equation looks like explicitly for star graphs. Define $$\label{dist}
d_{ij}(s,s') \equiv d_{ij}^{\beta}(s,s') = \sqrt{s^2\! +{s'}^2\!
-2ss' \cos|\vartheta_j -\vartheta_i|}$$ with $\vartheta_j -\vartheta_i = \sum_{l=i+1}^j \beta_l$, in particular, $d_{ii}(s,s')= |s-s'|$. By ${\mathcal{R}}^{\kappa}_{ij}(\beta) =
{\mathcal{R}}^{\kappa}_{ji}(\beta)$ we denote the operator $L^2({\mathbb{R}}^+) \to
L^2({\mathbb{R}}^+)$ with the integral kernel $${\mathcal{R}}^{\kappa}_{ij}(s,s';\beta) := {\alpha\over 2\kappa}\: K_0
\left(\kappa d_{ij}(s,s') \right)\,;$$ then the (discrete part of the) spectral problem for the operator $H_N(\beta)$ is by Proposition \[BS\] equivalent to the matrix integral-operator equation $$\label{speceq1}
\sum_{j=1}^N \left({\mathcal{R}}^{\kappa}_{ij}(\beta) - \delta_{ij}I \right)
\phi_j = 0\,, \quad i=1,\dots,N\,,$$ on $\bigoplus_{j=1}^N L^2({\mathbb{R}}^+)$. Notice that the “entries” of the above kernel have a monotonicity property, ${\mathcal{R}}^{\kappa}_{ij}(\beta) > {\mathcal{R}}^{\kappa}_{ij}(\beta')$ if $|\vartheta_j -\vartheta_i| < |\vartheta'_j -\vartheta'_i|$. This fact has the following easy consequence [@EN1]:

\[mono2\] Each isolated eigenvalue $\lambda_n(\beta)$ of $H_2(\beta)$ is an increasing function of the angle $\beta$ between the two half-lines in $(0,\pi)$.

Higher dimensions
-----------------

If we restrict ourselves to the physically interesting case of three-dimensional configuration space, there are two possible ways how to extend the above results. One refers to the situation when the interaction is supported by a surface. Here unfortunately only a particular result is known to the date, which we will mention in Sec. \[ss: 3strong\] below.

Let us thus 