The Spectrum of Relativistic Atoms According to Bethe and Salpeter and Beyond

We review Evans’ contributions to the spectral theory of operators describing relativistic particle systems. We will concentrate on no-pair operators and recent extensions of that work. 1. William Desmond Evans’ Papers on Relativistic Quantum Mechanics William Desmond Evans contributed to the spectral theory of operators describing relativistic particle systems as follows: (1) W. D. Evans. A problem in relativistic quantum mechanics. Quart. J. Math. Oxford Ser. (2), 17:345–358, 1966. (2) W. D. Evans. Eigenfunction expansions associated with the Dirac relativistic equations. Quart. J. Math. Oxford Ser. (2), 17:211–233, 1966. (3) W. D. Evans. Eigenfunction expansions associated with the Dirac relativistic equations. II. Quart. J. Math. Oxford Ser. (2), 18:239–262, 1967. (4) W. D. Evans. The Dirac equations with a spherically symmetrical scalar potential. Quart. J. Math. Oxford Ser. (2), 21:89–99, 1970. (5) W. D. Evans. On the decomposition method and the spectrum of the Dirac operator. Proc. London Math. Soc. (3), 21:651–673, 1970. (6) W. D. Evans. On the unique self-adjoint extension of the Dirac operator and the existence of the Green matrix. Proc. London Math. Soc. (3), 20:537–557, 1970. (7) W. Desmond Evans. Spectral theory of the Dirac operator. Math. Z., 121:1–23, 1971. (8) W. D. Evans. On the length of gaps in the essential spectrum of a generalised Dirac operator. Proc. Amer. Math. Soc., 35:137–146, 1972. (9) William Desmond Evans, Peter Perry, and Heinz Siedentop. The spectrum of relativistic one-electron atoms according to Bethe and Salpeter. Comm. Math. Phys., 178(3):733–746, 1996. (10) William Desmond Evans, Roger T. Lewis, Heinz Siedentop, and Jan Philip Solovej. Counting eigenvalues using coherent states with an application to Dirac and Schrödinger operators in the semi-classical limit. Ark. Mat., 34(2):265–283, 1996. (11) A. A. Balinsky and W. D. Evans. On the virial theorem for the relativistic operator of Brown and Ravenhall, and the absence of embedded eigenvalues. Lett. Math. Phys., 44(3):233–248, 1998. (12) V. I. Burenkov and W. D. Evans. On the evaluation of the norm of an integral operator associated with the stability of one-electron atoms. Proc. Roy. Soc. Edinburgh Sect. A, 128(5):993–1005, 1998. (13) A. A. Balinsky and W. D. Evans. Stability of one-electron molecules in the Brown-Ravenhall model. Comm. Math. Phys., 202(2):481–500, 1999. Date: February 21, 2006.