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Feshbach Resonance in a Tight-Binding Model

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).