Keller and Lieb–Thirring estimates of the eigenvalues in the gap of Dirac operators

• Jean Dolbeault ^[1] ; David Gontier ^[4] ; Fabio Pizzichillo ^[2] ; Hanne Van Den Bosch ^[3]
  1. [1] Paris Dauphine University

     Paris Dauphine University

     París, Francia

     
  2. [2] Universidad Politécnica de Madrid

     Universidad Politécnica de Madrid

     Madrid, España

     
  3. [3] Universidad de Chile

     Universidad de Chile

     Santiago, Chile

     
  4. [4] CNRS, PSL University, Université Paris-Dauphine; ENS/PSL University

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 40, Nº 2, 2024, págs. 649-692
• Idioma: inglés
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• Resumen

  • We estimate the lowest eigenvalue in the gap of the essential spectrum of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schrödinger operator, which are equivalent to some Gagliardo–Nirenberg–Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. The Keller estimate is then extended to a Lieb–Thirring inequality for the eigenvalues in the gap. Most of our result are established in the Birman–Schwinger reformulation.