Renormalized leaky-wire limit conjecture

Let $\Gamma$ be a graph with some semi-infinite, asymptotically straight edges, and suppose

$$\sigma_{\mathrm{ess}}(-\Delta_{\alpha,\Gamma})=\left[-\frac14\alpha^2,\infty\right).$$

Consider the renormalized operator $-\Delta_{\alpha,\Gamma}+\frac14\alpha^2$ as $\alpha\to\infty$. Renormalized leaky-wire limit conjecture. If $-\Delta_{\alpha,\Gamma}$ has a threshold resonance for some $\alpha>0$, then the operator-norm limit exists; denoting it by $-\Delta^{\mathrm{ren}}_\Gamma$, one has

$$\sigma_{\mathrm{disc}}(-\Delta^{\mathrm{ren}}_\Gamma)=\emptyset.$$

A nonempty discrete spectrum of the limiting operator could nevertheless be obtained if the geometry of $\Gamma$ is changed simultaneously in a suitable way. The conjecture concerns the strong-coupling limit of leaky-wire Hamiltonians and its relation to threshold resonances.