A compact hamiltonian with the same asymptotic mean spectral density as the Riemann zeros

For the classical hamiltonian (x + 1/x)(p + 1/p), with position x and conjugate momentum p, all orbits are bounded. After a symmetrization, the corresponding quantum integral equation possesses a family of self-adjoint extensions: compact operators on the entire positive x axis, labelled by an angle a specifying the boundary condition at the origin, with a discrete spectrum of real energies E. On the cylinder {-infinity < E < infinity, 0 <= alpha < 2 pi}, there is a single eigencurve in the form of a helix winding clockwise. The rise between successive windings gets sharper as the scaled Planck's constant decreases. This behaviour can be understood semiclassically. The first two terms of the asymptotic eigenvalue density are the same as those for the density of heights of the Riemann zeros.