  The physics of Feshbach resonance is analyzed using an analytic expression
for the $s$-wave scattering phase-shift and the scattering length $a$ which we
derive within a two-channel tight-binding model. Employing a unified treatment
of bound states and resonances in terms of the Jost function, it is shown that
for strong inter-channel coupling, Feshbach resonance can occur even when the
closed channel does not have a bound state. This may extend the range of
ultra-cold atomic systems that can be manipulated by Feshbach resonance. The
dependence of the sign of $a$ on the coupling strength in the unitary limit is
elucidated. As a by-product, analytic expressions are derived for the
background scattering length, the external magnetic field at which resonance
occurs, and the energy shift $\varepsilon-\varepsilon_B$, where $\varepsilon$
is the scattering energy and $\varepsilon_B$ is the bound state energy in the
closed channel (when there is one).

Quantum Gases