Editor R. de la Llave STABILITY OF THE BROWN-RAVENHALL OPERATOR.pdf

M P E JJMathematical Physics Electronic JournalISSN 1086-6655Volume 5, 1999Paper 6Received: Sep 21, 1999, Accepted: Dec 7, 1999Editor: R. de la Llave STABILITY OF THE BROWN-RAVENHALL OPERATOR GEORG HOEVER AND HEINZ SIEDENTOP Abstract. The Brown-Ravenhall Hamiltonian is a model for the behavior of N electrons in a field of K fixed nuclei having the atomic numbers Z = (Z1, . . . , ZK), which is written, in appropriate units, as B = Λ+,N NX n=1 D (n) 0 + αVc ! Λ+,N acting on the N-fold antisymmetric tensor product HN of Λ+(L2(R3)? C4 ), where D (n) 0 denotes the free Dirac operator D0 acting on the n-th particle, Λ+ denotes the projection onto the positive spectral subspace of D0, Λ+,N the projection onto HN and the potential Vc is the usual Coulomb interaction of the particles, coupled by the constant α. It is proved in the massless case that for any γ 2/(2/π + π/2) there exists an α0 such that for all α α0 and αZk ≤ γ (k = 1, . . .K) we have stability, i.e., B ≥ 0. Using numerical calculations we get stability for the physical value α ≈ 1/137 up to Zk ≤ 88 (k = 1, . . . K). 1. Introduction A basic requirement of thermodynamics is the extensivity of the energy. To be able to show this property, it is essential that on a microscopic level the energy per particle is bounded from below independently of the size of the considered system. This property, also referred to as stability of matter, as been proven in the literature for a wide class of models starting from the pioneering work of Dyson and Lenard [6, 7] in the non-relativistic case and of Conlon [3] and Fefferman and de la Llave [9] in the relativistic case. (See [10, 12, 14, 16] for an overview and more references.) A particular interesting work for our purposes is the work of Lieb and Yau [15] who consider the stability of matter of a relativistic system of N (spinless) electrons in a field of K fixed nuclei having atomic numbers Z = (Z1, . . . , ZK) and positions Key words and phrases. Dirac operator, stabi