  Recently, with Mussardo we defined a quantum mechanical problem of a single
particle scattering with impurities wherein the quantized energy levels $E_n
(\sigma)$ are exactly equal to the zeros of the Riemann $\zeta (s)$ where
$\sigma = \Re (s)$ in the limit $\sigma \to 1/2$. The S-matrix is based on the
Euler product and is unitary by construction, thus the underlying hamiltonian
is hermitian and all eigenvalues must be real. Motivated by the Hilbert-P\'olya
idea we study the spectral flows for $\{ E_n (\sigma) \}$. This leads to a
simple criterion for the validity of the Riemann Hypothesis. The spectral flow
arguments are simple enough that we present analogous results for the
Generalized and Grand Riemann Hypotheses. We also illustrate our results for a
counter example where the Riemann Hypothesis is violated since there is no
underlying unitary S-matrix due to the lack of an Euler product.
