A pedestrian introduction to Gamow vectors

Abstract

The Gamow vector description of resonances is compared with the S-matrix and the Green function descriptions using the example of the square barrier potential. By imposing different boundary conditions on the time independent Schrödinger equation, we obtain either eigenvectors corresponding to real eigenvalues and the physical spectrum or eigenvectors corresponding to complex eigenvalues (Gamow vectors) and the resonance spectrum. We show that the poles of the S matrix are the same as the poles of the Green function and are the complex eigenvalues of the Schrödinger equation subject to a purely outgoing boundary condition. The intrinsic time asymmetry of the purely outgoing boundary condition is discussed. Finally, we show that the probability of detecting the decay within a shell around the origin of the decaying state follows an exponential law if the Gamow vector (resonance) contribution to this probability is the only contribution that is taken into account.

Light-Quark Baryon Spectroscopy from ANL-Osaka Dynamical Coupled-Channels Analysis

“…This was a rather unsatisfactory situation [33] because the Breit-Wigner parameters are nothing more than "approximation" of the resonance parameters defined by poles of scattering amplitudes in the complex energy plane [34], where the latter has a clear physical meaning: the resonance states defined by poles are associated with the exact (complex) energy eigenstates of the full Hamiltonian of the system under the purely outgoing boundary condition (see, e.g., Refs. [35,36]). …”

Irreversible evolution of a wave packet in the rigged-Hilbert-space quantum mechanics

“…Despite these mathematical properties have been studied by several authors [18][19][20][21][22][23][24][25], a direct physical evidence of their implications is lacking, even in the simplest case of RHO. In this manuscript, we review the basics of TA-QM and of the GV approach to the RHO.…”

Estimation of the Spatial Structure of \(\Lambda \)(1405) with Energy-Dependent Complex \(\bar{K}N\) Potential

“…For an energy-dependent real potential, a consistent treatment was developed from the continuity equation [19]. For an energy-independent complex potential, it is known that the use of the Gamow vector is needed to normalize the wave function [20][21][22].…”