najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2023-14-3-295-303najo-301Research ArticleMATHEMATICSМАТЕМАТИКАOn eigenvalues and virtual levels of a two-particle Hamiltonian on a d-dimensional latticeО собственных значениях и виртуальных уровнях двухчастичного гамильтониана на $d$-мерной решеткеhttps://orcid.org/0000-0003-0335-6558МуминовМ. Э.MuminovM. I.

Mukhiddin I. Muminov

University blv., 15, Samarkand, 140104

Tashkent, 100174

mmuminov@mail.ruhttps://orcid.org/0000-0001-8565-3702ХуррамовА. М.KhurramovA. M.

Abdimajid M. Khurramov

University blv., 15, Samarkand, 140104

xurramov@mail.ruhttps://orcid.org/0000-0002-0502-8163БозоровИ. Н.BozorovI. N.

Islom N. Bozorov

University blv., 15, Samarkand, 140104

Tashkent, 100174

islomnb@mail.ruSamarkand State University; V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of SciencesUzbekistanSamarkand State UniversityUzbekistan202330062025143295303Copyright © Muminov M.I., Khurramov A.M., Bozorov I.N., 20252025Муминов М.Э., Хуррамов А.М., Бозоров И.Н.Muminov M.I., Khurramov A.M., Bozorov I.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/301

The two-particle Schrödinger operator h_(k); k 2 Td (where _ > 0, Td is a d-dimensional torus), associated to the Hamiltonian h of the system of two quantum particles moving on a d-dimensional lattice, is considered as a perturbation of free Hamiltonian h0(k) by the certain 3d rank potential operator _v. The existence conditions of eigenvalues and virtual levels of h_(k); are investigated in detail with respect to the particle interaction _ and total quasi-momentum k 2 Td.

Двухчастичный оператор Шредингера $h_\mu(k),$ $k\in\mathbb{T}^d$ (где $\mu>0$, $\mathbb{T}^d$ -- $d$-мерный тор), ассоциированный с гамильтонианом $\mathrm{h}$ системы двух квантовых частиц, движущихся на $d$-мерной решетке, рассматривается как возмущение свободного гамильтониана $h_0(k)$ потенциальным оператором $\mu{\bf v}$ ранга $3^d$. Условия существования собственных значений и виртуальных уровней оператора $h_\mu(k),$ подробно исследованы относительно энергии взаимодействия частиц $\mu$ и полный квазиимпульс системы двух частиц $k\in\mathbb{T}^d$.

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