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A continuum limit for the Kronig–Penney model

“…ρ diverges exponentially in N when |Φ| > 1). When E = E o , then 4.20 diverges as L 2 , in agreement with the item (2)i of theorem 3.1 in [11] (i.e. ρ diverges as N 2 when |Φ| = 1).…”

“…It is interesting to compare equation ( 4.20) with the result reported in theorem 3.1 of [11]. Observe that the factor V γ appearing in (4.20) can be interpreted as the (constant) intensity of the Dirac deltas in theorem 3.1 of [11], denoted therein by 'V ' (and corresponding to the constant Λ defined in formula (5.12) ahead). On the other hand, the limit N → ∞ in [11] corresponds to taking the limit L → ∞ in equation (4.20).…”

“…Such procedure leads to a limiting behaviour of the K-P model which differs, in general, from the thermodynamic limit discussed in the literature (see e.g. [11]).…”

“…Then, one can notice the following. When E < E o , the resistivity (4.20) diverges exponentially in L; this corresponds to the item (1) of theorem 3.1 in [11] (i.e. ρ diverges exponentially in N when |Φ| > 1).…”

“…Observe that the factor V γ appearing in (4.20) can be interpreted as the (constant) intensity of the Dirac deltas in theorem 3.1 of [11], denoted therein by 'V ' (and corresponding to the constant Λ defined in formula (5.12) ahead). On the other hand, the limit N → ∞ in [11] corresponds to taking the limit L → ∞ in equation (4.20). Then, one can notice the following.…”