Unicity of the integrated density of states for relativistic Schr?dinger operators with regular magnetic fields and singular electric potentials

摘要

We show coincidence of the two definitions of the integrated density of states (IDS) for a class of relativistic Schr?dinger operators with magnetic fields and scalar potentials introduced in Iftimie et al. (Publ Res Inst Math Sci 43(3):585-623, 2007; Topics in applied mathematics and mathematical physics, Editura Academiei Romane, 2008), the first one relying on the eigenvalue counting function of operators induced on open bounded sets with Dirichlet boundary conditions, the other one involving the spectral projections of the operator defined on the entire space. In this way one generalizes the results of Doi et al. (Math Z 237:335-371, 2001) and Iftimie (Publ Res Inst Math Sci 41(2):307-327, 2005) for non-relativistic operators. The proofs needs the magnetic pseudodifferential calculus developed in Iftimie et al. (Publ Res Inst Math Sci 43(3):585-623, 2007), as well as a Feynman-Kac-It? formula for Lévy processes (Ichinose and Tamura, Commun Math Phys 105(2):239-257, 1986; Iftimie et al. Topics in applied mathematics and mathematical physics, Editura Academiei Romane, 2008). In addition, in case when both the magnetic field and the scalar potential are periodic, one also proves the existence of the IDS.