najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2023-14-5-505-510najo-286Research ArticleMATHEMATICSМАТЕМАТИКАOn the spectrum of the two-particle Schrödinger operator with point potential: one dimensional caseО спектре двухчастичного оператора Шредингера с точечным потенциалом в одномерном случаеКулжановУ. Н.KuljanovU. N.

Utkir Nematovich Kuljanov

140104, University boulvare. 15, Samarkand

Professorlar street 51, Samarkand

uquljonov@bk.ruSamarkand State University; Samarkand branch of Tashkent State University of EconomicsUzbekistan202308062025145505510Copyright © Kuljanov U.N., 20252025Кулжанов У.Н.Kuljanov U.N.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/286

In the paper, a one-dimensional two-particle quantum system interacted by two identical point interactions is considered. The corresponding Schrödinger operator (energy operator) h" depending on " is constructed as a self-adjoint extension of the symmetric Laplace operator. The main results of the work are based on the study of the operator h". First, the essential spectrum is described. The existence of unique negative eigenvalue of the Schr¨odinger operator is proved. Further, this eigenvalue and the corresponding eigenfunction are found.

В работе рассматривается точечно-взаимодействующая одномерная двухчастичная квантовая система. Соответствующий оператор Шредингера (оператор энергии)  $h_\varepsilon$, зависящего от параметра расширения  строится как самосопряженное расширение $\varepsilon$ симметрического оператора Лапласа. Основные результаты работы основываются на изучение спектра оператора $h_\varepsilon$. Описан существенный спектр и доказано существование одного отрицательного собственного значения оператора Шредингера при положительных значениях параметра расширения. Более того, найдены отрицательные собственные значение и соответствующая собственная функция.

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