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

According to Berezin and Faddeev, a Schrödinger operator with point interactions−△+∑j=1mαjδ(x−xj),X={xj}1m⊂ℝ3,{αj}1m⊂ℝ, is any self-adjoint extension of the restriction−△xof the Laplace operator−△to the subset{f∈H2(R3):f(xj)=0,1≤j≤m}of the Sobolev spaceH2(ℝ3). The present paper studies the extensions (realizations) invariant under the symmetry group of the vertex setX={xj}1m of a regular m-gon. Such realizations HB are parametrized by special circulant matrices B∈ℂm×m. We describe all such realizations with non-trivial kernels. А Grinevich–Novikov conjecture on simplicity of a zero eigenvalue of the realization HB with a scalar matrixB=αI and an evenis proved. It is shown that for an odd m non-trivial kernels of all the realizationswith scalarare two-dimensional. Besides, for arbitrary realizations B≠αIthe estimatedimkerB≤m−1is proved, and all the invariant realizations of the maximal dimensiondimkerB=m−1are described. One of them is the Krein realization, which is the minimal positive extension of the operator−△x.