The Fredholm index of locally compact band-dominated operators on $L^p   (R)$
  We establish a necessary and sufficient criterion for the Fredholmness of a general locally compact band-dominated operator $A$ on $L^p(R)$ and solve the long-standing problem of computing its Fredholm index in terms of the limit operators of $A$. The results are applied to operators of convolution type with almost periodic symbol. 
Vladimir S. Rabinovich, Steffen Roch