consider the second possibility when $\Gamma$ is an infinite piecewise $C^1$ curve in ${\mathbb{R}}^3$. The argument is similar to that of the previous section but it needs care due to the more singular character of the interaction. If $\Gamma=\Gamma_0$ the spectrum of $H_{\alpha,\Gamma}$ is found by separation of variables; the known properties of the two-dimensional point interaction [@AGHH] imply that $$ \sigma (H_{\alpha,\Gamma})=\sigma_\mathrm{ac}
 (H_{\alpha,\Gamma})=[\zeta_{\alpha},\infty )\,,$$ where $\zeta_{\alpha}=-4 {\mathrm{e}}^{2(-2 \pi \alpha +\psi (1))}$; in this expression $-\psi(1)\approx 0.577$ is Euler-Mascheroni constant. To rephrase it in terms of the BS operator ${\mathcal{Q}}_0^\kappa:=
Q_0^{i\kappa}$ notice that the latter equals ${\mathcal{T}}_\kappa +\ln
2+\psi(1)$ where ${\mathcal{T}}_\kappa$ in the momentum representation acts as multiplication[^9] by $\ln(p^2+\kappa^2)^{1/2}$. Consequently, the spectrum of ${\mathcal{Q}}_0^\kappa$ is purely absolutely continuous and equal to $(-\infty ,s_{\kappa}]$ where $s_{\kappa
}:=\frac{1}{2\pi}(\psi(1)-\ln(\kappa/2))$.

To proceed we need assumptions about the curve $\Gamma$. We retain (a1) and (a3) from the previous section, while (a2) will be replaced by

[*(a2’) ASLS*]{}

:   there are positive $d,\, \mu>\frac12$, and $\omega\in(0,1)$ such that $$1-{|\gamma(s)-\gamma(s')|\over|s-s'|} \le d\:\frac{|s\!-\!s'|}
    {(1+|s\!-\!s'|) (1+(s^{2}\!+\!s^{\prime 2})^{\mu})^{1/2}}$$ holds true in the sector $S_\omega$, the same as before.

In addition to the asymptotic straightness[^10] we require newly some local smoothness of the curve. Now we can follow proof of 