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JOURNALS // Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia // Archive

Dokl. RAN. Math. Inf. Proc. Upr., 2024 Volume 516, Pages 31–37 (Mi danma510)

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. Taimanov
Received: 25.01.2024
Revised: 14.02.2024
Accepted: 16.02.2024

DOI: 10.31857/S2686954324020061


 English version:
Doklady Mathematics, 2024, 109:2, 125–129

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© Steklov Math. Inst. of RAS, 2025