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

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