  We study inelastic resonant scattering of a Gaussian wave packet with the
parameters close to a zero of the complex scattering coefficient. We
demonstrate, both theoretically and experimentally, that such near-zero
scattering can result in anomalously-large time delays and frequency shifts of
the scattered wave packet. Furthermore, we reveal a close analogy of these
anomalous shifts with the spatial and angular Goos-H\"anchen optical beam
shifts, which are amplified via quantum weak measurements. However, in contrast
to other beam-shift and weak-measurement systems, we deal with a
one-dimensional scalar wave without any intrinsic degrees of freedom. It is the
non-Hermitian nature of the system that produces its rich and non-trivial
behaviour. Our results are generic for any scattering problem, either quantum
or classical. As an example, we consider the transmission of an optical pulse
through a nano-fiber with a side-coupled toroidal micro-resonator. The zero of
the transmission coefficient corresponds to the critical coupling conditions.
Experimental measurements of the time delays near the critical-coupling
parameters verify our weak-measurement theory and demonstrate amplification of
the time delay from the typical inverse resonator linewidth scale to the pulse
duration scale.
