Population dynamics and non-Hermitian localization

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27. The argument is best understood after taking the continuum limit by letting the lattice constant l ₀ tend to zero. Suppose the continuum eigenfunction φ _n(x, g) with complex eigenvalue were in fact exactly zero at position x=x ₀. By a similarity or “gauge” transformation similar to Eq. (17), one could shift all effects of the non-Hermiticity to the position x ₀. The transformed eigenfunction would still vanish at x ₀, because this gauge transformation is non-singular. All effects of non-Hermiticity would then disappear for this eigenfunction, which would thus have to be localized with a real eigenvalue, leading to a contradiction. We are grateful to Ady Stern for discussions on this point.

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