mathematicsВестник российских университетов. МатематикаRussian Universities Reports. Mathematics2686-96672782-3342Тамбовский государственный университет имени Г.Р. Державинаmathematics-38Research ArticleСТАТЬИARTICLESСпектральные свойства и эффект Ефимова для ограниченных самосопряженных частично интегральных операторовSpectral properties and the Efimov effect for bounded self-adjoint partial integral operatorshttps://orcid.org/https://orcid.org/0000-0002-0728-9340КучаровРамзиддин РузимурадовичKucharovRamziddin Ruzimuradovichramz3364647@yahoo.comhttps://orcid.org/https://orcid.org/0009-0002-1721-880XТухтамуродоваТиллохон МансуржонTuxtamurodovaTillohon Mansurjonmirzayevatilloxon13@gmail.comНациональный университет Узбекистана; Ташкентский государственный технический университет имени Ислама КаримоваРоссияNational University of Uzbekistan; Tashkent State Technical University named after Islam KarimovRussian FederationНациональный университет УзбекистанаРоссияNational University of UzbekistanRussian Federation20261004202631153521Copyright © Кучаров Р.Р., Тухтамуродова Т.М., 20262026Кучаров Р.Р., Тухтамуродова Т.М.Kucharov R.R., Tuxtamurodova T.M.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathematics.elpub.ru/jour/article/view/38

В статье изучается эффект Ефимова для класса ограниченных самосопряженных частичных интегральных операторов с вырожденными ядрами, возникающих в модели Хаббарда. Эти операторы типа Фредгольма действуют в гильбертовом пространстве L2(1×2) и реализуются как некомпактные возмущения оператора умножения. Учитывая спектральное поведение операторов типа Шредингера, мы изучаем возникновение бесконечного числа собственных значений, расположенных ниже существенного спектра. Отличительной особенностью модели является структура ядра, представленная в виде сходящегося ряда, которая существенно влияет на спектр. Приведено подробное описание существенного спектра и доказана теорема, устанавливающая достаточные условия для существования бесконечного дискретного спектра ниже его нижней границы. Подход основан на инструментах спектральной теории, включая принцип минимакса и свойства положительных интегральных операторов. Для подтверждения теоретических результатов построен явный пример, демонстрирующий эффект Ефимова в данной постановке.

This paper investigates the Efimov effect for a class of bounded self-adjoint partial integral operators with degenerate kernels arising in the Hubbard model. These Fredholm-type operators act in the Hilbert space L2(1×2) and are realized as non-compact perturbations of a multiplication operator. Motivated by the spectral behavior of Schrödinger-type operators, we study the emergence of infinitely many eigenvalues located below the essential spectrum. A distinctive aspect of the model is the kernel structure, expressed as a convergent series, which significantly influences the spectrum. We present a detailed description of the essential spectrum and prove a theorem that gives sufficient conditions for the presence of an infinite discrete spectrum beneath its lower bound. The approach relies on tools from spectral theory, including the minimax principle and the properties of positive integral operators. To support the theoretical findings, we also construct an explicit example that demonstrates the Efimov effect in this framework.

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The authors declare that there are no conflicts of interest present.