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

TL;DR

This paper investigates a unique spectral singularity in continuum energies where two bound states coalesce, leading to resonances characterized by phase shifts and cross section features, revealing novel spectral phenomena.

Contribution

It introduces the concept of a spectral singularity in continuum energies caused by coalescence of bound states and analyzes its manifestation as resonances with distinctive interference effects.

Findings

Phase shift exhibits a 2π jump at the singularity.

Cross section shows two inverted peaks due to interference.

Perturbation breaks the singularity into two 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.