where γ = arcosh | cos(αa) + λ α sin(αa)| is an auxiliary variable introduced to simplify the expression. The above Green's function describes the propagation between 0 ⩽ x0 ⩽ a and r points in a crystal of the lattice constant a, m = ⌊r/a⌋ is the integer index, and P = λa/2 is the renormalized crystal-potential strength. Except for this very formula, to the best of our knowledge, there is no other analytical description of the Kronig-Penney models in d ⩾ 2-dimensions.