Spectral comparison between Dirac and Schrödinger operators

Abstract

We show a general comparison theorem for the eigenvalues of two self-adjoint semibounded operators acting on two different Hilbert spaces, which are related by a suitable mapping. As a particular case, we get estimates of the eigenvalues of the classical Dirac operator acting on spinors in terms of the eigenvalues of the Laplace-Beltrami operator. These estimates are sharp, in the sense that they give Friedrich’s inequality for the minimal eigenvalue.

Keywords

• comparison of eigenvalues
• dirac and schrödinger operators