Calogero Type Bounds in Two Dimensions View Full Text


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Article Info

DATE

2022-07-23

AUTHORS

Ari Laptev, Larry Read, Lukas Schimmer

ABSTRACT

For a Schrödinger operator on the plane R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document} with electric potential V and an Aharonov–Bohm magnetic field, we obtain an upper bound on the number of its negative eigenvalues in terms of the L1(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1(\mathbb {R}^2)$$\end{document}-norm of V. Similar to Calogero’s bound in one dimension, the result is true under monotonicity assumptions on V. Our method of proof relies on a generalisation of Calogero’s bound to operator-valued potentials. We also establish a similar bound for the Schrödinger operator (without magnetic field) on the half-plane when a Dirichlet boundary condition is imposed and on the whole plane when restricted to antisymmetric functions. More... »

PAGES

1491-1505

References to SciGraph publications

  • 2012-06-20. Bargmann type estimates of the counting function for general Schrödinger operators in JOURNAL OF MATHEMATICAL SCIENCES
  • 2012-06-07. On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential in COMMUNICATIONS IN MATHEMATICAL PHYSICS
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  • 1999. Hardy inequalities for magnetic Dirichlet forms in MATHEMATICAL RESULTS IN QUANTUM MECHANICS
  • 1994-10. Piecewise-polynomial approximation of functions fromHℓ((0, 1)d), 2ℓ=d, and applications to the spectral theory of the Schrödinger operator in ISRAEL JOURNAL OF MATHEMATICS
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  • 1996-05. On the Lieb-Thirring constantsLγ,1 for γ≧1/2 in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2021-04. Hardy Inequality for Antisymmetric Functions in FUNCTIONAL ANALYSIS AND ITS APPLICATIONS
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  • 2000. Sharp Lieb-Thirring inequalities in high dimensions in ACTA MATHEMATICA
  • 2015-02-03. Negative Eigenvalues of Two-Dimensional Schrödinger Operators in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00205-022-01811-2

    DOI

    http://dx.doi.org/10.1007/s00205-022-01811-2

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1149710596


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