Particular spectral singularity in the continuum energies: a manifestation as resonances

Abstract

We study the coalescence of two bound energy eigenstates embedded in the continuous spectrum of a real Hamiltonian $H[4]$ and the singular point produced by this coalescence. At the singular point, the two unnormalized Jost eigenfunctions are no longer linearly independent but coalesce to give rise to a bound state eigenfunction embedded in the continuum. We disturb the potential $V[4]$ by means of a truncation, this perturbation breaks the singular point in two resonances. The phase shift shows a jump of magnitude $2\pi$ and the shape of the cross section shows two inverted peaks, this behaviour is due to the interference between the two nearly degenerate resonances and the background component of the Jost function.