\right) - K_0 \left(
\kappa|s\!-\!s'| \right) \bigg) \ge 0$$ in view of (a1) and the monotonicity of $K_0$, the inequality being strict for at least some values of the variables $s,s'$. We shall then argue in three steps:

1.  a variational argument in combination with (\[Dkern\]) shows that the spectrum is “pushed up” by the perturbation, $\sup \sigma\left( {\mathcal{R}}^{\kappa}_{\alpha,\Gamma} \right)
    > {\alpha\over 2\kappa}\,$ if $\,\Gamma$ is not straight.

2.  in view of (a2), $\,{\mathcal{D}}_{\kappa}\,$ is Hilbert-Schmidt for $\mu>\,{1\over 2}\,$, and therefore compact.

3.  the map $\,\kappa\mapsto {\mathcal{R}}^{\kappa}_{\alpha, \Gamma}\,$ is operator-norm continuous and $\,{\mathcal{R}}^{\kappa}_{\alpha,\gamma}\to 0
    $ as $\kappa\to\infty$.

The compactness of ${\mathcal{D}}_{\kappa}$ implies, in particular, in combination with Proposition \[BS\] the claim about the negative part of the essential spectrum.

The discrete spectrum part can be also derived from the generalized BS principle. It follows from the above claims that there are spectral points of ${\mathcal{R}}^{\kappa}_{\alpha,\Gamma}$ above $\alpha/2\kappa$ and they cannot be anything but eigenvalues of a finite multiplicity. Moreover, every such eigenvalue depends continuously on $\kappa$ and tends to zero as $\kappa\to\infty$. Hence it crosses one at a value $\kappa_0
>\, {1\over 2}\alpha$ giving rise to the sought eigenvalue of the operator $H_{\alpha,\Gamma}$; for details of the argument see [@EI].

