najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2025-16-5-577-585najo-1528Research ArticleMATHEMATICSМАТЕМАТИКАSpectral analysis of two-particle Hamiltonians with short-range interactionsСпектральный анализ двухчастичных гамильтонианов с короткодействующими взаимодействиямиhttps://orcid.org/0009-0000-9082-5986АхмадоваМ. О.AkhmadovaM. O.

Ахмадова Мухайё Олимжон кизи

 

Mukhayyo O. Akhmadova

140104, Samarkand

amukhayyo.akhmadova@mail.ruАзизоваМ. А.AzizovaM. A.

Азизова Мукаммал Амриддин кизи

Mukammal A. Azizova

140104, Samarkand

bmukammal.azizova@gmail.comSamarkand State UniversityUzbekistan202505112025165577585Copyright © Akhmadova M.O., Azizova M.A., 20252025Ахмадова М.О., Азизова М.А.Akhmadova M.O., Azizova M.A.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/1528

We analyze the spectral characteristics of lattice Schrodinger operators, denoted as Hγλµ(K), K ∈ (−π, π] 3 , which represent a system of two identical bosons existing on Z 3 lattice. The model includes onsite and nearest-neighbor interactions, parameterized by γ, λ, µ ∈ R. Our study of Hγλµ(0) reveals an invariant subspace on which its restricted form, Hea λµ(0), is solely dependent on λ and µ. To elucidate the mechanisms of eigenvalue birth and annihilation for Hea λµ(0), we define a critical operator. A detailed criterion is subsequently developed within the plane spanned by λ and µ. This involves: (i) the derivation of smooth critical curves that mark the onset of criticality for the operator, and (ii) the proof of exact conditions for the existence of precisely α eigenvalues below and β eigenvalues above the essential spectrum, where α, β ∈ {0, 1, 2} and α + β ≤ 2.

Мы анализируем спектральные характеристики решеточных операторов Шрёдингера Hγλμ(K), K∈(−π,π]3, которые представляют систему двух идентичных бозонов, расположенных на решетке Z3. Модель включает в себя взаимодействия на узлах решетки и взаимодействия ближайших соседей, параметризованные γ,λ,μ∈R. Наше исследование оператора Hγλμ(0) выявляет инвариантное подпространство, на котором его ограниченная форма Heaλμ(0) зависит исключительно от λ и μ. Чтобы прояснить механизмы рождения и уничтожения собственных значений для Heaλμ(0), мы определяем критический оператор. Впоследствии на плоскости, натянутой на λ и μ, разрабатывается подробный критерий, который включает: (i) вывод гладких критических кривых, которые отмечают наступление критичности для оператора, и (ii) доказательство точных условий существования именно α собственных значений ниже и β собственных значений выше существенного спектра, где α,β∈{0,1,2} и α+β≤2.

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