Spectral Analysis of the Deformed Riemann Operator and the De Bruijn–Newman Constant

Abstract

We investigate the spectral theory of a family of Schrödinger operators associated with the heat-flow deformation of the Riemann xi-function.

Spectral Theory and the Riemann Hypothesis

This paper introduces a novel operator-theoretic framework connecting the de Bruijn–Newman constant Λ to the spectral properties of a family of Schrödinger operators. The central object is the heat-flow deformation H_t of the Riemann xi-function, which interpolates between the original xi-function at t = 0 and a purely Gaussian function as t → ∞. The reality of the zeros of H_t is equivalent to the negativity of the spectrum of an associated differential operator L_t, providing a bridge between analytic number theory and spectral analysis.

The De Bruijn–Newman Operator

We define the operator L_t = −∂²/∂x² + V_t(x) on L²(ℝ), where the potential V_t is derived from the logarithmic derivative of H_t. The key insight is that the zeros of ζ(s) on the critical line Re(s) = 1/2 correspond to eigenvalues of this operator. The paper proves that for t ≥ 1, the essential spectrum σ_ess(L_t) lies strictly in the left half-plane, specifically satisfying Re(λ) ≤ −1/4 for all λ in the essential spectrum. This localization result provides strong constraints on the possible movement of zeros under the heat flow.

Spectral Gap and the Riemann Hypothesis

The main theorem establishes a conditional result: if the spectral gap γ(t) — defined as the distance between the first two eigenvalues of L_t — satisfies γ(t) ≫ t⁻¹ as t → 0⁺, then Λ = 0. Since the Riemann Hypothesis is equivalent to the statement that Λ ≤ 0, and recent work has established Λ ≥ 0, proving this spectral gap bound would confirm RH. The paper provides variational estimates suggesting this decay rate holds, supported by high-precision numerical computation of the first hundred eigenvalues for t ∈ [0.1, 1].

Computational Evidence

The computational section presents Wolfram Language algorithms for evaluating H_t(z) using the Riemann–Siegel formula and computing the eigenvalues of the associated Jacobi matrix discretization of L_t. These calculations confirm the theoretical predictions to within 10⁻¹⁰ precision and reveal a striking pattern: the eigenvalue spacing appears to follow a distribution consistent with random matrix theory, mirroring the Montgomery–Odlyzko law for the zeta zeros. This suggests that the operator L_t captures the universal spectral statistics underlying the Riemann zeros.

Implications for Zero-Free Regions

By relating the existence of non-real zeros to the spectral properties of L_t, the paper offers a new mechanism for establishing zero-free regions. If one can demonstrate that the operator remains dissipative (all eigenvalues have negative real part) for all t > 0, the reality of the zeros of H_t follows immediately. The techniques developed here — particularly the bounds on the essential spectrum — may extend to broader classes of L-functions, potentially providing a unified approach to the Generalized Riemann Hypothesis through the study of parabolic evolution equations and their spectral asymptotics.