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

Abstract

According to Berezin and Faddeev, a Schrödinger operator with point interactions –Δ + \(\sum\limits_{j = 1}^m {{\alpha }_{j}}\delta (x - {{x}_{j}}),X = \{ {{x}_{j}}\} _{1}^{m} \subset {{\mathbb{R}}^{3}},\{ {{\alpha }_{j}}\} _{1}^{m} \subset \mathbb{R},\) is any self-adjoint extension of the restriction \({{\Delta }_{X}}\) of the Laplace operator \( - \Delta \) to the subset \(\{ f \in {{H}^{2}}({{\mathbb{R}}^{3}}):f({{x}_{j}}) = 0,\;1 \leqslant j \leqslant m\} \) 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 \((B \ne \alpha I)\) the estimate \(\dim (\ker {{{\mathbf{H}}}_{B}}) \leqslant m - 1\) is proved, and all invariant realizations of the maximal dimension \(\dim (\ker {{{\mathbf{H}}}_{B}}) = m - 1\) are described. One of them is the Krein realization, which is the minimal positive extension of the operator \({{\Delta }_{X}}\).