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

Research article - Functional analysis

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

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

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

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

Abstract (Original language)
Résumé

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

\[ -\Delta + \sum _{j=1}^m\alpha _j\delta (x - x_j), \quad X = \lbrace x_j\rbrace ^m_1\subset {\mathbb{R}}^3, \quad \lbrace \alpha _j\rbrace ^m_1\subset {\mathbb{R}}, \]

one means any selfadjoint extension of the restriction $\Delta _X$ of the Laplace operator $-\Delta $ to the subset $\bigl \lbrace f \in H^{2}({\mathbb{R}}^3): f(x_j)=0, \ 1\le j\le m \bigr \rbrace $ of the Sobolev space $H^{2}({\mathbb{R}}^3)$.

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 $X = \lbrace x_j\rbrace _1^m$ are studied.

Such realizations $\operatorname{H}_B$ are parametrized by special matrices $B = B^*\in {\mathbb{C}}^{m\times m}$. 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 $\dim (\ker \operatorname{H}_B) \le m-1$ is proved, and realizations with all feasible $\dim (\ker {\mathbb{H}_B})$ are described.

Particular attention is paid to realizations with maximum value $\dim (\ker \operatorname{H}_B) = m-1$. One of them is the Krein realization, which is the minimal positive selfadjoint extension of the operator $\Delta _X \ge 0$.

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

\[ -\Delta + \sum _{j=1}^m\alpha _j\delta (x - x_j), \quad X = \lbrace x_j\rbrace ^m_1\subset {\mathbb{R}}^3, \quad \lbrace \alpha _j\rbrace ^m_1\subset {\mathbb{R}}, \]

on entend toute extension autoadjointe de la restriction $\Delta _X$ de l’opérateur de Laplace $-\Delta $ au sous-ensemble $\bigl \lbrace f \in H^{2}({\mathbb{R}}^3): f(x_j)=0, \ 1\le j\le m \bigr \rbrace $ de l’espace de Sobolev $H^{2}({\mathbb{R}}^3)$.

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 $X = \lbrace x_j\rbrace _1^m$ sont étudiées.

De telles réalisations $\operatorname{H}_B$ sont paramétrées par des matrices spéciales $B = B^*\in {\mathbb{C}}^{m\times m}$. 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 $\dim (\ker \operatorname{H}_B) \le m-1$ est prouvée, et des réalisations avec tous les $\dim (\ker {\mathbb{H}_B})$ réalisables sont décrites.

Une attention particulière est accordée aux réalisations avec la valeur maximale $\dim (\ker \operatorname{H}_B) = m-1$. L’une d’elles est la réalisation de Krein, qui est l’extension autoadjointe positive minimale de l’opérateur $\Delta _X \ge 0$.

Received: 2024-11-12
Revised: 2025-01-08
Accepted: 2025-01-16
Published online: 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

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     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

References
Cited by

[1] Sergio Albeverio; Friedrich Gesztesy; Raphael Høegh-Krohn; Helge Holden Solvable models in quantum mechanics, Texts and Monographs in Physics, Springer, 1988, xiv+452 pages | DOI | MR | Zbl

[2] Feliks Aleksandrovich Berezin; Lyudvig Dmitrievich Faddeev A remark on Schrödinger’s equation with a singular potential, Sov. Math., Dokl., Volume 2 (1961), pp. 372-375 | Zbl

[3] Volodymyr Oleksandrovych Derkach; Mark M. Malamud Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal., Volume 95 (1991) no. 1, pp. 1-95 | DOI | MR | Zbl

[4] Volodymyr Oleksandrovych Derkach; Mark M. Malamud Extension theory of symmetric operators and boundary value problems, Institute of Mathematics, NAS of Ukraine, 2017, 573 pages

[5] Natalʹya I. Goloshchapova; Mark M. Malamud; Viktor P. Zastavnyi Radial positive definite functions and spectral theory of the Schrödinger operators with point interactions, Math. Nachr., Volume 285 (2012) no. 14-15, pp. 1839-1859 | DOI | MR | Zbl

[6] Valentyna Ivanivna Gorbachuk; Myroslav Lvovych Gorbachuk Boundary value problems for operator differential equations, Mathematics and its Applications (Soviet Series), 48, Kluwer Academic Publishers, 1991, xii+347 pages (Translated and revised from the 1984 Russian original) | DOI | MR | Zbl

[7] Petr G. Grinevich; Roman G. Novikov Multipoint scatterers with bound states at zero energy, Theor. Math. Phys., Volume 193 (2017) no. 2, pp. 309-314 | DOI | MR

[8] Petr G. Grinevich; Roman G. Novikov Spectral inequality for Schrödinger’s equation with multipoint potential, Russ. Math. Surv., Volume 77 (2022) no. 6, pp. 69-76 | DOI | MR

[9] Tosio Kato Perturbation theory for linear operators, Grundlehren der Mathematischen Wissenschaften, 132, Springer, 1966, xix+592 pages | MR | Zbl

[10] Mark M. Malamud; Vladimir V. Marchenko Invariant Schrödinger operators with point interactions at the vertices of a regular polyhedron, Math. Notes, Volume 110 (2021) no. 3-4, pp. 463-469 | DOI | MR | Zbl

[11] Mark M. Malamud; Vladimir V. Marchenko On kernels of invariant Schrödinger operators with point interactions. Grinevich-Novikov conjecture, Dokl. Math., Volume 109 (2024) no. 2, pp. 125-129 | MR | DOI | Zbl

[12] Mark M. Malamud; Konrad Schmüdgen Spectral theory of Schrödinger operators with infinitely many point interactions and radial positive definite functions, J. Funct. Anal., Volume 263 (2012) no. 10, pp. 3144-3194 | DOI | MR | Zbl

[13] Michael Reed; Barry Simon Methods of modern mathematical physics. I. Functional analysis, Academic Press Inc., 1972, xvii+325 pages | MR | Zbl

[14] Iskander Asanovich Taĭmanov; S. P. Tsarëv Two-dimensional Schrödinger operators with rapidly decaying rational potential and multidimensional $L_{2}$ -kernel, Russ. Math. Surv., Volume 62 (2007) no. 3, pp. 631-633 | DOI | MR | Zbl

[15] Iskander Asanovich Taĭmanov; S. P. Tsarëv Two-dimensional rational solitons and their blowup via the Moutard transformation, Theor. Math. Phys., Volume 157 (2008) no. 2, pp. 1525-1541 | DOI | Zbl

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