A. Jáuregui

Degeneracy of resonances in a double barrier potential

Degeneracy of resonant states and double poles in the scattering matrix of a double barrier potential are contrived by adjusting the parameters of the system. The cross section, scattering wavefunction and Gamow eigenfunction are computed at degeneracy. Some general properties of the degeneracy of resonances are exhibited and discussed in this simple quantum system.

Non-Hermitian degeneracy of two unbound states

We numerically solved the implicit, transcendental equation that defines the eigenenergy surface of a degenerating isolated doublet of unbound states in the simple but illustrative case of the scattering of a beam of particles by a double barrier potential. Unfolding the degeneracy point with the help of a contact equivalent approximant, crossings and anticrossings of energies and widths, as well as the changes of identity of the poles of the S-matrix, are explained in terms of sections of the eigenenergy surfaces.

Gamow vectors for barrier wells

Abstract We study four examples of Gamow vectors in one-dimensional potential barriers, namely square barriers and delta barriers. We show that resonances appear when the potential has at least two relative maxima and investigate the emergence of double resonances given rise to Gamow–Jordan vectors as well.

Bound states at exceptional points in the continuum

In this work, an example of exceptional points in the continuous spectrum of a Hamiltonian of von Neumann-Wigner type is presented and discussed. Remarkably, these exceptional points are not associated with a double pole in the scattering matrix but with a double pole in the normalization factor of the Jost eigenfunctions normalized to unit flux. At the exceptional points the two unnormalized Jost eigenfunctions are no longer linearly independent but coalesce to give rise to two Jordan cycles of generalized bound state energy eigenfunctions in the continuum and a Jordan block representation of the Hamiltonian. The regular scattering eigenfunction vanishes at the exceptional points and the irregular scattering eigenfunction has a double pole at the exceptional points. The scattering matrix is a regular analytical function of the wave number k for all k real including the exceptional points.

Exceptional points of a Hamiltonian of von Neumann–Wigner type

In this paper, we present and discuss an example of exceptional points in the continuous spectrum of a Hamiltonian of von Neumann–Wigner type associated with a double pole in the normalized Jost eigenfunctions, as functions of the wave number. The two unnormalized Jost energy eigenfunctions are analytic functions of the wave number k for all k complex, but at the exceptional points k = ±q, they coalesce to give rise to Jordan cycles of two generalized bound state quadratically integrable eigenfunctions. The Green function has simple poles at the exceptional points, but the regular scattering eigenfunction vanishes at those points. The scattering matrix is a regular function of the wave number at the exceptional points, that is, S(k) does not have a pole at the exceptional points.

Doublets and Accidental Degeneracy of Resonances

By means of two simple examples, we show that degeneracy of two resonances and the concomitant occurrence of a complex double pole in the S-matrix of a quantum system may be contrived by adjusting the control parameters of the system.