3. Universal transition patterns

When the particle's energy is equal exactly to one of transverse eigenenergies, i.e., when E ! EQ , then the wavefunction reduces to a simple but universal expression

$$\Psi(\mathbf{r}) = \sum\_{m=1}^{\infty} \exp\left(ik\_m \mathbf{x}\right) \chi\_m(\mathbf{y}) \left\{ \delta\_{n,m} - \delta\_{\mathbf{Q},m} \frac{\chi\_n(\varepsilon)}{\chi\_\mathbf{Q}(\varepsilon)} \right\} = \exp\left(i\sqrt{E\_\mathbf{Q} - E\_n} \mathbf{x}\right) \chi\_n(\mathbf{y}) - \chi\_\mathbf{Q}(\mathbf{y}) \frac{\chi\_n(\varepsilon)}{\chi\_\mathbf{Q}(\varepsilon)}.\tag{21}$$

A similar universality was shown for zero-field wire [23] (for other patterns, see [24])

$$\Psi(\mathbf{r}) = \exp\left(i\pi\sqrt{Q^2 - n^2}\mathbf{x}\right)\sin\left(\frac{n\pi y}{w}\right) - \sin\left(\frac{Q\pi y}{w}\right)\frac{\sin\left(n\pi e/w\right)}{\sin\left(Q\pi e/w\right)},\tag{22}$$

but Eq. (21) solution is valid in the presence of an electric field as well.

This solution is universal in the sense that it is totally independent of the point defect's strength (potential), which is manifested by the parameter ρ0—a parameter that does not appear in Eq. (21). This pattern is presented in the upper panel of Figure 2.

Clearly, when the defect is close to the surface, i.e., ε=w < < 1, then the solution is even independent of ε

$$\Psi(\mathbf{r}) \cong \exp\left(i\pi\sqrt{Q^2 - n^2\chi}\right)\sin\left(\frac{n\pi\eta}{w}\right) - \sin\left(\frac{Q\pi\eta}{w}\right)\frac{n}{Q}.\tag{23}$$

### Figure 2.

A false color presentation of the conductance pattern. The spatial distribution in the wire of the probability <sup>2</sup> density <sup>j</sup>ψð Þ<sup>r</sup> <sup>j</sup> <sup>2</sup> is plotted at the transition energy <sup>E</sup> <sup>¼</sup> <sup>E</sup><sup>2</sup> ffi ð2π=wÞ þ Fw=<sup>2</sup> (upper panel) and at the zerocurrent energy <sup>E</sup> <sup>¼</sup> <sup>E</sup>min (lower panel). The crosses represent the point defect'<sup>s</sup> location. <sup>2</sup>

At this operation point, the transistor experiences maximum transmission with maximum current, which is universal and is independent of the point defect parameters. The defect deforms the conducting pattern, but it does not transfer any current to the mth mode. Consequently, the current remains in the initial nth mode, as if the defect is absent.
