School of Mathematics, Cardiff University, 23 Senghennydd Rd., Cardiff CF24 4YH, United Kingdom
Abstract:
A perturbation decaying to at and not too irregular at introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues in a compact subset of a gap in the essential spectrum is given by a quasi-semiclassical asymptotic formula in the slow-decay limit, which for power-decaying perturbations is equivalent to the large-coupling limit. This asymptotic behaviour elucidates the origin of the dense point spectrum observed in spherically symmetric, radially periodic three-dimensional Dirac operators.