∂u + V(u).
Closed the mathematical gap in Dirac operator self-adjointness in the DAS theory.
Indicated that under the DAS Spectral Conjecture, the Riemann Hypothesis follows.

Abstract

We prove essential self-adjointness of the first-order differential operator A = −i∂u + V (u) on L² (ℝ), where V (u) = Σ Λ (n) δ (u − log n) is the Mangoldt potential with spikes on a dense subset of ℝ. The key tool is an explicit construction of a strongly continuous one-parameter unitary group followed by Stone's theorem. The proof relies on the unitarity of jump conditions for first-order operators with real potential, and on the Prime Number Theorem providing uniform control over the phase. The result closes the main mathematical gap in the proof of self-adjointness of the Dirac operator on the world tube of the Dynamic Abstract Sphere (DAS) theory. Combined with the three previously established pillars (Weyl limit point, reality of potential, direct sum stability), this yields: Spec (𝒟DAS) ⊂ ℝ. Under the DAS Spectral Conjecture — Spec (𝒟DAS) = γk — the Riemann Hypothesis follows. Related publications: DOI 10. 5281/zenodo. 20205436 · DOI 10. 5281/zenodo. 20254441

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