Pohozaev identity and Virial Theorem for the Dirac–Coulomb operator

Abstract

In this note, we give an alternative proof of the Virial Theorem for the Dirac equation perturbed with a Coulomb-like potential, result which goes back to Albeverio (Ann Phys 71:167–276, 1972), Kalf (J Funct Anal 21:389–396, 1976) and refined by Leinfelder (Integral Equ Oper Theory 4(2):226–244, 1981). We will deduce it proving a Pohozaev-like identity for a Neumann boundary value problem for an elliptic equation in \(\mathbb {R}^{4}_{+}\) which, following ideas going back to Caffarelli and Silvestre (Commun Partial Differ Equ 32(7–9):1245–1260, 2007) can be related to the eigenvalue problem for the Dirac equation in \(\mathbb {R}^{3}\).