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Pré-Publication, Document De Travail Année : 2022

Keller estimates of the eigenvalues in the gap of Dirac operators

Résumé

We estimate the lowest eigenvalue in the gap 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 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. Most of our result are established in the Birman-Schwinger reformulation of the problem.
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Dates et versions

hal-03803758 , version 1 (06-10-2022)
hal-03803758 , version 2 (24-07-2023)

Identifiants

  • HAL Id : hal-03803758 , version 1

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Jean Dolbeault, David Gontier, Fabio Pizzichillo, Hanne Van Den Bosch. Keller estimates of the eigenvalues in the gap of Dirac operators. 2022. ⟨hal-03803758v1⟩
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