Green’s Functions at Exceptional Points

Abstract

The Green’s function is an essential tool for the description of scattering. Spectral singularities such as exceptional points have a very specific effect upon the structure of Green’s functions. It is well known that the coalescence of two eigenvalues gives rise to a pole of second order in addition to the usual first order pole. The present paper describes the general patterns of Green’s functions at exceptional points of arbitrary order. The higher orders of the pole terms as well as their respective coefficients - being matrices - are presented in terms of the underlying Hamiltonian. For the coalescence of three eigenvalues this appears to be of immediate physical interest while the coalescence of four or more levels is still awaiting experimental realisation in the laboratory.