On the number of the discrete spectrum of two-particle discrete Schröodinger operators

Authors

Zahriddin Muminov, National University of UzbekistanFollow
Utkir Kuljanov, Samarkand State UniversityFollow
Shukhrat Alladustov, National University of UzbekistanFollow

Abstract

We consider a family of discrete Schrö dinger operators h^d(k), where k is the two-particle quasi-momentum varying in 𝕋^d=(−π,π]^d , associated to a system of two-particles on the d - dimensional lattice ℤ^d, d>1. The CwikelLieb-Rozenblum (CLR)-type estimates are written for h^d(k) when the Fermi surface E_k^-1(𝔢_m(k)) of the associated dispersion relation is a one point set at em(k), the bottom of the essential spectrum. Moreover, when the Fermi surface E_k^-1(𝔢_m(k)) is of dimension d−1 or d−2, we obtain the necessary and sufficient conditions for the existence of infinite discrete spectrum of h^d(k) , while in the case dimE_k^-1(𝔢_m(k)) ≤ d-3, the discrete spectrum of h^d(k) is finite.

First Page

33

Last Page

46

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Recommended Citation

Muminov, Zahriddin; Kuljanov, Utkir; and Alladustov, Shukhrat (2020) "On the number of the discrete spectrum of two-particle discrete Schröodinger operators," Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 3: Iss. 1, Article 3.
DOI: https://doi.org/10.56017/2181-1318.1051