Universality of the discrete spectrum asymptotics of the three-particle Schrödinger operator on a lattice
https://doi.org/10.17586/2220-8054-2015-6-2-280-293
Abstract
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.
Keywords
Three-particle Schr¨odinger operator,
zero-range pair attractive potentials,
Birman-Schwinger principle,
the Efimov effect,
discrete spectrum asymptotics
Supporting agencies: The authors would like to thank Prof. A. Teta for helpful discussions about the results of the paper. This work was supported by the TOSCA Erasmus Mundus grant. T. H. Rasulov wishes to thank the University of L’Aquila for the invitation and hospitality.
About the Authors
M. I. Muminov
Universiti Teknologi Malaysia (UTM)
Malaysia
Malaysia
Faculty of Scince
81310 Skudai, Johor Bahru
T. H. Rasulov
Bukhara State University
Uzbekistan
Uzbekistan
Faculty of Physics and Mathematics
M. Ikbol str. 11, 200100 Bukhara
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Review
For citations:
Muminov M.I., Rasulov T.H. Universality of the discrete spectrum asymptotics of the three-particle Schrödinger operator on a lattice. Nanosystems: Physics, Chemistry, Mathematics. 2015;6(2):280-293. https://doi.org/10.17586/2220-8054-2015-6-2-280-293
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