najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2023-14-5-518-529najo-288Research ArticleMATHEMATICSМАТЕМАТИКАOn the discrete spectrum of the Schrödinger operator using the 2+1 fermionic trimer on the latticeО дискретном спектре оператора шредингера, соответствующего 2+1 фермионному тримеру на решеткеhttps://orcid.org/0009-0003-0569-6780ХалхужаевA. M.KhalkhuzhaevA. M.

 Халхужаев Ахмад Мияссарович - аспирант Самаркандского государственного университета

г. Самарканд,  Университетский бульвар 15

Ahmad M. Khalkhuzhaev

Mirzo Ulugbek 81, 100170

ahmad_x@mail.ruhttps://orcid.org/0000-0001-5439-8729ХужамиеровИ. А.KhujamiyorovI. A.

Islom A. Khujamiyorov

University Boulevard 15, Samarkand 140104

xujamiyorov1990@mail.ruИнститут Математики имени В.И.Романовского Академии наук Республики Узбекистан, начальник Самаркандского отделенияV. I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of UzbekistanSamarkand State UniversityUzbekistan202308062025145518529Copyright © Khalkhuzhaev A.M., Khujamiyorov I.A., 20252025Халхужаев A.M., Хужамиеров И.А.Khalkhuzhaev A.M., Khujamiyorov I.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/288

We consider the three-particle discrete Schrödinger operator H; (K); K 2 T3, associated with the three-particle Hamiltonian (two of them are fermions with mass 1 and one of them is arbitrary with mass m = 1=  < 1), interacting via pair of repulsive contact potentials > 0 on a three-dimensional lattice Z3. It is proved that there are critical values of mass ratios  = 1 and  = 2 such that if  2 (0; 1), then the operator H; (0) has no eigenvalues. If  2 ( 1; 2), then the operator H; (0) has a unique eigenvalue; if  > 2, then the operator H; (0) has three eigenvalues lying to the right of the essential spectrum for all sufficiently large values of the interaction energy.

Рассматривается трехчастичный дискретный оператор Шредингера 〖 H〗_(μ,γ)(K), K∈T^3, ассоциированный с гамильтонианом системы трех частиц (две - фермионы с массой 1 и одна - произвольная с массой m=1/γ<1), взаимодействующих с помощью парных отталкивающих контактных потенциалов μ>0 на трехмерной решетке Z^3. Доказано, что существуют критические значения отношений масс γ= γ_1 и γ= γ_2такие, что если  γ∈(0, γ_1), то оператор 〖 H〗_(μ,γ)(0) не имеет собственных значений, если γϵ( γ_1, γ_2), то имеет единственное собственное значение, если  γ>γ_2 то имеет три собственных значения, лежащих правее существенного спектра при всех достаточно больших значениях энергии взаимодействия μ.

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