On Kernels of Invariant Schrödinger Operators with Point Interactions. Grinevich–Novikov Conjecture

According to Berezin and Faddeev, a Schrödinger operator with point interactions –Δ + is any self-adjoint extension of the restriction of the Laplace operator to the subset of the Sobolev space. The present paper studies the extensions (realizations) invariant under the symmetry group of the vertex set of a regular m-gon. Such realizations HB are parametrized by special circulant matrices. We describe all such realizations with non-trivial kernels. А Grinevich–Novikov conjecture on simplicity of the zero eigenvalue of the realization HB with a scalar matrix and an even m is proved. It is shown that for an odd m non-trivial kernels of all realizations HB with scalar are two-dimensional. Besides, for arbitrary realizations the estimate is proved, and all invariant realizations of the maximal dimension are described. One of them is the Krein realization, which is the minimal positive extension of the operator. © Pleiades Publishing, Ltd. 2024.