  Starting from a quantized version of the classical Hamiltonian H = x p, we
add a non local interaction which depends on two potentials. The model is
solved exactly in terms of a Jost like function which is analytic in the
complex upper half plane. This function vanishes, either on the real axis,
corresponding to bound states, or below it, corresponding to resonances. We
find potentials for which the resonances converge asymptotically toward the
average position of the Riemann zeros. These potentials realize, at the quantum
level, the semiclassical regularization of H = x p proposed by Berry and
Keating. Furthermore, a linear superposition of them, obtained by the action of
integer dilations, yields a Jost function whose real part vanishes at the
Riemann zeros and whose imaginary part resembles the one of the zeta function.
Our results suggest the existence of a quantum mechanical model where the
Riemann zeros would make a point like spectrum embbeded in the continuum. The
associated spectral interpretation would resolve the emission/absortion debate
between Berry-Keating and Connes. Finally, we indicate how our results can be
extended to the Dirichlet L-functions constructed with real characters.
