Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice View Full Text


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Article Info

DATE

2014-03

AUTHORS

S. N. Lakaev, Sh. U. Alladustov

ABSTRACT

We consider the family H(k) of two-particle discrete Schrödinger operators depending on the quasimomentum of a two-particle system k ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}^d $\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}^d $\end{document} is a d-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the d-dimensional lattice ℤd, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrödinger operator H(k) below the essential spectrum are positive for all nonzero values of the quasimomentum k ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}^d $\end{document} if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrödinger operator H+(k), k ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}^d $\end{document}, corresponding to a two-particle system with repulsive interaction. More... »

PAGES

336-346

References to SciGraph publications

  • 2005-11-24. The Threshold Effects for the Two-Particle Hamiltonians on Lattices in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1993-07. On Efimov's effect in a system of three identical quantum particles in FUNCTIONAL ANALYSIS AND ITS APPLICATIONS
  • 2004-08. Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics in ANNALES HENRI POINCARÉ
  • 2006-06. Repulsively bound atom pairs in an optical lattice in NATURE
  • 2007-12. Positivity of the two-particle Hamiltonian on a lattice in THEORETICAL AND MATHEMATICAL PHYSICS
  • 2012-03. Existence and analyticity of bound states of a two-particle Schrödinger operator on a lattice in THEORETICAL AND MATHEMATICAL PHYSICS
  • 1993-09. The Efimov effect. Discrete spectrum asymptotics in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1979-03. The virtual level of the Schrödinger equation in JOURNAL OF MATHEMATICAL SCIENCES
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    http://scigraph.springernature.com/pub.10.1007/s11232-014-0146-1

    DOI

    http://dx.doi.org/10.1007/s11232-014-0146-1

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