najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2025-16-6-737-748najo-1612Research ArticleMATHEMATICSМАТЕМАТИКАOn the existence of the maximum number of isolated eigenvalues for a lattice Schrödinger operatorО существовании максимального числа изолированных собственных значений для решёточного оператора Шрёдингераhttps://orcid.org/0000-0003-4951-9340ЛакаевС. Н.LakaevS. N.

Saidakhmat N. Lakaev 

140104, Samarkand

s.lakaev@mail.ruhttps://orcid.org/0009-0005-0159-8002ЛатиповаД. А.LatipovaD. A.

Dildora A. Latipova

140104, Samarkand

ms.dlatipova@mail.ruhttps://orcid.org/0009-0000-9082-5986АхмадоваМ. О.AkhmadovaM. O.

Mukhayyo O. Akhmadova

140104, Samarkand

mukhayyo.akhmadova@mail.ruSamarkand State Pedagogical InstituteUzbekistanSamarkand State UniversityUzbekistan202506012026166737748Copyright © Lakaev S.N., Latipova D.A., Akhmadova M.O., 20262026Лакаев С.Н., Латипова Д.А., Ахмадова М.О.Lakaev S.N., Latipova D.A., Akhmadova M.O.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/1612

This paper presents a detailed spectral analysis of the discrete Schrodinger operator¨ Hγλµ(K), which describes a system of two identical bosons on a two-dimensional lattice, Z2. The operator’s family is parameterized by the quasi-momentum K ∈ T2 and real interaction strengths: γ for on-site, λ for nearest-neighbor, and µ for next-nearest-neighbor interactions. A key finding of our study is that, under specific conditions on the interaction parameters, the operator Hγλµ(K) consistently possesses a total of seven eigenvalues that lie either below the bottom or above the top of its essential spectrum, over all K ∈ T2.

В данной работе представлен подробный спектральный анализ дискретного оператора Шрёдингера Hγλµ(K), который описывает систему двух одинаковых бозонов на двумерной решётке Z2. Семейство операторов параметризовано квазиимпульсом K ∈ T2 и вещественными константами взаимодействия: γ (для взаимодействия на узле), λ (для взаимодействия с ближайшими соседями) и µ (для взаимодействия со следующими ближайшими соседями). Ключевым результатом нашего исследования является то, что при определённых условиях на параметры взаимодействия (γ, λ, µ) оператор Hγλµ(K) для всех K ∈ T2 всегда имеет ровно семь собственных значений, лежащих либо ниже нижней границы, либо выше верхней границы его существенного спектра.

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