Dirac operators and domain walls

We study the eigenvalue problem for a one-dimensional Dirac operator with a spatially varying "mass" term. It is well-known that when the mass function has the form of a kink, or domain wall, transitioning between strictly positive and strictly negative asymptotic mass, ±κ_∞, at ±∞, the Dirac operator has a simple eigenvalue of zero energy (geometric multiplicity equal to one) within a gap in the continuous spectrum, with corresponding exponentially localized zero mode. We consider the eigenvalue problem for the one-dimensional Dirac operator with mass function defined by "gluing" together n domain wall-type transitions, assuming that the distance between transitions, 2δ, is sufficiently large, focusing on the illustrative cases n = 2 and 3. When n = 2 we prove that the Dirac operator has two real simple eigenvalues of opposite sign and of order e^-2|κ_∞|δ. The associated eigenfunctions are, up to L² error of order e^-2|κ_∞|δ, linear combinations of shifted copies of the single domain wall zero mode. For the case n = 3, we prove the Dirac operator has two nonzero simple eigenvalues as in the two domain wall case and a simple eigenvalue at energy zero. The associated eigenfunctions of these eigenvalues can again, up to small error, be expressed as linear combinations of shifted copies of the single domain wall zero mode. When n > 3 no new technical difficulty arises and the result is similar. Our methods are based on a Lyapunov-Schmidt reduction/ Schur complement strategy, which maps the Dirac operator eigenvalue problem for eigenstates with near-zero energies to the problem of determining the kernel of an n×n matrix reduction, which depends nonlinearly on the eigenvalue parameter. The class of Dirac operators we consider controls the bifurcation of topologically protected "edge states" from Dirac points (linear band crossings) for classes of Schrödinger operators with domain wall modulated periodic potentials in one and two space dimensions. The present results may be used to construct a rich class of defect modes in periodic structures modulated by multiple domain walls.