ABSTRACT
We show that when an isolated doublet of unbound states of a physical system becomes degenerate for some values of the control parameters of the system, the energy hypersurfaces representing the complex resonance energy eigenvalues as functions of the control parameters have an algebraic branch point of rank one in parameter space. Associated with this singularity in parameter space, the scattering matrix, https://www.w3.org/1998/Math/MathML"> S l ( E ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq6444.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and the Green's function, https://www.w3.org/1998/Math/MathML"> G l ( + ) k ; r , r ' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq6445.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> have one double pole in the unphysical sheet of the complex energy plane. We characterize the universal unfolding or deformation of any degeneracy point of two unbound states in parameter space by means of a universal 2-parameter family of functions which is contact equivalent to the pole position function of the isolated doublet of resonances at the exceptional point and includes all small perturbations of the degeneracy condition up to contact equivalence.
