  We study the two-body problem with a spatially modulated interaction
potential using a two-channel model, in which the inter-channel coupling is
provided by an optical standing wave and its strength modulates periodically in
space. As the modulation amplitudes increases, there will appear a sequence of
bound states. Part of them will cause divergence of the effective scattering
length, defined through the phase shift in the asymptotic behavior of
scattering states. We also discuss how the local scattering length, defined
through short-range behavior of scattering states, modulates spatially in
different regimes. These results provide a theoretical guideline for new
control technique in cold atom toolbox, in particular, for alkali-earth-(like)
atoms where the inelastic loss is small.
