$\Gamma=\Gamma_0$ is a straight line corresponding to $\gamma_0(s)= as+b$ for some $a,b\in{\mathbb{R}}^2$ with $|a|=1$, we can separate variables and show that $$\label{straightspec}
\sigma(H_{\alpha,\Gamma_0}) = \Big\lbrack - \frac14\, \alpha^2,
\infty \Big)$$ is purely absolutely continuous. We are going to show that a bend or deformation produces, within a wide class of curves $\Gamma$, a non-void discrete spectrum. To be specific, we assume that the generating function $\gamma:\:{\mathbb{R}}\to{\mathbb{R}}^2$ is continuous and piecewise $C^1$ (or, in terms of the assumption (g1)–(g3), a graph which may have vertices but no branchings) satisfying the following conditions[^6]:

[*(a1) a lower distance bound*]{}

:   there is $c\in(0,1)$ such that $|\gamma(s)-\gamma(s')| \ge c|s-s'|$. In particular, $\Gamma$ has no cusps and self-intersections, and its possible asymptotes are not parallel to each other.

[*(a2) asymptotic straightness*]{}

:   there are positive $d,\, \mu>\frac12$, and $\omega\in(0,1)$ such that the inequality $$1-\, {|\gamma(s)-\gamma(s')|\over|s-s'|} \le d \left\lbrack
    1+|s+s'|^{2\mu} \right\rbrack^{-1/2}$$ holds true in the sector $S_\omega:= \left\{ (s,s'):\: \omega <
     {s\over s'} < \omega^{-1}\, \right\}$.

[*(a3) non-triviality*]{}

:   we excluded the case $\Gamma=\Gamma_0$. Recall that $$|\gamma(s)-\gamma(s')| \le |s-s'|$$ holds for any $s,s'\in {\mathbb{R}}$, hence we request in other words that the last inequality is sharp at least for some $s,s'\in {\mathbb{R}}$.

Then we have the 