Quasi-one-dimensional systems demonstrate Van Hove singularities in the density of states and the resistivity, occurring when the Fermi level E crosses a bottom E_N of some (resonant) subband of transverse quantization. Taking scattering at short-range impurities into account smears the singularities. However, in the immediate vicinity of the Van Hove singularity the electron wave functions in the resonant subband are strongly localized. Therefore, the plane waves do not provide any good approximation for them and one has to employ exact 1D-results instead.
 
We show that the character of smearing crucially depends on the concentration of impurities. Namely, there is a crossover concentration n_c \propto |\lambda|, \lambda being the dimensionless amplitude of scattering. For n \gg nc the positions of singularities are shifted by the average impurity potential \overline{U} \propto n\lambda and the
singularities are simply rounded at \varepsilon\equiv E-E_N+\overline{U}\sim\tau^{-1} – the Born scattering
rate. The overall density of states and resistivity in this case are given by the direct sum of nonresonant and resonant subbands contribution.
 
However, for n\ll n_c the result is more complicated because of strong hybridization between resonant and nonresonant subbands (non-Born effect) and generally depends on the sign of \lambda. For simplicity, in this work we consider only the case \lambda>0. In this case the resonant subband states at low energies are localized on stretches between any two adjacent impurities. The resistivity peak is then asymmetrically split - it has a broad maximum at \varepsilon\propto\lambda^2, a deep minimum at \varepsilon\propto n^2\ll\lambda^2 and saturates for \varepsilon<0, |\varepsilon|\gg\lambda^2.