Non-Born effects in scattering of electrons in a clean conducting tube

Quasi-one-dimensional systems demonstrate Van Hove singularities in the density of states ${\nu }_F$ and the resistivity $\rho$, occurring when the Fermi level $E$ crosses a bottom $E_N$ of some subband of transverse quantization. We demonstrate that the character of smearing of the singularities crucially depends on the concentration of impurities. There is a crossover concentration $n_c\propto \mathrm{\mid }\lambda \mathrm{\mid }$, $\lambda \ll 1$ being the dimensionless amplitude of scattering. For $n\gg n_c$ the singularities are simply rounded at $\epsilon \equiv E-E_N\sim {\tau }^{-1}$ – the Born scattering rate. For $n\ll n_c$thenon-Borneffectsinscatteringbecomeessentialdespite$\lambda\ll 1$. The peak of the resistivity is asymmetrically split in a Fano-resonance manner (however with a more complex structure). Namely, for $\epsilon >0$ there is a broad maximum at $\epsilon \propto {\lambda }^2$ while for $\epsilon <0$ there is a deep minimum at $|\varepsilon|\propto n^2\ll \lambda^2$\mathrm{.}Thebehaviourof$\rho$belowtheminimumdependsonthesignof$\lambda$\mathrm{.}Incaseofrepulsion$\rho$monotonicallygrowswith$|\varepsilon|$andsaturatesfor$|\varepsilon|\gg \lambda^2$\mathrm{.}Incaseofattraction$\rho$hassharpmaximumat$|\varepsilon|\propto \lambda^2$. The latter feature is due to resonant scattering by quasistationary bound states that inevitably arise just below the bottom of each subband for any attracting impurity.