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Since the functions γ1 (s), F01 (s), and ΔE(s) are supposed to be well-behaved near the real s axis, there are no small (or large) numbers in the problem apart from those generated by the minimum of the gap ΔE(s). Thus, the significant changes of the eigenvectors are also localized around this minimum. In the complex plane, this minimum along the real axis becomes a saddle point. For analytic functions, the characteristic length scale of variation must be the same along the real axis and into the complex plane (of order Im (s) 1) and is determined by the lowest nontrivial Taylor coefficient at that point.
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Since the functions γ1 (s), F01 (s), and ΔE(s) are supposed to be well-behaved near the real s axis, there are no small (or large) numbers in the problem apart from those generated by the minimum of the gap ΔE(s). Thus, the significant changes of the eigenvectors are also localized around this minimum. In the complex plane, this minimum along the real axis becomes a saddle point. For analytic functions, the characteristic length scale of variation must be the same along the real axis and into the complex plane (of order Im (s1) and is determined by the lowest nontrivial Taylor coefficient at that point.
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