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

TL;DR

This paper extends Keller and Lieb-Thirring estimates to Dirac operators, providing bounds on eigenvalues in the spectral gap using Lebesgue norms of the potential, with implications for spectral theory and nonlinear equations.

Contribution

It introduces new bounds for Dirac eigenvalues in the spectral gap, including a critical potential norm and an extension to Lieb-Thirring inequalities, using Birman-Schwinger analysis.

Findings

Derived Keller estimates for Dirac operators in terms of Lebesgue norms.

Identified a critical potential norm where eigenvalues reach the spectral gap edge.

Extended Keller estimates to Lieb-Thirring inequalities for Dirac eigenvalues.

Abstract

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 Schr\"odinger 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. 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.