The Berry-Keating Hamiltonian and the Local Riemann Hypothesis

Abstract

The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry-Keating hamiltonian H=(xp+px)/2 projected onto the subspace of oscillator eigenfunctions of lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert-Polya conjecture. The p-adic case is also discussed.

9 pages, no figures; v2 included more mathematical background, v3 has minor edits for clarity

References in corpus (2)

• Landau levels and Riemann zeros
• A quantum mechanical model of the Riemann zeros

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