  We consider systems $(M,\omega,g)$ with $M$ a closed smooth manifold,
$\omega$ a real valued closed one form and $g$ a Riemannian metric, so that
$(\omega,g)$ is a Morse-Smale pair, Definition~2. We introduce a numerical
invariant $\rho(\omega,g)\in[0,\infty]$ and improve Morse-Novikov theory by
showing that the Novikov complex comes from a cochain complex of free modules
over a subring $\Lambda'_{[\omega],\rho}$ of the Novikov ring
$\Lambda_{[\omega]}$ which admits surjective ring homomorphisms
$\ev_s:\Lambda'_{[\omega],\rho}\to\C$ for any complex number $s$ whose real
part is larger than $\rho$. We extend Witten-Helffer-Sj\"ostrand results from a
pair $(h,g)$ where $h$ is a Morse function to a pair $(\omega,g)$ where
$\omega$ is a Morse one form. As a consequence we show that if $\rho<\infty$
the Novikov complex can be entirely recovered from the spectral geometry of
$(M,\omega,g)$.
