The zero modes and zero resonances of massless Dirac operators

Abstract

The zero modes and zero resonances of the Dirac operatorWe shall show that every zero mode f (x) is continuous on R 3 and decays at infinity with the decay rate |x| −2 . Also, we shall show that H has no zero resonance if ρ > 3/2.

Some new results about the massless Dirac operator

“…16, also Saitō and Umeda used it to show that Theorem A is the best for ρ ∈ (1, 2] in Ref. 21. For more detailed, we easily see that…”

On the Global Limiting Absorption Principle for Massless Dirac Operators

“…The first such result for free, massless Dirac operators H 0 = α · (−i∇) (cf. (3.2)), again in dimensions n = 3, was proved by Saitō and Umeda [42] in 2008 upon relying on quite different methods as part of their study of zero-energy eigenvalues and resonances. Since, apparently, no other results are available in the massless case, we now fill this gap, and by suitably modifying the approach of Iftimovici and Mȃntoiu, we prove a global limiting absorption principle on R n for free, massless Dirac operators in all dimensions n ∈ N, n 2.…”

“…In 1999, Iftimovici and Mȃntoiu [28] proved a global limiting absorption principle, that is, one on the entire real axis and hence including threshold energies ±m for the free, massive Dirac operator H 0 (m) with mass m > 0 in three dimensions. The first proof of a global limiting absorption principle for massless Dirac operators H 0 in three dimensions is due to Saitō and Umeda [42] in 2008. As no other result on a global limiting absorption principle in the massless case is known to us, we now fill this gap and upon modifying the approach by Iftimovici and Mȃntoiu for m > 0, we treat free, massless (m = 0) Dirac operators H 0 in all dimensions n ∈ N, n 2.…”

Non-Existence of Zero Resonances of Pauli Operators

“…[4], [5], [6] and, Saitō and Umeda [9], [10], for example. On the non-existence of zero resonances of Weyl-Dirac operators, we refer the reader to the paper [10].…”