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 a 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 projected onto the subspace of oscillator eigenfunctions of a lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert–Pólya conjecture. The p-adic case is also discussed.