A. Appendix A: derivation of Eq. (39)

The expression (26), i.e.,

$$R(\varepsilon) = \exp\lim\_{\rho \to 0} \left[ \ln \rho + \pi \sum\_{q=3}^{\infty} \frac{\left| \chi\_q(\varepsilon) \right|^2}{\sqrt{E\_q - E}} \exp\left( - \left| E - E\_q \right| \rho^2 / 4 \right) \right]$$

as a function of the defect's location ε is plotted for several energy values in the energy range π<sup>2</sup>=w<sup>2</sup> < E < 4π<sup>2</sup>=w<sup>2</sup> in Figure A1.

As can be seen from this figure, while there are considerably large variations around ε ffi 0:2, the differences in the value of Rð Þε for ε ffi 0:5, i.e., when the defect is located at the center of the wire, are relatively mild, and in which case RðεÞ ffi 0:3. Moreover, in the case of a surface defect, i.e., ε < < 1, Rð Þε is independent of the particles' energy.

Using the definition ξ � πqρ=w and the weak field approximation

$$\ln R(\varepsilon) \equiv \lim\_{\rho \to 0} \left[ \ln \rho + \frac{2}{w} \pi \exp \left( \pi^2 \rho^2 / w^2 \right) \sum\_{q=3}^{\infty} \rho \frac{\sin^2(\xi \varepsilon/\rho)}{\sqrt{\xi^2 - \left(4\pi\rho\right)^2/w^2}} \exp \left(-\xi^2 / 4\right) \right] \tag{A1}$$

which can be written as an integral
