najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2015-6-2-280-293najo-941Research ArticleREGULAR PAPERSREGULAR PAPERSUniversality of the discrete spectrum asymptotics of the three-particle Schrödinger operator on a latticeMuminovM. I.

Faculty of Scince

81310 Skudai, Johor Bahru

mmuminov@mail.ruRasulovT. H.

Faculty of Physics and Mathematics

M. Ikbol str. 11, 200100 Bukhara

rth@mail.ruUniversiti Teknologi Malaysia (UTM)MalaysiaBukhara State UniversityUzbekistan20151408202562280293Copyright © Muminov M.I., Rasulov T.H., 20252025Muminov M.I., Rasulov T.H.Muminov M.I., Rasulov T.H.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/941

In the present paper, we consider the Hamiltonian H(K), K ∈ T3 := (−π; π]3 of a system of three arbitrary quantum mechanical particles moving on the three-dimensional lattice and interacting via zero range potentials. We find a finite set Λ ⊂ T3 such that for all values of the total quasi-momentum K ∈ Λ the operator H(K) has infinitely many negative eigenvalues accumulating at zero. It is found that for every K ∈ Λ, the number N(K; z) of eigenvalues of H(K) lying on the left of z, z < 0, satisfies the asymptotic relation lim z→−0 N(K; z)| log |z||−1 = U0 with 0 < U0 < ∞, independently on the cardinality of Λ.

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