Self-adjointness and spectral properties of Dirac operators with magnetic links

Abstract

We define Dirac operators on S 3 (and R 3 ) with magnetic fields supported on smooth, oriented links and prove selfadjointness of certain (natural) extensions. We then analyze their spectral properties and show, among other things, that these operators have discrete spectrum. Certain examples, such as circles in S 3 , are investigated in detail and we compute the dimension of the zero-energy eigenspace.

Analysis of zero modes for Dirac operators with magnetic links

“…We emphasize that these sections are different from the sections (ξ + , ξ − ) considered in [41,42], but on γ, the sections ξ ± (γ(s)) and η ± (γ(s)) coincide (up to a common phase e iϕ(s) ).…”

“…In two previous papers [41,42] we have introduced Dirac operators on S 3 and R 3 with magnetic fields supported on links, and we investigated the spectral properties of these operators. These rather singular magnetic fields, which we call magnetic links, are described by an oriented link γ = ∪ K k=1 γ k together with a collection of fluxes (2πα k ) K k=1 carried by its connected components.…”

“…Main results. In this paper we study the Dirac operator with a magnetic field supported on links introduced in [41]. More precisely, given a smooth, oriented link γ = ∪ K k=1 γ k ⊂ S 3 and a set of fluxes (2πα k ) K k=1 ∈ (0, 1) K , the one-current B := K k=1 2πα k [γ k ] can be seen as a magnetic field on S 3 .…”

“…where S k is a Seifert surface for γ k such that S k and γ k ′ are transverse in S 3 for k = k ′ (a Seifert surface for a link is a compact oriented surface whose boundary is the given link). The transversality property is generic and as shown in [41] can always be assumed. We recall that a n-current is a continuous linear form on the set of smooth n-forms with compact support [43,Chapter 3].…”

“…Its self-adjoint extensions are investigated in [41]; the domain of each is characterized by the behavior of the wave functions of its domain in the vicinity of the knots γ k . We are interested in one particular extension, denoted by D…”