  We construct a Hamiltonian H whose discrete spectrum contains, in a certain
limit, the Riemann zeros. H is derived from the action of a massless Dirac
fermion living in a domain of Rindler spacetime, in 1+1 dimensions, that has a
boundary given by the world line of a uniformly accelerated observer.
  The action contains a sum of delta function potentials that can be viewed as
partially reflecting moving mirrors. An appropriate choice of the accelerations
of the mirrors, provide primitive periodic orbits associated to the prime
numbers $p$, whose periods, measured by the observer's clock, are log p. Acting
on the chiral components of the fermion, H becomes the Berry-Keating
Hamiltonian (x p + p x)/2, where x is identified with the Rindler spatial
coordinate and p with the conjugate momentum.
  The delta function potentials give the matching conditions of the fermion
wave functions on both sides of the mirrors. There is also a phase shift for
the reflection of the fermions at the boundary where the observer sits. The
eigenvalue problem is solved by transfer matrix methods in the limit where the
reflection amplitudes become infinitesimally small. We find that for generic
values of the phase shift the spectrum is a continuum, where the Riemann zeros
are missing, as in the adelic Connes model. However, for some values of phase
shift, related to the phase of the zeta function, the Riemann zeros appear as
discrete eigenvalues immersed in the continuum.
  We generalize this result to the zeros of Dirichlet L-functions, associated
to primitive characters, that are encoded in the reflection coefficients of the
mirrors. Finally, we show that the Hamiltonian associated to the Riemann zeros
belongs to class AIII, or chiral GUE, of Random Matrix Theory.
