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4 - The Schrödinger Operator

from Part Two - The Laplace and Schrödinger Operators

Published online by Cambridge University Press:  03 November 2022

Rupert L. Frank
Affiliation:
Ludwig-Maximilians-Universität München
Ari Laptev
Affiliation:
Imperial College of Science, Technology and Medicine, London
Timo Weidl
Affiliation:
Universität Stuttgart
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Summary

We discuss the definition of the Schrödinger operator on Euclidean space as a self-adjoint operator and discuss basic properties of its spectrum, including criteria for discreteness and finiteness of its negative spectrum. We analyze in detail the classical examples of the harmonic oscillator, the Coulomb Hamiltonian and the Pöschl–Teller potential, which can be solved using a commutation method. Returning to general potentials, we use Dirichlet–Neumann bracketing to prove Weyl asymptotics for the number and Riesz means of negative eigenvalues in the strong coupling constant limit. These asymptotic results are complemented by the nonasymptotic results of Lieb–Thirring, Cwikel–Lieb–Rozenblum, and Weidl. We present a unified method of proof of these bounds, based on Sobolev inequalities and the Besicovitch covering lemma. As an application of these bounds, we extend Weyl asymptotics to a large class of potentials.

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Publisher: Cambridge University Press
Print publication year: 2022

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  • The Schrödinger Operator
  • Rupert L. Frank, Ludwig-Maximilians-Universität München, Ari Laptev, Imperial College of Science, Technology and Medicine, London, Timo Weidl, Universität Stuttgart
  • Book: Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities
  • Online publication: 03 November 2022
  • Chapter DOI: https://doi.org/10.1017/9781009218436.008
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  • The Schrödinger Operator
  • Rupert L. Frank, Ludwig-Maximilians-Universität München, Ari Laptev, Imperial College of Science, Technology and Medicine, London, Timo Weidl, Universität Stuttgart
  • Book: Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities
  • Online publication: 03 November 2022
  • Chapter DOI: https://doi.org/10.1017/9781009218436.008
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • The Schrödinger Operator
  • Rupert L. Frank, Ludwig-Maximilians-Universität München, Ari Laptev, Imperial College of Science, Technology and Medicine, London, Timo Weidl, Universität Stuttgart
  • Book: Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities
  • Online publication: 03 November 2022
  • Chapter DOI: https://doi.org/10.1017/9781009218436.008
Available formats
×