  In this paper we describe a remarkable new property of the self-adjoint
extension W of the prolate spheroidal operator introduced in
\cite{college98},\cite{CMbook}. The restriction of this operator to the
interval J whose characteristic function commutes with it is well known, has
discrete positive spectrum and is well understood. What we have discovered is
that the restriction of W to the complement of J admits (besides a replica of
the above positive spectrum) negative eigenvalues whose ultraviolet behavior
reproduce that of the squares of zeros of the Riemann zeta function.
Furthermore, their corresponding eigenfunctions belong to the Sonin space. This
feature fits with the proof \cite{weilpos} of Weil's positivity at the
archimedean place, which uses the compression of the scaling action to the
Sonin space. As a byproduct we construct an isospectral family of Dirac
operators whose spectra have the same ultraviolet behavior as the zeros of the
Riemann zeta function.
