• DOI:10.1007/s00205-022-01811-2
• Corpus ID: 244709791

Calogero Type Bounds in Two Dimensions

@article{Laptev2021CalogeroTB,
  title={Calogero Type Bounds in Two Dimensions},
  author={Ari Laptev and Larry Read and Lukas Schimmer},
  journal={Archive for Rational Mechanics and Analysis},
  year={2021},
  volume={245},
  pages={1491 - 1505}
}

• A. Laptev, L. Read, L. Schimmer
• Published 26 November 2021
• Mathematics
• Archive for Rational Mechanics and Analysis

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.