Crossings and Anticrossings of Unbound States

Abstract

We investigate the characteristic crossings and anticrossings of energies and widths of a doublet of resonances, observed in the vicinity of, and at a degeneracy of unbound states, when the control parameters of the system are varied. This characteristic behavior is explained in terms of the local, topological structure of the surfaces that represent the complex energy eigenvalues in parameter space in the vicinity of a degeneracy point. In the simple but illustrative case of the scattering of a beam of particles by a double barrier potential well with two regions of trapping, we solved numerically the implicit, transcendental equation that defines the eigenwave numbers of a degenerate isolated doublet of resonances as functions of the real, control parameters of the system. We found that, at a degeneracy of unbound states, the surface representing the resonance eigenwave numbers as functions of the control parameters has an algebraic branch point of rank one. Unfolding the degeneracy point, crossings and anticrossings of energies and widths are obtained as projections of sections of the eigenwave number surfaces.

Degeneracy of Resonances: Branch Point and Branch Cuts in Parameter Space

“…The resulting Re k n,n+1 (ξ 1 , ξ 2 ), Im k n,n+1 (ξ 1 , ξ 2 ) and the trajectories C n (π i ) and C n+1 (π i ) are shown as three dimensional graphs Figs. 8, 9 and 10, of that paper (Hernández et al, 2007).…”

“…There is a joint crossing of energies ( E = 0) and widths ( = 0) at exact degeneracy where the two simple poles meet and coalesce to form one double pole of S(E). The full lines are the exact trajectories obtained from a numerical computation of the energypole position function (Hernández et al, 2003a(Hernández et al, , 2007. The dashed lines were obtained from the contact approximantsÊ n (d, V 3 ) andÊ n+1 (d, V 3 ) described in the text.…”

“…In Figs. 1, 2 and 3, in this paper we show the trajectoriesĈ n (π i ) andĈ n+1 (π i ) for the example discussed in Hernández et al (2007). In these figures, the numerically computed exact trajectories C n (π i ) and C n,n+1 (π i ) are not shown, because they can hardly be distinguished from the contact aproximantsĈ n (π i ) andĈ n+1 (π i ).…”

“…In a companion paper (Hernández et al, 2007), we give the results of solving numerically the implicit Eqs. (16)-(18) for the pole position function, k n,n+1 (ξ 1 , ξ 2 ), in the case of a degeneracy of an isolated doublet of resonances in the scattering of a beam of particles by a double barrier potential well with two regions of trapping.…”

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

“…These points are the beginning of branch points and branch cuts on the surfaces of the energy as a function of the control parameters of the system [24,57]. However, in very few cases [19,58] emphasis is placed on the important role of the eigenvectors associated with these exceptional points.…”