Spherically Symmetric Dirac Operators with Variable Mass and Potentials Infinite at Infinity

Abstract

We study the spectrum of spherically symmetric Dirac operators m three-dimensional space with potentials tending to infinity at infinity under weak regularity assumptions. We prove that purely absolutely continuous spectrum covers the whole real line if the potential dominates the mass, or scalar potential, term. In the situation where the potential and the scalar potential are identical, the positive part of the spectrum is purely discrete : we show that the negative half-line is filled with purely absolutely continuous spectrum m this case. § 1. IntroductionIn a recent paper [20] the spectral properties of the three-dimensional Dirac operator (where p -~i F, i 2 -~~ 1, l n is the n Xn unit matrix, and ao -j8, a\, a^ 0.3 are Hermitian 4x4 matrices satisfying the anti-commutation relations aja k +a k aj = 25 Jk (/, *e{0, 1, 2, 3})) were studied under the condition that the real-valued coefficient function m tends to °° (or -°°) as \x -*°°. For constant m, H is the Hamilton operator describing a relativistic quantum mechanical particle of mass m moving in an external force field of (real-valued) potential q. As a non-constant function, m can also take the role of a so-called scalar potential, which has been discussed in the physical literature as a model of quark confinement (cf. the references in [20], Thaller [14] p. 305).

The Dirac System on the Anti-de Sitter Universe

“…There are many mathematical works on the one-half spin field on curved space-time, in particular [4], [17], [18], [25], [26], [27], [28]. The gravitational potential plays the role of a variable mass that tends to the infinity at the space infinity ; the rather similar Dirac equation on Minkowski space with increasing potential has been considered in [23], [34], [38]. The litterature on the boundary value problems for the Dirac system is huge ; among important contributions, we can cite [5], [6], [9], [10], [16], [20].…”

The Dirac equation on the Anti-de-Sitter Universe

“…The analogous situation of the infinite mass at the infinity of the Minkowski space has been investigated in [6,7]. In our case, the key result is the asymptotic behaviour, near the boundary, of the spinors of D(H M ).…”

“…(i) Given Ψ 0 ∈ L 2 , there exist solutions of (1)- (3), and all the solutions of (1), (2), are equal for It would be interesting to study the role of the asymptotic conditions for the propagation of the singularities and the energy beyond the maximal domain of global hyperbolicity (7).…”

Absence of high-energy spectral concentration for Dirac systems with divergent potentials

“…This includes eventually monotonic though not necessarily continuous q, thus vindicating a conjecture of Rose and Newton [18]. This type of criterion also extends to spherically symmetric three-dimensional Dirac operators, even with a variable mass term, provided the latter is dominated by the potential [21]. On the other hand, (1.4) alone, without a condition of type (1.6), is not sufficient for the spectrum to fill the real line [20].…”