On Kernels of Invariant Schrodinger Operators with Point Interactions. Grinevich-Novikov Conjecture

According to Berezin and Faddeev, a Schrodinger operator with point interactions -Delta + Sigma(m)(j=1) alpha(j) delta(x-x(j)), X = {x(j)}(1)(m) subset of R-3 , {alpha(j)}(1)(m) subset of R is any self-adjoint extension of the restriction Delta(X) of the Laplace operator -Delta to the subset {f is an element of H-2 (R-3) : f(x(j)) = 0, 1 <= j <= m} of the Sobolev space H-2 (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 is an element of C-mxm. We describe all such realizations H(B)with non-trivial kernels. A 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 are two-dimensional. Besides, for arbitrary realizations (B not equal alpha I) the estimate dim(ker H-B) <= m-1 is proved, and all invariant realizations of the maximal dimension m(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)