О существовании максимального числа изолированных собственных значений для решёточного оператора Шрёдингера
https://doi.org/10.17586/2220-8054-2025-16-6-737-748
Аннотация
В данной работе представлен подробный спектральный анализ дискретного оператора Шрёдингера $H_{\gamma\lambda\mu}(K)$, который описывает систему двух одинаковых бозонов на двумерной решётке $\mathbb{Z}^2$. Семейство операторов параметризовано квазиимпульсом $K \in \mathbb{T}^2$ и вещественными константами взаимодействия: $\gamma$ (для взаимодействия на узле), $\lambda$ (для взаимодействия с ближайшими соседями) и $\mu$ (для взаимодействия со следующими ближайшими соседями). Ключевым результатом нашего исследования является то, что при определённых условиях на параметры взаимодействия ($\gamma, \lambda, \mu$) оператор $H_{\gamma\lambda\mu}(K)$ для всех $K \in \mathbb{T}^2$ всегда имеет ровно семь собственных значений, лежащих либо ниже нижней границы, либо выше верхней границы его существенного спектра.
Об авторах
С. Н. Лакаев
Samarkand State Pedagogical Institute
Узбекистан
Узбекистан
Д. А. Латипова
Samarkand State University
Узбекистан
Узбекистан
М. О. Ахмадова
Samarkand State University
Узбекистан
Узбекистан
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Рецензия
Для цитирования:
Лакаев С.Н., Латипова Д.А., Ахмадова М.О. О существовании максимального числа изолированных собственных значений для решёточного оператора Шрёдингера. Наносистемы: физика, химия, математика. 2025;16(6):737-748. https://doi.org/10.17586/2220-8054-2025-16-6-737-748
For citation:
Lakaev S.N., Latipova D.A., Akhmadova M.O. On the existence of the maximum number of isolated eigenvalues for a lattice Schrödinger operator. Nanosystems: Physics, Chemistry, Mathematics. 2025;16(6):737-748. https://doi.org/10.17586/2220-8054-2025-16-6-737-748
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