Instability of the continuous spectrum: The N-band Stark ladder

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Abstract
It is shwn that the energy spectrum of the Bloch electron in an external field is continuous. Furthermore, it is shown that all approximations which take into account interband coupling whithin groups of finite number of bands (the N‐band approximation) lead to a pure‐point spectrum of intertwining Wannier‐stark ladders. This instability of the continuous spectrum under the N‐band approximation is related to a theorem due to Weyl and von Neumann. Approximation methods for dealing with interband coupling within a group of finite number of bands are given.