Definitive Proof of the Riemann Hypothesis via Strong Convergence of Quantum Operators

The Riemann Hypothesis (RH), formulated by Bernhard Riemann in 1859, posits that all non-trivial zeros of the function ζ(𝑠) reside on the critical line Re(𝑠) = 1/2. This work addresses the problem within the framework of the Hilbert-Pólya Spectral Hypothesis (HP), transforming the RH into a question of spectral stability: the RH is equivalent to proving that the Riemann Operator, 𝐻∞, is self-adjoint (Hermitian). The main obstacle in previous spectral approaches was demonstrating the preservation of Hermiticity during the transition to the limit, given that the analytical convergence of ζ(𝑠) only guarantees weak spectrum convergence. Rigorous demonstration requires strong convergence from the operator 𝐻𝑛 → 𝐻∞. This research solves this problem by proving the Hermitian Convergence Theorem. By decomposing the Hamiltonian as 𝐻𝑛 = 𝑇 + 𝑉𝑛(𝑥), it is established that strong convergence is forced by the uniform convergence of the Quantum Potential. It is rigorously demonstrated that the sequence of potentials 𝑉𝑛(𝑥) converges uniformly to 𝑉∞(𝑥) in the supremum normAcceptance of the Alternative Hypothesis 𝐻𝑎 (continuity of potential), combined with Kato's Perturbation Theorem, ensures strong convergence 𝐻𝑛 → 𝐻∞. This convergence preserves the self-adjoint property, concluding that 𝐻∞ it is Hermitian. By the Spectral Theorem, the eigenvalues (the non-trivial zeros) must be strictly real, which validates the Riemann Hypothesis.