On kernels of invariant Schrödinger operators with point interactions. Grinevich–Novikov conjecture

According to Berezin and Faddeev, a Schrödinger operator with point interactions
$$ -\Delta+\sum\limits_{j=1}^m\alpha_j\delta(x-x_j), \, X=\{x_j\}_1^m\subset\mathbb R^3, \, \{\alpha_j\}_1^m\subset\mathbb R, $$
is any self-adjoint extension of the restriction $-\Delta_X$ of the Laplace operator $-\Delta$ to the subset $\{f\in H^2(\mathbb R^3): f(x_j)=0,1\leq j\leq m\}$ of the Sobolev space $H^2(\mathbb R^3)$. The present paper studies the extensions (realizations) invariant under the symmetry group of the vertex set $X=\{x_j\}_1^m$ of a regular $m$-gon. Such realizations $H_B$ are parametrized by special circulant matrices $B\in\mathbb C^{m\times m}$. We describe all such realizations with non-trivial kernels. А Grinevich–Novikov conjecture on simplicity of the zero eigenvalue of the realization $H_B$ with a scalar matrix $B=\alpha I$ and an even $m$ is proved. It is shown that for an odd $m$ non-trivial kernels of all realizations $H_B$ with scalar $B=\alpha I$ are two-dimensional. Besides, for arbitrary realizations ($B\neq \alpha I$) the estimate $\operatorname{dim}(\operatorname{ker} H_B)\leq m-1$ is proved, and all invariant realizations of the maximal dimension $\operatorname{dim}(\operatorname{ker} H_B)=m-1$ are described. One of them is the Krein realization, which is the minimal positive extension of the operator $-\Delta_X$.