%O This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
%T Spectral analysis of the Dirac operator on a 3-sphere
%P 501-527
%I ELEMENT
%L discovery1498494
%D 2018
%J Operators and Matrices
%N 2
%K Science & Technology, Physical Sciences, Mathematics, Dirac operator, spectral asymmetry, generalized Berger spheres, RIEMANNIAN GEOMETRY, THEORETIC CHARACTERIZATION, 1ST-ORDER SYSTEMS, HARMONIC SPINORS, ASYMMETRY
%V 12
%A Y-L Fang
%A M Levitin
%A D Vassiliev
%X We study the (massless) Dirac operator on a 3-sphere equipped with Riemannian metric. For
the standard metric the spectrum is known. In particular, the eigenvalues closest to zero are the
two double eigenvalues +3/2 and −3/2. Our aim is to analyse the behaviour of eigenvalues when
the metric is perturbed in an arbitrary smooth fashion from the standard one. We derive explicit
perturbation formulae for the two eigenvalues closest to zero, taking account of the second variations.
Note that these eigenvalues remain double eigenvalues under perturbations of the metric:
they cannot split because of a particular symmetry of the Dirac operator in dimension three
(it commutes with the antilinear operator of charge conjugation). Our perturbation formulae
show that in the first approximation our two eigenvalues maintain symmetry about zero and are
completely determined by the increment of Riemannian volume. Spectral asymmetry is observed
only in the second approximation of the perturbation process. As an example we consider a
special family of metrics, the so-called generalized Berger spheres, for which the eigenvalues can
be evaluated explicitly.