Abstract

LOHR, D.; HERNANDEZ, E.; JAUREGUI, A.  and  MONDRAGON, A.. Bound states in the continuum and time evolution of the generalized eigenfunctions. Rev. mex. fis. [online]. 2018, vol.64, n.5, pp.464-471.  Epub Nov 30, 2019. ISSN 0035-001X.

We study the Jost solutions for the scattering problem of a von Neumann-Wigner type potential, constructed by means of a two times iterated and completely degenerated Darboux transformation. We show that for a particular energy the unnormalized Jost solutions coalesce to give rise to a Jordan cycle of rank two. Performing a pole decomposition of the normalized Jost solutions we find the generalized eigenfunctions: one is a normalizable function corresponding to the bound state in the continuum and the other is a bounded, non-normalizable function. We obtain the time evolution of these functions as pseudo-unitary, characteristic of a pseudo-Hermitian system. An explicit calculation of the cross section as a function of the wave number k reveals no sign of the bound state in the continuum.

Keywords : Bound states in the continuum; Darboux transformations; Jordan chain; 03.65.-w; 03.65.Ge; 03.65.Nk.