  We prove a new upper bound for the first eigenvalue of the Dirac operator of
a compact hypersurface in any Riemannian spin manifold carrying a non-trivial
twistor spinor without zeros on the hypersurface. The upper bound is expressed
as the first eigenvalue of a drifting Schr\"odinger operator on the
hypersurface. Moreover, using a recent approach developed by O. Hijazi and S.
Montiel, we completely characterize the equality case when the ambient manifold
is the standard hyperbolic space.
