MATHEMATICS
On kernels of invariant Schrödinger operators with point interactions. Grinevich–Novikov conjecture
M. M. Malamuda,
V. V. Marchenkob a Peoples' Friendship University of Russia named after Patrice Lumumba, Moscow, Russia
b Bauman Moscow State Technical University, Moscow, Russia
Abstract:
According to Berezin and Faddeev, a Schrödinger operator with point interactions
$$
-\Delta+\sum\limits_{j=1}^m\alpha_j\delta(x-x_j), \, X=\{x_j\}_1^m\subset\mathbb R^3, \, \{\alpha_j\}_1^m\subset\mathbb R,
$$
is any self-adjoint extension of the restriction
$-\Delta_X$ of the Laplace operator
$-\Delta$ to the subset $\{f\in H^2(\mathbb R^3): f(x_j)=0,1\leq j\leq m\}$ 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 (
$B\neq \alpha I$) the estimate $\operatorname{dim}(\operatorname{ker} H_B)\leq m-1$ is proved, and all invariant realizations of the maximal dimension
$\operatorname{dim}(\operatorname{ker} H_B)=m-1$ are described. One of them is the Krein realization, which is the minimal positive extension of the operator
$-\Delta_X$.
Keywords:
Schrödinger operators with point interactions, invariant operators, Krein realization, multiplicity of zero eigenvalue.
UDC:
517.98 Presented: I. A. TaimanovReceived: 25.01.2024
Revised: 14.02.2024
Accepted: 16.02.2024
DOI:
10.31857/S2686954324020061