A continuum limit for the Kronig-Penney model

We investigate the transmission properties of a quantum one-dimensional periodic system of fixed length L, with N barriers of constant height V and width λ and N wells of width δ. In particular, we study the behaviour of the transmission coefficient in the limit N → ∞, with L fixed. This is achieved by letting δ and λ both scale as 1/N, in such a way that their ratio γ= λ/δ is a fixed parameter characterizing the model. In this continuum limit, the multi-barrier system behaves as it were constituted by a unique barrier of constant height E<inf>o</inf> = (γV)/(1 + γ). The analysis of the dispersion relation of the model shows the presence of forbidden energy bands at any finite N.