A quantum mechanical model of the Riemann zeros

Understand

In 1999 Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp gives semiclassically the smooth counting function of the Riemann zeros.

• In this paper we first generalize this result by considering a phase space delimited by two boundary functions in position and momenta, which induce a fluctuation term in the counting of energy levels.
• We next quantize the xp Hamiltonian, adding an interaction term that depends on two wave functions associated to the classical boundaries in phase space.
• The general model is solved exactly, obtaining a continuum spectrum with discrete bound states embbeded in it.