  We show that a narrow resonance produces strong interaction effects far
beyond its width on the side of the resonance where the bound state has not
been formed. This is due to a resonance structure of its phase shift, which
shifts the phase of a large number of scattering states by $\pi$ before the
bound state emerges. As a result, the magnitude of the interaction energy when
approaching the resonance on the "upper" and "lower" branch from different side
of the resonance is highly asymmetric, unlike their counter part in wide
resonances. Measurements of these effects are experimentally feasible.
