Theorem \[dsexist\] step by step; after checking that the essential spectrum is preserved, and denoting ${\mathcal{Q}}^\kappa:=
Q^{i\kappa}$, we prove that

1.  $\sup \sigma ({\mathcal{Q}}^{\kappa})>s_{\kappa}$ by a variational argument using the sign definitess, $$ {\mathcal{D}}_{\kappa }(s,s^{\prime})= G_{i\kappa}(\gamma
     (s)\!-\!\gamma (s^{\prime}) )- G_{i\kappa }(
     s\!-\!s^{\prime })\ge 0\,,$$ with a sharp inequlity at least for some values of the variables $s,s'$.

2.  in view of (a2’) $\,{\mathcal{D}}_{\kappa}\,$ is Hilbert-Schmidt for $\mu>\,{1\over 2}\,$, hence compact, and the corresponding norm $
    \left\| {\mathcal{D}}_{\kappa}\right\|_\mathrm{HS}$ is uniformly bounded w.r.t. $\kappa \geq \left| \zeta_{\alpha}\right|^{1/2}$.

3.  the function $\,\kappa \to {\mathcal{Q}}^{\kappa}\,$ is operator-norm continuous in $(\left| \zeta_{\alpha}\right|^{1/2}, \infty)$ and $\,{\mathcal{Q}}^{\kappa}\to -\infty $ holds as $\kappa\to\infty$.

Working out details of this scheme [@EK2] we arrive at the following conclusion:

\[ds3exist\] Fix $\alpha>0$ and suppose that the generating function $\gamma:
{\mathbb{R}}\to{\mathbb{R}}^3$ of $\Gamma$ satisfies the stated assumptions. Then $\sigma_\mathrm{ess}(H_{\alpha,\Gamma}) = \left\lbrack
\zeta_{\alpha}, \infty \right)$, and the operator $H_{\alpha,\Gamma}$ has at least one isolated eigenvalue in the interval $(-\infty ,\zeta_{\alpha})$.

Let us remark that the strengthened hypothesis in (a2’) is not needed to prove the existence of the geometrically induced spectrum, but rather to determine its character; due to the more strongly singular character of the interaction in the codimension two case the 