Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics View Full Text


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Article Info

DATE

2004-08

AUTHORS

Sergio Albeverio, Saidakhmat N. Lakaev, Zahriddin I. Muminov

ABSTRACT

. The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \mathbb{Z}^3 $$ \end{document} and interacting via zero-range attractive potentials is considered. For the two-particle energy operator h(k), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ k \in \mathbb{T}^3 = (-\pi, \pi]^3 $$ \end{document} the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of h(k) for k ≠ 0 is proven, provided that h(0) has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ k \in \mathbb{T}^3 $$ \end{document} being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number N(0, z) of eigenvalues of H(0) lying below \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ z < 0 $$ \end{document} the following limit exists\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \lim_{z \to 0-}\, {N(0, z)\over |\log|z\|} = {\mathcal U}_0 $$ \end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ {\mathcal U}_0 > 0. $$ \end{document} Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum K the finiteness of the number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ N(K, \tau_{ess}(K))$$ \end{document} of eigenvalues of H(K) below the essential spectrum is established and the asymptotics for the number N(K, 0) of eigenvalues lying below zero is given. More... »

PAGES

743-772

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00023-004-0181-9

DOI

http://dx.doi.org/10.1007/s00023-004-0181-9

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1017127152


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