Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states

Abstract

An accidental degeneracy of resonances gives rise to a double pole in the scattering matrix, a double zero in the Jost function and a Jordan chain of length two of generalized Gamow-Jordan eigenfunctions of the radial Schrödinger equation. The generalized Gamow-Jordan eigenfunctions are basis elements of an expansion in bound and resonant energy eigenfunctions plus a continuum of scattering wave functions of complex wave number. In this biorthonormal basis, any operator f (H (ℓ) r ) which is a regular function of the Hamiltonian is represented by a complex matrix wich is diagonal except for a Jordan block of rank two. The occurrence of a double pole in the Green's function, as well as the non-exponential time evolution of the Gamow-Jordan generalized eigenfunctions are associated to the Jordan block in the complex energy representation.

General theory of spontaneous emission near exceptional points

“…By using such a modified expansion, we obtain a quantitative formula for the LDOS, which is a measure of how much power a dipole source can radiate [41] or, equivalently, a measure of its spontaneous emission rate. Note that similar expansion methods were previously used to evaluate the Green's functions at EPs [42][43][44]; however, these works were limited to one-dimensional and paraxial systems, and were not applied to study spontaneous emission. An alternative semi-analytic approach was presented in [25].…”

Bound states of dipolar molecules studied with the Berggren expansion method

“…The complex-energy GamowSiegert states [53,59] states have been used in various contexts in nuclear, atomic, and molecular physics [60][61][62][63][64][65][66][67][68][69][70][71][72][73]. Some recent applications of Gamow-Siegert states, also in the context of a CC formalism relevant to the problem of dipole anions, can be found in, e.g., Refs.…”

Resonance expansions in quantum mechanics

“…When double poles come into play, the effective Hamiltonians have non-diagonal terms [17]. Note that, contrary to PT-symmetric Hamiltonians, the eigenvalues of this effective Hamiltonian are complex.…”