The Exact Number of Eigenvalues of the Discrete Schrödinger Operators in One-Dimensional Case View Full Text


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Article Info

DATE

2021-06

AUTHORS

S. N. Lakaev, I. U. Alladustova

ABSTRACT

We study two-particle Schrödinger operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{\lambda\mu}(k)$$\end{document}, with the fixed quasi-momentum of particles pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in\mathbb{T}$$\end{document}, on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2,o}(\mathbb{T},\eta)$$\end{document}. These operators are associated to the Bose–Hubbard Hamiltonian \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\mathbb{H}}_{\lambda\mu}$$\end{document} of a system of two identical quantum-mechanical particles (fermions) interacting via zero-range potential \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu\in\mathbb{R}$$\end{document} on one site and potential \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda\in\mathbb{R}$$\end{document} on neighboring sites. We establish a partition of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda-\mu$$\end{document} parameter-plane into several connected components where the Schrödinger operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\lambda\mu}(k)$$\end{document} can have only a definite (constant) number of eigenvalues. The eigenvalues may locate and below the bottom of the essential spectrum and above its top. More... »

PAGES

1294-1303

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1134/s1995080221060172

DOI

http://dx.doi.org/10.1134/s1995080221060172

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1139456397


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