On the Zero Modes of Weyl–dirac Operators and Their Multiplicity

Abstract

It is proved that the existence of zero modes of Weyl-Dirac operators is a rare phenomenon. An estimate of the multiplicity is given in term of the magnetic potential.

The Asymptotic Limits of Zero Modes of Massless Dirac Operators

“…See Balinsky and Evans [5]. The paper by Fröhlich, Lieb and Loss [12] revealed that the existence of zero modes of a Weyl-Dirac operator plays a crucial role in the study of stability of Coulomb systems with magnetic fields.…”

Dirac–Sobolev inequalities and estimates for the zero modes of massless Dirac operators

“…The first objective of the research reported here was to confirm that the results in [1] can be extended to Dirac-type operators with matrixvalued potentials. This then set the scene for the main goal which was to determine the decay rates of zero modes whenever they occur.…”

“…Assuming that Q(·) C 4 ∈ L 3 (R 3 ), where · C 4 denotes any matrix norm on C 4 , it was shown that Q is a small perturbation of α · p and hence (1.1) defines a self-adjoint operator H as an operator 4 with weak first derivatives in (L 2 (R 3 )) 4 . The technique in [1] readily applied to (1.1) to meet the first objective and yield the following result:…”

Asymptotics for Erdős–Solovej Zero Modes in Strong Fields

“…Furthermore, fields B ∈ B ES are smooth and satisfy bounds of the form |B(x)| = O(|x| −4 ) as |x| → ∞, while it is always possible to find a smooth potential A with B = dA which satisfies the bounds of the form |A(x)| = O(|x| −3 ) as |x| → ∞. It follows that fields in B ES (and their associated potentials) fall into the classes considered in [4,5,7].…”

“…Most early works concentrated on the construction of explicit examples, including the original example [17], examples with arbitrary multiplicity [2], compact support [6] and a certain rotational type of symmetry ( [10]; further details below). Some subsequent work moved towards studying the set of all zero mode producing fields (or potentials) within a given class; in particular, this set is nowhere dense [4,5] and is generically a sub-manifold of co-dimension one ( [7]; slightly different classes of potentials were considered in these works).…”