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\title{On zero eigenvalue of invariant Schr\"odinger operators with point interactions at vertices of some regular polyhedra}
\alttitle{Sur la valeur propre nulle des op\'erateurs de Schr\"odinger invariants avec interactions ponctuelles aux sommets de certains poly\`edres r\'eguliers}

\author{\firstname{Mark} \middlename{M.} \lastname{Malamud}\CDRorcid{0000-0001-9118-448X}}
\address{Saint Petersburg State University, Saint Petersburg, Russia}
\email{malamud3m@gmail.com}

\author{\firstname{Vladimir} \middlename{V.} \lastname{Marchenko}\CDRorcid{0000-0001-8077-3887}}
\address{Bauman Moscow State Technical University, Moscow, Russia}
\address{Institute of Applied Mathematics and Mechanics, Donetsk, Russia}
\email{wmarchenko@rambler.ru}

\thanks{The study of the first author in Sections~3--5 was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation in the framework of a scientific project under agreement No.~075-15-2024-631. His research on Section~1, 2, and~6 was carried out by grant of the Russian Science Foundation No.~23-11-00153}
\CDRGrant[Ministry of Science and Higher Education of the Russian Federation]{075-15-2024-631}
\CDRGrant[Russian Science Foundation]{23-11-00153}

\subjclass{47A57}

\keywords{\kwd{Schr\"odinger operators with point interactions} \kwd{invariant operators} \kwd{Krein realization} \kwd{multiplicity of zero eigenvalue}}
\altkeywords{\kwd{OpÃ©rateurs de SchrÃ¶dinger avec interactions ponctuelles} \kwd{opÃ©rateurs invariants} \kwd{rÃ©alisation de Krein} \kwd{multiplicitÃ© de valeur propre nulle}}

\begin{abstract}
Following Berezin and Faddeev, by a Schr\"odinger operator with point interactions:
\[
-\Delta + \sum_{j=1}^m\alpha_j\delta(x - x_j), \quad X = \{x_j\}^m_1\subset{\mathbb R}^3, \quad \{\alpha_j\}^m_1\subset{\mathbb R},
\]
one means any selfadjoint extension of the restriction $\Delta_X$ of the Laplace operator $-\Delta$ to the subset $\bigl\{f \in H^{2}({\mathbb R}^3): f(x_j)=0, \ 1\le j\le m \bigr\}$ of the Sobolev space $H^{2}({\mathbb R}^3)$.

In the present paper the above set of interactions $X$ is assumed to be a vertex set of a certain regular polyhedron, and selfadjoint extensions (realizations) invariant under the symmetry group of $X = \{x_j\}_1^m$ are studied.

Such realizations $\boldsymbol{\operatorname{H}}_B$ are parametrized by special matrices $B = B^*\in {\mathbb C}^{m\times m}$. We describe all such selfadjoint realizations with non-trivial kernels. By this we continue investigation by Grinevich--Novikov and ours relating to regular polygons. Besides, for arbitrary realizations the estimate $\dim(\ker\boldsymbol{\operatorname{H}}_B) \le m-1$ is proved, and realizations with all feasible $\dim(\ker{\mathbb H_B})$ are described.

Particular attention is paid to realizations with maximum value $\dim(\ker\boldsymbol{\operatorname{H}}_B) = m-1$. One of them is the Krein realization, which is the minimal positive selfadjoint extension of the operator $\Delta_X \ge 0$.
\end{abstract}

\begin{altabstract}
D'apr\`es Berezin et Faddeev, par un op\'erateur de Schr\"odinger avec des interactions ponctuelles:
\[
-\Delta + \sum_{j=1}^m\alpha_j\delta(x - x_j), \quad X = \{x_j\}^m_1\subset{\mathbb R}^3, \quad \{\alpha_j\}^m_1\subset{\mathbb R},
\]
on entend toute extension autoadjointe de la restriction $\Delta_X$ de l'op\'erateur de Laplace $-\Delta$ au sous-ensemble $\bigl\{f \in H^{2}({\mathbb R}^3): f(x_j)=0, \ 1\le j\le m \bigr\}$ de l'espace de Sobolev $H^{2}({\mathbb R}^3)$.

Dans le pr\'esent article, l'ensemble d'interactions $X$ ci-dessus est suppos\'e \^etre un ensemble de sommets d'un certain poly\`edre r\'egulier, et des extensions (r\'ealisations) autoadjointes invariantes sous le groupe de sym\'etrie de $X = \{x_j\}_1^m$ sont \'etudi\'ees.

De telles r\'ealisations $\boldsymbol{\operatorname{H}}_B$ sont param\'etr\'ees par des matrices sp\'eciales $B = B^*\in {\mathbb C}^{m\times m}$. Nous d\'ecrivons toutes ces r\'ealisations autoadjointes avec des noyaux non triviaux. Nous poursuivons ainsi l'investigation de Grinevich--Novikov et la n\^otre concernant les polygones r\'eguliers. De plus, pour des r\'ealisations arbitraires, l'estimation $\dim(\ker\boldsymbol{\operatorname{H}}_B) \le m-1$ est prouv\'ee, et des r\'ealisations avec tous les $\dim(\ker{\mathbb H_B})$ r\'ealisables sont d\'ecrites.

Une attention particuli\`ere est accord\'ee aux r\'ealisations avec la valeur maximale $\dim(\ker\boldsymbol{\operatorname{H}}_B) = m-1$. L'une d'elles est la r\'ealisation de Krein, qui est l'extension autoadjointe positive minimale de l'op\'erateur $\Delta_X \ge 0$.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

It is well-known (see~\cite{Kato, Saimon}) that positive spectrum of the Schr\"odinger operator $-\Delta+q$ in $L^2(\mathbb R^n)$, $n=2,3$, with decaying potential $q(x) = O(1+\abs{x})^{-1-\varepsilon}$, $\varepsilon &gt; 0$, is purely absolutely continuous. However, the operator $-\Delta+q$ may have the (embedded) zero eigenvalue. This effect for rational potentials $q(x_1, x_2)$ was discovered by Taimanov and Tsarev in~\cite{TaimTsar,TaimTsar2008} by means of the Moutard transformation.

For the Schr\"odinger operator with point interactions in the set $X=\{x_j\}^m_1(\subset{\mathbb R}^3)$:
\begin{equation}\label{eq:Intro,Lapl+d-int-s}
-\Delta + \sum_{j=1}^m\alpha_ j\delta(x - x_j),\quad \{\alpha_ j\}_1^m \subset \mathbb{R},
\end{equation}
this effect was discovered by Grinevich and Novikov in~\cite{GriNov17}.

Recall that, following Berezin and Faddeev~\cite{BF61}, by a Schr\"odinger operator with point interactions in $X=\{x_j\}^m_1(\subset{\mathbb R}^3)$ one means any realization (self-adjoint extension) $\Hrlztn_B$ of the three-dimensional Laplace operator in $\mathfrak{H}= L^2({\mathbb R}^3)$ restricted to the (dense in $\mathfrak{H}$) domain
\begin{equation}\label{min}
\Hrlztn = -\Delta \mathbin{\upharpoonright} \dom(\Delta_X), \quad
\dom(\Delta_X) \coloneqq \braces[\big]{f \in W^{2,2}({\mathbb R}^3): f(x_j)=0,\ j\in \{1,\dotsc, m\}}.
\end{equation}

The operator~\eqref{min} is a closed nonnegative symmetric operator in $\mathfrak{H}$ with defect numbers $n_\pm(\Hrlztn)=m$ with its adjoint $\Hrlztn^*$ given by~\eqref{eq3}--\eqref{eq3'}. A large number of papers are devoted to studying spectral properties of extensions of such operators (see, eg., \cite{GriNov22, AGHH88, GMZ11, MalSchm2012}, and references therein). It is well-known (see~\cite{AGHH88,GMZ11}) that positive spectrum of each realization is purely absolutely continuous, although negative spectrum is finite. In particular, it has no positive (embedded) eigenvalues. Assuming that $X=\{x_j\}^m_1(\subset{\mathbb R}^3)$ is a vertex set of a regular polygon or polyhedron, we described in~\cite{MalMar21} all realizations invariant under the symmetry group of $X$. Namely, we describe in~\cite{MalMar21} all ``boundary'' matrices $B = B^*\in{\mathbb C}^{m\times m}$ that parametrize all invariant realizations by formula $\Hrlztn_B \coloneqq \Hrlztn^*\!\!\mathbin{\upharpoonright}\ker\parens{\Gamma_1 - B\Gamma_0}$ (see~\eqref{bijop}), where $\Gamma_0$ and $\Gamma_1$ are given by~\eqref{g0}--\eqref{g1}.

The paper~\cite{MalMar24} considers the case of a regular $m$-gon $X = \{x_j\}_1^m$ and describes all invariant realizations $\Hrlztn_B$ satisfying $\ker\Hrlztn_B\neq\{0\}$. In particular, \cite{MalMar24} solves the problem, posed by Grinevich and Novikov in~\cite{GriNov17}, on multiplicity of zero eigenvalue of $\Hrlztn_B$.

In this note we continue the research initiated in~\cite{GriNov17,MalMar24} as well as describe all invariant realizations $\HrlztnB$ with non-trivial kernels, assuming that $X$ is the vertex set of some regular polyhedra (tetrahedron, hexagedron, octahedron). As in~\cite{MalMar24} we study this problem in the framework of extension theory of the operator $\Hrlztn$ by applying the boundary triples technique and the corresponding Weyl functions (see definitions in Section~\ref{BtrfV}). We describe all invariant Schr\"odinger operators (extensions $\Hrlztn_B$ of $\Hrlztn$) with non-trivial kernels.

In particular, we complement the result of~\cite{GriNov17}, where, in the case of a tetrahedron, invariant realizations $\HrlztnB$ with non-trivial kernels and scalar matrices $B = \alpha I$, $\alpha\in{\mathbb R}$, are described.

Note that consideration of all ``boundary'' (not only scalar) matrices $B$ leads to curious effects. Thus, it leads to a description of all invariant realizations $\HrlztnB$ with maximal possible kernels: $\dim(\ker\Hrlztn_B) = m-1$. Moreover, among them are contained non-negative realizations $\Hrlztn_B \ge 0$, in particular, the Krein realization (the smallest one) $\widehat\Hrlztn_K (\ge 0)$. Finally, we mention interesting features which essentially distinguish the considered case from the one of a regular $m$-gon:
\begin{enumerate}\romanenumi
\item ``boundary'' matrices $B$ are not necessarily Toeplitz;
\item $C^*$-algebras generated by ``boundary'' matrices contain more than one generator;
\item dimensions of $\ker\Hrlztn_B$ have forbidden values missing some numbers from $1$ to $m-2$ (see Section~\ref{final}).
\end{enumerate}

Besides, for $B \not = \alpha I$ there appear non-selfadjoint realizations $\Hrlztn_B$ ($\Longleftrightarrow B \not = B^*$) with non-trivial kernels.

\section{Boundary triples and Weyl functions}\label{BtrfV}

In this paragraph, following~\cite{GMZ11, DerMal17, DerMal91, GG}, we summarize the basic notions and facts of the boundary triples theory.

Let $A$ be a symmetric operator in a separable Hilbert space $\mathfrak{H}$ with dense domain $\dom(A)$ and equal defect numbers $n_+(A) = n_-(A)\le \infty$.

\begin{defi}
The totality $\Pi=\{\kH,\gG_0,\gG_1\}$, in which $\kH$ is an auxiliary Hilbert space, and $\Gamma_0,\Gamma_1 \colon \dom(A^*)\to\kH$ are linear mappings, is called a \emph{boundary triple} of the operator $A^*$ if:
\begin{enumerate}\romanenumi
\item the following Green formula holds:
\begin{equation}\label{GI}
(A^*f,g)_\gotH - (f,A^*g)_\gotH = (\gG_1f,\gG_0g)_\kH - (\gG_0f,\gG_1g)_\kH,\quad f,g\in\dom(A^*);
\end{equation}
\item the mapping $\gG\coloneqq (\Gamma_0,\Gamma_1)^\top \colon \dom(A^*) \to \kH \oplus\kH$ is surjective.
\end{enumerate}
\end{defi}

\begin{defi}
An extension $\tilde{A} \supset A$ is called\/ \emph{proper} if $A\subset \tilde{A} \subset A^*$.
\end{defi}

\begin{prop}[\cite{DerMal91}]\label{bij}
Let $A$ be a symmetric operator in a Hilbert space $\mathfrak{H}$ with defect numbers $n_\pm(A) = m$, and let\/ $\Pi=\{\mathcal{H},\Gamma_0,\Gamma_1\}$ be a boundary triple of the operator $A^*$. Then the mapping
\begin{equation}\label{bijop}
B\to A_B=A^*\!\!\mathbin{\upharpoonright}\ker\parens{\Gamma_1-B\Gamma_0}
\end{equation}
establishes a bijective correspondence between the set $\mathcal C(\kH)$ of closed linear operators $B$ in $\kH$ and the set of closed proper extensions $A_B$ of the operator $A$ which are disjoint with $A_0\coloneqq A^*\!\!\mathbin{\upharpoonright}\ker(\gG_0)$, i.\,e., such that $\dom A_B \cap\dom A_0 = \dom A$.
\end{prop}

\begin{defi}[\cite{DerMal91}]\label{WeylfuncGamma}
Let $\Pi=\{\kH,\gG_0,\gG_1\}$ be a boundary triple for $A^*$. The operator function $M(\,\cdot\,)$ defined by the equality
\begin{equation}
M(z)\Gamma_0f = \Gamma_1f, \quad \text{where} \ f\in\mathfrak{N}_z = \ker\parens{A^* - zI},
\end{equation}
is called the \emph{Weyl function corresponding to the boundary triple $\Pi$}.
\end{defi}

\section{Three-dimensional Schr\"odinger operators with point interactions}\label{Schr}

\begin{prop}[\cite{GMZ11}]\label{pr1}
Let $\Hrlztn$ be the minimal Schr\"odinger operator~\eqref{min} and let $\xi_0\coloneqq \{\xi_{0j}\}_{j=1}^m$, $\xi_1\coloneqq \{\xi_{1j}\}_{j=1}^m
\in\mathbb{C}^m$. Then the following statements hold.
\begin{enumerate}\romanenumi
\item $\Hrlztn$ is a closed non-negative symmetric operator with deficiency indices $n_\pm(\Hrlztn)=m$.
\item The adjoint operator\/ $\Hrlztn^*$ is given by the equalities
\begin{gather}
\dom(\Hrlztn^*) =\braces[\Bigg]{ f=
\sum^m_{j=1}\parens[\Bigg]{\xi_{0j}\frac{e^{-\abs{x-x_j}}}{\abs{x-x_j}}+\xi_{1j}e^{-\abs{x-x_j}}} + f_H:\ \xi_{0},\xi_{1}\in{\mathbb C}^m,\, f_H\in\dom(\Hrlztn)}, \label{eq3} \\
\Hrlztn^*f=-\sum^m_{j=1}\parens[\Bigg]{\xi_{0j}\frac{e^{-\abs{x-x_j}}}{\abs{x-x_j}}+
\xi_{1j}\parens[\Bigg]{e^{-\abs{x-x_j}}-\frac{2e^{-\abs{x-x_j}}}{\abs{x-x_j}}}}-\Delta f_H. \label{eq3'}
\end{gather}
\item The totality $\Pi =\{\kH,\Gamma_0,\Gamma_1\}$ with
\begin{align}
\kH={\mathbb C}^m,\quad \Gamma_0 f &amp;\coloneqq \{\Gamma_{0j}f\}^m_{j=1}= 4\pi\,\braces[\Big]{\lim_{x\to x_j}f(x)\abs{x-x_j}}^m_{j=1}=4\pi\{\xi_{0j}\}^m_{j=1}, \label{g0} \\
\Gamma_1 f &amp; \coloneqq \{\Gamma_{1j}f\}^m_{j=1}=\braces[\Big]{\lim_{x\to x_j}\parens[\Big]{f(x)-\tfrac{\xi_{0j}}{\abs{x-x_j}}}}^m_{j=1}, \label{g1}
\end{align}
forms a boundary triple for the operator\/ $\Hrlztn^*$.
\item The corresponding Weyl function $M(\,\cdot\,)$ has the form
\begin{equation}\label{W3}
M(z)=\parens[\Bigg]{\frac{i\sqrt{z}}{4\pi}\delta_{jk}+\widetilde{G}_{\sqrt{z}}(x_j-x_{k})}_{j,k=1}^m, \quad z\in\mathbb{C}_+,
\end{equation}
where
\[
\widetilde{G}_{\sqrt{z}}(x)=
\begin{cases}
\frac{e^{i\sqrt{z}\abs{x}}}{4\pi\abs{x}}, 	&amp; x\neq0, 	\\
0, 											&amp; x=0. 		\\
\end{cases}
\]
Here $\delta_{jk}$ is the Kronecker symbol, $\sqrt{\,\cdot\,}$ is a branch of the root defined on ${\mathbb C}\setminus {\mathbb R}_+$ by the condition $\sqrt{1}=1$.
\item There exists the strong limit $M(0)\coloneqq \stronglim_{z\uparrow 0} M(z)$ in ${\mathbb C}^m$, and $M(0) = M(0)^*$.
\item $\Hrlztn_{0} \coloneqq \Hrlztn^*\!\!\mathbin{\upharpoonright}\ker \Gamma_0 = \widehat\Hrlztn_F = -\Delta$ coincides with the Friedrichs (maximal positive) extension of the operator\/ $\Hrlztn$. Besides, $\Hrlztn_{M(0)} = \Hrlztn^*\!\!\mathbin{\upharpoonright}\ker\parens[\big]{\Gamma_1-M(0)\Gamma_0} = \widehat\Hrlztn_K$ is the Krein (minimal positive) extension of $\Hrlztn$.
\end{enumerate}
\end{prop}

\begin{rema}
The expression~\eqref{eq:Intro,Lapl+d-int-s} is often identified with the realization $\Hrlztn_B$ given in the boundary triple~\eqref{g0}--\eqref{g1} by the equality~\eqref{bijop} with the diagonal matrix $B = \diag(\alpha_1,\dotsc, \alpha_m)$.
\end{rema}

\section{Invariant Schr\"odinger operators}

The concepts of this section are presented following~\cite{MalMar21}.

For a set $X\subset {\mathbb R}^3$ by $\mathcal S_X$ we denote the symmetry group of $X$, i.\,e., the motions of the space ${\mathbb R}^3$ which preserve $X$.

A symmetry $u (\in \mathcal S_X)$ induces a unitary operator $U$ in $L^2({\mathbb R}^3)$ of the form
\begin{equation}\label{op}
U f(x) = f\parens[\big]{u(x)}, \qquad U\colon L^2({\mathbb R}^3)\to L^2({\mathbb R}^3).
\end{equation}

\begin{defi}\label{invar}
An operator $A\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$ is called invariant under a symmetry $u$ if the equality $AU = UA$ holds on $\dom(A)$. If this equality holds for all symmetries $u\in \mathcal S_X$, then $A$ is called invariant under the group $\mathcal{S}_X$ (or simply invariant).
\end{defi}

It is easily seen that the minimal operator $\Hrlztn$ is invariant.

\section{Regular tetrahedron}

Let $X$ be a vertex set of a regular tetrahedron. According to Proposition~\ref{bij}, each realization $\Hrlztn_B$ disjoint with the operator $\Hrlztn_0 =-\Delta$ is the restriction of the operator $\Hrlztn^*$ given by~\eqref{eq3}--\eqref{eq3'} to the domain $\dom\Hrlztn_B = \ker\parens{\Gamma_1 - B\Gamma_0}$, where $\Gamma_0$, $\Gamma_1$ are given by~\eqref{g0}--\eqref{g1}, and $B\in{\mathbb C}^{4\times 4}$.

Meanwhile, according to~\cite{MalMar21}, the extension $\Hrlztn_B$ is invariant (under the group $\mathcal{S}_{X}$) if and only if $B$ is a special Toeplitz matrix of the following form:
\begin{equation}\label{matrtetr}
B =
\begin{pmatrix}
a &amp; b &amp; b &amp; b\\
b &amp; a &amp; b &amp; b\\
b &amp; b &amp; a &amp; b\\
b &amp; b &amp; b &amp; a
\end{pmatrix}\in{\mathbb C}^{4\times 4}.
\end{equation}

According to~\cite{MalMar21}, the Weyl function corresponding to the triple~\eqref{g0}--\eqref{g1} in this case has the following form:
\begin{equation}
M(z) = \frac{1}{4\pi}
{\begin{pmatrix}
i\sqrt{z} &amp; \frac{e^{ir\sqrt{z}}}{r} &amp; \frac{e^{ir\sqrt{z}}}{r} &amp; \frac{e^{ir\sqrt{z}}}{r}\\
\frac{e^{ir\sqrt{z}}}{r} &amp; i\sqrt{z} &amp; \frac{e^{ir\sqrt{z}}}{r} &amp; \frac{e^{ir\sqrt{z}}}{r}\\
\frac{e^{ir\sqrt{z}}}{r} &amp; \frac{e^{ir\sqrt{z}}}{r} &amp; i\sqrt{z} &amp; \frac{e^{ir\sqrt{z}}}{r}\\
\frac{e^{ir\sqrt{z}}}{r} &amp; \frac{e^{ir\sqrt{z}}}{r} &amp; \frac{e^{ir\sqrt{z}}}{r} &amp; i\sqrt{z}
\end{pmatrix}}, \quad r = \abs{x_j-x_k}, \ j\neq k, \quad z\in \mathbb{C}\setminus \mathbb{R}_+.
\end{equation}

\begin{theo}\label{th:tetraedr}
An invariant realization\/ $\Hrlztn_B$ defined by a boundary matrix $B$ of the form~\eqref{matrtetr} has a non-trivial kernel if and only if
\begin{equation}\label{tetrmax}
a - b = -\frac{1}{4\pi r}.
\end{equation}
Moreover, $\dim(\ker\Hrlztn_B) = 3$, and\/ $\ker\Hrlztn_B$ is spanned by the functions
\begin{equation}\label{tetreigf}
\begin{gathered}
\textstyle \psi_1(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_1}} -\frac{1}{\abs{x-x_2}}}, \\
\textstyle \psi_2(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_1}} -\frac{1}{\abs{x-x_3}}}, \\
\textstyle \psi_3(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_1}} -\frac{1}{\abs{x-x_4}}}.
\end{gathered}
\end{equation}
\end{theo}

\begin{rema}
In the case of a scalar matrix $B = aI\in{\mathbb R}^{4\times 4}$, i.e.\ when $b=0$, this result in a different way was obtained by Grinevich and Novikov in~\cite[Example~1]{GriNov17}.
\end{rema}

\begin{theo}
Let $\Hrlztn_B$ be an invariant realization with a matrix $B$ of the form~\eqref{matrtetr}. Let also $E_4 (\in {\mathbb R}^{4\times 4})$ be a matrix with all entries equal to $1$. Then $\ker\Hrlztn_B\neq\{0\}$ if and only if the matrix $B$ is of the form
\[
B = B(a) \coloneqq aE_4 + M(0), \qquad a\in {\mathbb C}.
\]
Moreover, the following statements hold true:
\begin{enumerate}\romanenumi
\item $\Hrlztn_{B(a)}= \Hrlztn_{B(a)}^*$ precisely when $a\in {\mathbb R}$;
\item $\Hrlztn_{B(a)}\ge 0$ precisely when $a\ge 0$;
\item in particular, for $a=0$ the operator\/ $\Hrlztn_{B(0)} = \Hrlztn_{M(0)}$ coincides with the Krein (smallest positive) realization $\widehat\Hrlztn_K$ of the operator\/ $\Hrlztn$.
\end{enumerate}
\end{theo}

\begin{coro}
In the case $a&lt;0$ the number of negative eigenvalues (counting multiplicity) of the operator\/ $\Hrlztn_{B(a)}$ equals
\[
\kappa_-(\Hrlztn_{B(a)}) = \kappa_-\parens[\big]{B(a) - M(0)} = \kappa_-(aE_4) = 1.
\]
\end{coro}

Now we specify the rate of decay at infinity of the eigenfunctions~\eqref{tetreigf}.

\begin{prop}
For the eigenfunctions~\eqref{tetreigf} the following estimates hold:
\begin{equation}
\psi_j(x) = O\parens[\Bigg]{\frac{1}{\abs{x}^{2}}}, \quad \abs{x}\rightarrow+\infty, \quad j\in\{1, 2, 3\}.
\end{equation}
\end{prop}

\section{Regular hexahedron (cube)}

Let the set of interactions $X$ be a vertex set of a cube.

According to Proposition~\ref{bij}, each realization $\Hrlztn_B$ disjoint with the operator $\Hrlztn_0 =-\Delta$ is the restriction of the operator $\Hrlztn^*$ given by~\eqref{eq3}--\eqref{eq3'} to the domain $\dom\Hrlztn_B = \ker\parens{\Gamma_1 - B\Gamma_0}$, where $\Gamma_0$, $\Gamma_1$ are defined by the equalities~\eqref{g0}--\eqref{g1}, and $B\in{\mathbb C}^{8\times 8}$.

Meanwhile, according to~\cite{MalMar21}, extension $\Hrlztn_B$ is invariant (under the group $\mathcal{S}_{X}$) if and only if $B$ is a special block Toeplitz matrix of the following form:
\begin{equation}\label{matrcube}
B=
\parens*{
\begin{array}{cccc|cccc}
a &amp; b &amp; c &amp; b \,&amp;\, b &amp; c &amp; d &amp; c\\
b &amp; a &amp; b &amp; c \,&amp;\, c &amp; b &amp; c &amp; d\\
c &amp; b &amp; a &amp; b \,&amp;\, d &amp; c &amp; b &amp; c\\
b &amp; c &amp; b &amp; a \,&amp;\, c &amp; d &amp; c &amp; b\\
\hline
b &amp; c &amp; d &amp; c \,&amp;\, a &amp; b &amp; c &amp; b\\
c &amp; b &amp; c &amp; d \,&amp;\, b &amp; a &amp; b &amp; c\\
d &amp; c &amp; b &amp; c \,&amp;\, c &amp; b &amp; a &amp; b\\
c &amp; d &amp; c &amp; b \,&amp;\, b &amp; c &amp; b &amp; a
\end{array}
}\in {\mathbb C}^{8\times 8}.
\end{equation}

According to~\cite{MalMar21}, the Weyl function corresponding to the triple~\eqref{g0}--\eqref{g1} in this case has the following form:
\begin{equation}\label{weylcube}
M(z)=\frac{1}{4\pi}
\parens*{
\begin{array}{cccc|cccc}
i\sqrt{z} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_3\sqrt{z}}}{r_3} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2}\\
\frac{e^{ir_1\sqrt{z}}}{r_1} &amp; i\sqrt{z} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_3\sqrt{z}}}{r_3}\\
\frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; i\sqrt{z} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_3\sqrt{z}}}{r_3} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2}\\
\frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; i\sqrt{z} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_3\sqrt{z}}}{r_3} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1}\\[.7ex]
\hline
&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\[-2.2ex]
\frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_3\sqrt{z}}}{r_3} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; i\sqrt{z} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1}\\
\frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_3\sqrt{z}}}{r_3} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; i\sqrt{z} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2}\\
\frac{e^{ir_3\sqrt{z}}}{r_3} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; i\sqrt{z} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1}\\
\frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_3\sqrt{z}}}{r_3} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; i\sqrt{z}
\end{array}
},
\end{equation}
where $r_1$, $r_2$, $r_3$ are three distinct distances between the cube vertices ($r_1&lt;r_2&lt;r_3$).

\begin{theo}\label{th:geksaedr}
An invariant realization\/ $\Hrlztn_B$ defined by a boundary matrix $B$ of the form~\eqref{matrcube} has a non-trivial kernel precisely when at least one of the following conditions holds:
\begin{equation}\label{condcube}
\begin{varwidth}{.9\displaywidth}
\begin{enumerate}\romanenumi
\item $a-b-c+d = \frac{1}{4\pi}\parens[\bigg]{-\frac{1}{r_1}-\frac{1}{r_2} +\frac{1}{r_3}}$;
\item $a+b-c-d = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{r_1}-\frac{1}{r_2}-\frac{1}{r_3}}$;
\item $a-3b+3c-d = \frac{1}{4\pi}\parens[\bigg]{-\frac{3}{r_1}+\frac{3}{r_2} -\frac{1}{r_3}}$.
\end{enumerate}
\end{varwidth}
\end{equation}
Moreover, in the first case, $\ker\Hrlztn_B \supseteq \mathcal H_1^h$, where $\mathcal H_1^h$ is the subspace spanned by the functions
\begin{equation}\label{cubeigf1}
\begin{gathered}
\textstyle \psi_{1,1}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_3}} -\frac{1}{\abs{x-x_4}}+\frac{1}{\abs{x-x_5}} -\frac{1}{\abs{x-x_6}}},\\
\textstyle \psi_{2,1}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_1}} -\frac{1}{\abs{x-x_3}}-\frac{1}{\abs{x-x_5}} +\frac{1}{\abs{x-x_7}}},\\
\textstyle \psi_{3,1}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_2}} -\frac{1}{\abs{x-x_3}}-\frac{1}{\abs{x-x_5}} +\frac{1}{\abs{x-x_8}}}.
\end{gathered}
\end{equation}
In the second case, $\ker\Hrlztn_B \supseteq \mathcal H_2^h$, where $\mathcal H_2^h$ is the subspace spanned by the functions
\begin{equation}\label{cubeigf2}
\begin{gathered}
\textstyle \psi_{1,2}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_3}} +\frac{1}{\abs{x-x_4}}-\frac{1}{\abs{x-x_5}} -\frac{1}{\abs{x-x_6}}},\\
\textstyle \psi_{2,2}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_1}} -\frac{1}{\abs{x-x_3}}+\frac{1}{\abs{x-x_5}} -\frac{1}{\abs{x-x_7}}},\\
\textstyle \psi_{3,2}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_2}} +\frac{1}{\abs{x-x_3}}-\frac{1}{\abs{x-x_5}} -\frac{1}{\abs{x-x_8}}}.
\end{gathered}
\end{equation}
In the third case, $\ker\Hrlztn_B \supseteq \mathcal H_3^h$, where $\mathcal H_3^h$ is the subspace spanned by the function
\begin{equation}\label{cubeigf3}
\textstyle
\psi_{1,3}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_1}} -\frac{1}{\abs{x-x_2}}+\frac{1}{\abs{x-x_3}} -\frac{1}{\abs{x-x_4}}-\frac{1}{\abs{x-x_5}} +\frac{1}{\abs{x-x_6}}-\frac{1}{\abs{x-x_7}} +\frac{1}{\abs{x-x_8}}}.
\end{equation}
Besides, $\ker\Hrlztn_B = \mathcal H_k^h$ if and only if the $k$th condition in~\eqref{condcube} holds, and the other two fail. In this case, $\dim(\ker \Hrlztn_B) = 3$ for $k=1$, $k=2$, and $\dim(\ker \Hrlztn_B) = 1$ for $k= 3$.
\end{theo}

\begin{rema}
The conditions~\eqrefcondcubei, \eqrefcondcubeii, \eqrefcondcubeiii\ can hold either in pairs (in any combinations) or all three at once. However, in the case of a scalar matrix $B = a I$ only one of the above conditions is feasible.
\end{rema}

The full list of all possible dimensions of $\ker\Hrlztn_B$ is derived from Theorem~\ref{th:geksaedr}.

\begin{coro}\label{corgeks}
Under the conditions of Theorem~\ref{th:geksaedr} the following description of\/ $\dim(\ker \Hrlztn_B)$ holds:
\[
\dim(\ker \Hrlztn_B) =
\begin{cases}
1 &amp; \text{if\/ \eqrefcondcubeiii\ holds, but both~\eqrefcondcubei\ and~\eqrefcondcubeii\ fail};				\\
3 &amp; \text{if precisely one of\/ \eqrefcondcubei\ and~\eqrefcondcubeii\ holds, but~\eqrefcondcubeiii\ fails};	\\
4 &amp; \text{if\/ \eqrefcondcubeiii\ and precisely one of\/ \eqrefcondcubei\ and~\eqrefcondcubeii\ hold};			\\
6 &amp; \text{if both~\eqrefcondcubei\ and~\eqrefcondcubeii\ hold, but~\eqrefcondcubeiii\ fails};				\\
7 &amp; \text{if all three conditions~\eqrefcondcubei, \eqrefcondcubeii\ and~\eqrefcondcubeiii\ hold}.
\end{cases}
\]
Meanwhile, in the case of a scalar matrix $B = aI$ ($b = c = d = 0$) the dimension of\/ $\ker\Hrlztn_B$ takes only two values:
\[
\dim(\ker \Hrlztn_B) =
\begin{cases}
1 &amp; \text{if\/ \eqrefcondcubeiii\ holds};						\\
3 &amp; \text{if\/ \eqrefcondcubei\ or~\eqrefcondcubeii\ holds}.
\end{cases}
\]
\end{coro}

The following theorem explicitly describes all realizations $\Hrlztn_B$ with maximum value of $\dim\parens{\ker\Hrlztn_B}$.

\begin{theo}\label{maxcube}
Let $\Hrlztn_B$ be an invariant realization with a matrix $B$ of the form~\eqref{matrcube}. Let also $E_8 (\in {\mathbb R}^{8\times 8})$ be a matrix with all entries equal to $1$. Then the zero eigenvalue of the operator $\Hrlztn_B$ is of maximal multiplicity
\begin{equation}\label{maxeigcube}
\dim\parens{\ker\Hrlztn_B} = 7
\end{equation}
precisely when any of the following equivalent conditions holds:
\begin{itemize}
\item $b = a + \frac{1}{4\pi r_1}$, $c = a + \frac{1}{4\pi r_2}$, $d = a + \frac{1}{4\pi r_3}$;
\item $B = B(a) \coloneqq aE_m + M(0)$, $a\in {\mathbb C}$.
\end{itemize}
Moreover, the following statements hold true:
\begin{enumerate}\romanenumi
\item $\Hrlztn_{B(a)}= \Hrlztn_{B(a)}^*$ precisely when $a\in {\mathbb R}$;
\item $\Hrlztn_{B(a)}\ge 0$ precisely when $a\ge 0$;
\item in particular, for $a=0$ the operator\/ $\Hrlztn_{B(0)} = \Hrlztn_{M(0)}$ coincides with the Krein (smallest positive) realization $\widehat\Hrlztn_K$ of the operator\/ $\Hrlztn$.
\end{enumerate}
\end{theo}

\begin{coro}
In the case $a&lt;0$ the number of negative eigenvalues (counting multiplicity) of the operator\/ $\Hrlztn_{B(a)}$ equals
\[
\kappa_-(\Hrlztn_{B(a)}) = \kappa_-\parens[\big]{B(a) - M(0)} = \kappa_-(aE_8) = 1.
\]
\end{coro}

Now we specify the rate of decay at infinity of the eigenfunctions~\eqref{cubeigf1}--\eqref{cubeigf3}.

\begin{prop}\label{cubedecay}
For the eigenfunctions~\eqref{cubeigf1}--\eqref{cubeigf3} the following (distinct) estimates hold:
\begin{equation}
\psi_{j,k}(x) = O\parens[\Bigg]{\frac{1}{\abs{x}^{3}}}, \ j\in\{1, 2, 3\},
\ k\in\{1, 2\}, \qquad
\psi_{1,3}(x) = O\parens[\Bigg]{\frac{1}{\abs{x}^{4}}}, \qquad \abs{x}\rightarrow+\infty.
\end{equation}
\end{prop}

\section{Regular octahedron}

Let $X$ be a vertex set of a regular octahedron. According to Proposition~\ref{bij}, each realization $\Hrlztn_B$ disjoint with the operator $\Hrlztn_0 =-\Delta$ is the restriction of the operator $\Hrlztn^*$ given by~\eqref{eq3}--\eqref{eq3'} to the domain $\dom\Hrlztn_B = \ker\parens{\Gamma_1 - B\Gamma_0}$, where $\Gamma_0$, $\Gamma_1$ are given by the equalities~\eqref{g0}--\eqref{g1}, and $B\in{\mathbb C}^{6\times 6}$.

Similarly to the cases of a tetrahedron and cube, which were considered in~\cite{MalMar21}, we show that in the case of an octahedron the extension $\Hrlztn_B$ is invariant (under the group $\mathcal{S}_{X}$) if and only if $B$ is of the following form
\begin{equation}\label{matroct}
B =
\begin{pmatrix}
a &amp; b &amp; c &amp; b &amp; b &amp; b\\
b &amp; a &amp; b &amp; c &amp; b &amp; b\\
c &amp; b &amp; a &amp; b &amp; b &amp; b\\
b &amp; c &amp; b &amp; a &amp; b &amp; b\\
b &amp; b &amp; b &amp; b &amp; a &amp; c\\
b &amp; b &amp; b &amp; b &amp; c &amp; a
\end{pmatrix}\in{\mathbb C}^{6\times 6}.
\end{equation}

Moreover, it is also shown that the Weyl function corresponding to the triple~\eqref{g0}--\eqref{g1} in this case has the following form:
\begin{equation}
M(z) = \frac{1}{4\pi}
{\begin{pmatrix}
i\sqrt{z} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1}\\
\frac{e^{ir_1\sqrt{z}}}{r_1} &amp; i\sqrt{z} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2}&amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1}\\
\frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; i\sqrt{z} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1}&amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1}\\
\frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1}&amp; i\sqrt{z} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1}\\
\frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; i\sqrt{z} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2}\\
\frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_1\sqrt{z}}}{r_1} &amp; \frac{e^{ir_2\sqrt{z}}}{r_2} &amp; i\sqrt{z}
\end{pmatrix}}.
\end{equation}
Here $r_1$ and $r_2$ are distinct distances between the octahedron vertices ($r_1&lt;r_2$). Hence, for each $z\in \mathbb{C}_{\pm}$ the matrix $M(z)$ is of the form~\eqref{matroct} with certain $a$, $b$, $c$.

\begin{theo}\label{th:octaedr}
An invariant realization\/ $\Hrlztn_B$ with a boundary matrix $B$ of the form~\eqref{matroct} has a non-trivial kernel precisely when at least one of the following conditions holds:
\begin{equation}\label{condoct}
\begin{varwidth}{.9\displaywidth}
\begin{enumerate}\romanenumi
\item $a - c = -\frac{1}{4\pi r_2}$;
\item $a - 2b + c = \frac{1}{4\pi}\parens[\bigg]{-\frac{2}{r_1}+\frac{1}{r_2}}$. 
\end{enumerate}
\end{varwidth}
\end{equation}
Moreover, in the first case, $\ker\Hrlztn_B \supseteq \mathcal H_1^o$, where $\mathcal H_1^o$ is the subspace spanned by the functions
\begin{equation}\label{octeigf1}
\begin{aligned}
\textstyle \psi_{1,1}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_1}} -\frac{1}{\abs{x-x_3}}},\\
\textstyle \psi_{2,1}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_2}} -\frac{1}{\abs{x-x_4}}},\\
\textstyle \psi_{3,1}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_5}} -\frac{1}{\abs{x-x_6}}}.
\end{aligned}
\end{equation}
In the second case, $\ker\Hrlztn_B \supseteq \mathcal H_2^o$, where $\mathcal H_2^o$ is the subspace spanned by the functions
\begin{equation}\label{octeigf2}
\begin{aligned}
\textstyle \psi_{1,2}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_1}} -\frac{1}{\abs{x-x_2}}+\frac{1}{\abs{x-x_3}} -\frac{1}{\abs{x-x_4}}},\\
\textstyle \psi_{2,2}(x) = \frac{1}{4\pi}\parens[\bigg]{\frac{1}{\abs{x-x_1}} +\frac{1}{\abs{x-x_3}}-\frac{1}{\abs{x-x_5}} -\frac{1}{\abs{x-x_6}}}.
\end{aligned}
\end{equation}
Besides, $\ker\Hrlztn_B = \mathcal H_k^o$ if and only if the $k$th condition in~\eqref{condoct} holds, and the other fails. In this case, $\dim(\ker \Hrlztn_B) = 3$ for $k=1$, and $\dim(\ker \Hrlztn_B) = 2$ for $k=2$.
\end{theo}

\begin{rema}
The conditions~\eqrefcondocti\ and \eqrefcondoctii\ can hold simultaneously. However, in the case of a scalar matrix $B = a I$ only one of the above conditions is feasible.
\end{rema}

The following description of all possible dimensions of $\ker\Hrlztn_B$ is easily derived from Theorem~\ref{th:octaedr}.

\begin{coro}\label{coroct}
Under the conditions of Theorem~\ref{th:octaedr} the dimension of\/ $\ker \Hrlztn_B$ takes the following values:
\[
\dim(\ker \Hrlztn_B) =
\begin{cases}
2 &amp; \text{if\/ \eqrefcondoctii\ holds, but~\eqrefcondocti\ fails},\\
3 &amp; \text{if\/ \eqrefcondocti\ holds, but~\eqrefcondoctii\ fails},\\
5 &amp; \text{if both~\eqrefcondocti\ and~\eqrefcondoctii\ hold}.
\end{cases}
\]
Meanwhile, in the case of a scalar matrix $B = aI$ ($b = c = 0$) the dimension $\ker\Hrlztn_B$ takes only two values:
\[
\dim(\ker \Hrlztn_B) =
\begin{cases}
2 &amp; \text{if\/ \eqrefcondoctii\ holds};\\
3 &amp; \text{if\/ \eqrefcondocti\ holds}.
\end{cases}
\]
\end{coro}

The following theorem explicitly describes all realizations $\Hrlztn_B$ with maximum value of $\dim\parens{\ker\Hrlztn_B}$.

\begin{theo}\label{maxoct}
Let $\Hrlztn_B$ be an invariant realization with a matrix $B$ of the form~\eqref{matroct}. Let also $E_6 (\in {\mathbb R}^{6\times 6})$ be a matrix with all entries equal to $1$. Then the zero eigenvalue of the operator\/ $\Hrlztn_B$ is of maximum multiplicity
\begin{equation}\label{maxeigoct}
\dim\parens{\ker\Hrlztn_B} = 5
\end{equation}
if and only if any of the following equivalent conditions hold:
\begin{itemize}
\item $b = a + \frac{1}{4\pi r_1}$, $c = a + \frac{1}{4\pi r_2}$;
\item $B = B(a) \coloneqq aE_6 + M(0)$, $a\in {\mathbb C}$.
\end{itemize}
Moreover, the following statements hold true:
\begin{enumerate}\romanenumi
\item $\Hrlztn_{B(a)}= \Hrlztn_{B(a)}^*$ precisely when $a\in {\mathbb R}$;
\item $\Hrlztn_{B(a)}\ge 0$ precisely when $a\ge 0$;
\item in particular, for $a=0$ the operator\/ $\Hrlztn_{B(0)} = \Hrlztn_{M(0)}$ coincides with the Krein (smallest positive) realization\/ $\widehat\Hrlztn_K$ of the operator\/ $\Hrlztn$.
\end{enumerate}
\end{theo}

\begin{coro}
In the case $a&lt;0$ the number of negative eigenvalues (counting multiplicity) of the operator\/ $\Hrlztn_{B(a)}$ is given by
\[
\kappa_-(\Hrlztn_{B(a)}) = \kappa_-\parens[\big]{B(a)-M(0)} = \kappa_-(aE_6) = 1.
\]
\end{coro}

Now we specify the rate of decay at infinity of the eigenfunctions~\eqref{octeigf1}, \eqref{octeigf2}.

\begin{prop}
For the eigenfunctions~\eqref{octeigf1}, \eqref{octeigf2} the following (distinct) estimates hold:
\begin{equation}
\psi_{j,1}(x) = O\parens[\Bigg]{\frac{1}{\abs{x}^{2}}}, \ j\in\{1, 2, 3\}, \qquad
\psi_{k,2}(x) = O\parens[\Bigg]{\frac{1}{\abs{x}^{3}}}, \ k\in\{1, 2\}, \qquad \abs{x}\rightarrow+\infty.
\end{equation}
\end{prop}

\section{Concluding remarks}\label{final}

We emphasize that the entries of the matrices $B$ in Theorems~\ref{th:tetraedr}, \ref{th:geksaedr}, and~\ref{th:octaedr} might take complex values. Therefore, the matrices $B$ and, thus, the realizations induced by them for all three considered cases with the property $\ker \Hrlztn_B \not =\{0\}$ are not necessarily selfadjoint. But for scalar matrices $B = \alpha I$ the condition $\alpha \in \mathbb{R}$, i.e.\ $B = B^*$, follows from the property $\ker \Hrlztn_B \not =\{0\}$.

It is interesting to note that for regular polyhedra, in contrast to the case of a regular $2l$-gon, $\dim\parens{\ker\Hrlztn_B}$ miss some values from $1$ to $m-2$. Thus, it does not take the values $1$ and $2$ in the case of a tetrahedron, $2$ and $5$ in the case of a cube, and $1$ and $4$ in the case of an octahedron.

We also emphasize that the maximal multiplicity of zero eigenvalue of local interactions parameterized by scalar matrices $B$ does not exceed $3$, while the multiplicity $3$ is attained for every polyhedron. This follows from Theorem~\ref{th:tetraedr}, Corollary~\ref{corgeks}, and Corollary~\ref{coroct}.

In this paper we have considered three of the five regular polyhedra in ${\mathbb R}^3$. The remaining two (icosahedron and dodecahedron) seem to be more complicated due to certain computational difficulties. Nevertheless, we hope to consider them in one of the upcoming publications.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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