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The integer quantum Hall effect is derived for a finite rectangular sample and rather general boundary conditions using a Kubo formula approach. The precise quantization of the Hall resistance in units of h/e2 is subject to constraints on the size of the sample, the strength of the magnetic field and the shape of the confining potential at the sample boundaries. The derivation assumed a (bulk) mobility gap but not necessarily a gap in the density of states in between Landau levels.
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M. Jonson (1985) studied this question.
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