On zero eigenvalue of invariant Schrödinger operators with point interactions at vertices of some regular polyhedra

Mark M. Malamud; Vladimir V. Marchenko

doi:10.5802/crmath.719

Article de recherche - Analyse fonctionnelle

On zero eigenvalue of invariant Schrödinger operators with point interactions at vertices of some regular polyhedra
[Sur la valeur propre nulle des opérateurs de Schrödinger invariants avec interactions ponctuelles aux sommets de certains polyèdres réguliers]

Mark M. Malamud ¹ ; Vladimir V. Marchenko ^2,³

¹ Saint Petersburg State University, Saint Petersburg, Russia
² Bauman Moscow State Technical University, Moscow, Russia
³ Institute of Applied Mathematics and Mechanics, Donetsk, Russia

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 629-639.

Abstract (VO)
Résumé

Following Berezin and Faddeev, by a Schrödinger operator with point interactions:

- Δ + \sum_{j = 1}^{m} α_{j} δ (x - x_{j}), X = {x_{j}}_{1}^{m} \subset R^{3}, {α_{j}}_{1}^{m} \subset R,

one means any selfadjoint extension of the restriction $Δ_{X}$ of the Laplace operator $- Δ$ to the subset ${f \in H^{2} (R^{3}) : f (x_{j}) = 0, 1 \leq j \leq m}$ of the Sobolev space $H^{2} (R^{3})$ .

In the present paper the above set of interactions $X$ is assumed to be a vertex set of a certain regular polyhedron, and selfadjoint extensions (realizations) invariant under the symmetry group of $X = {x_{j}}_{1}^{m}$ are studied.

Such realizations $H_{B}$ are parametrized by special matrices $B = B^{*} \in C^{m \times m}$ . We describe all such selfadjoint realizations with non-trivial kernels. By this we continue investigation by Grinevich–Novikov and ours relating to regular polygons. Besides, for arbitrary realizations the estimate $\dim (\ker H_{B}) \leq m - 1$ is proved, and realizations with all feasible $\dim (\ker H_{B})$ are described.

Particular attention is paid to realizations with maximum value $\dim (\ker H_{B}) = m - 1$ . One of them is the Krein realization, which is the minimal positive selfadjoint extension of the operator $Δ_{X} \geq 0$ .

D’après Berezin et Faddeev, par un opérateur de Schrödinger avec des interactions ponctuelles :

- Δ + \sum_{j = 1}^{m} α_{j} δ (x - x_{j}), X = {x_{j}}_{1}^{m} \subset R^{3}, {α_{j}}_{1}^{m} \subset R,

on entend toute extension autoadjointe de la restriction $Δ_{X}$ de l’opérateur de Laplace $- Δ$ au sous-ensemble ${f \in H^{2} (R^{3}) : f (x_{j}) = 0, 1 \leq j \leq m}$ de l’espace de Sobolev $H^{2} (R^{3})$ .

Dans le présent article, l’ensemble d’interactions $X$ ci-dessus est supposé être un ensemble de sommets d’un certain polyèdre régulier, et des extensions (réalisations) autoadjointes invariantes sous le groupe de symétrie de $X = {x_{j}}_{1}^{m}$ sont étudiées.

De telles réalisations $H_{B}$ sont paramétrées par des matrices spéciales $B = B^{*} \in C^{m \times m}$ . Nous décrivons toutes ces réalisations autoadjointes avec des noyaux non triviaux. Nous poursuivons ainsi l’investigation de Grinevich–Novikov et la nôtre concernant les polygones réguliers. De plus, pour des réalisations arbitraires, l’estimation $\dim (\ker H_{B}) \leq m - 1$ est prouvée, et des réalisations avec tous les $\dim (\ker H_{B})$ réalisables sont décrites.

Une attention particulière est accordée aux réalisations avec la valeur maximale $\dim (\ker H_{B}) = m - 1$ . L’une d’elles est la réalisation de Krein, qui est l’extension autoadjointe positive minimale de l’opérateur $Δ_{X} \geq 0$ .

Reçu le : 2024-11-12
Révisé le : 2025-01-08
Accepté le : 2025-01-16
Publié le : 2025-06-23

DOI : 10.5802/crmath.719

Classification : 47A57
Keywords: Schrödinger operators with point interactions, invariant operators, Krein realization, multiplicity of zero eigenvalue
Mots-clés : Opérateurs de Schrödinger avec interactions ponctuelles, opérateurs invariants, réalisation de Krein, multiplicité de valeur propre nulle

Affiliations des auteurs :

Mark M. Malamud ¹ ; Vladimir V. Marchenko ^{2, 3}

¹ Saint Petersburg State University, Saint Petersburg, Russia
² Bauman Moscow State Technical University, Moscow, Russia
³ Institute of Applied Mathematics and Mechanics, Donetsk, Russia

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Mark M. Malamud and Vladimir V. Marchenko},
     title = {On zero eigenvalue of invariant {Schr\"odinger} operators with point interactions at vertices of some regular polyhedra},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {629--639},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.719},
     language = {en},
}

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Mark M. Malamud; Vladimir V. Marchenko. On zero eigenvalue of invariant Schrödinger operators with point interactions at vertices of some regular polyhedra. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 629-639. doi : 10.5802/crmath.719. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.719/

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