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1 results

  • 5 - Sharp Lieb–Thirring Inequalities

  • from Part Three - Sharp Constants in Lieb–Thirring Inequalities
  • 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
    Published online:
    03 November 2022
    Print publication:
    17 November 2022, pp 321-384

  • Summary

    We discuss the problem of finding the optimal constant in Lieb–Thirring and Cwikel–Lieb–Rozenblum inequalities, thereby introducing, in particular, the semiclassical constant and the one-particle constants, which appear in the Lieb–Thirring conjecture. We discuss Keller's problem of minimizing the lowest eigenvalue of a Schrödinger operator among all potentials with a given L^p norm. We present the Aizenman–Lieb monotonicity argument, as well as semiexplicit computations for eigenvalues of the harmonic oscillator (including the counterexample of Helffer and Robert) and the Pöschl–Teller potential. In the one-dimensional case, we present the optimal bounds due to Hundertmark–Lieb–Thomas and Gardner–Greene–Kruskal–Miura. We provide two proofs of the latter bound, namely, the original one based on trace formulas and a more recent one by Benguria and Loss based on the commutation method.