Complete separation of resonance and nonresonance channel spaces

• Published in: Journal of physics. B, Atomic, molecular, and optical physics Atomic, molecular, and optical physics, 2011-10, Vol.44 (20), p.201002-1-4
• Main Author: Shimamura, Isao
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Atomic and molecular physics; Calculations and mathematical techniques in atomic and molecular physics (excluding electron correlation calculations); Channels; Complement; Conjugates; Decay; Eigenvalues; Electronic structure of atoms, molecules and their ions: theory; Exact sciences and technology; Physics; Separation; Symmetry
• ISSN: 0953-4075; 1361-6455
• DOI: 10.1088/0953-4075/44/20/201002

The concept of 'resonance channel space' is proposed. Resonance states can decay into any open channels of common symmetry. They can also decay into any of the eigenchannels denned by diagonalizing the S matrix. However, an isolated resonance is known to decay only into a particular one of the eigenchannels (referred to as Q-eigenchannels) defined by diagonalizing the time-delay matrx Q = ihSdS super([Display omitted])/dE, where the dagger indicates the Hermitian conjugate. This particular Q-eigenchannel corresponds to the eigenvalue of Lorentzian shape as a function of the energy E. A theorem generalizing this for N overlapping resonances is proved; only N eigenvalues of the Q matrix are nonzero and exhibit N Lorentzian profiles, avoiding each other around their crossing points, provided that the background S matrix is independent of E. Any of the overlapping resonances can decay only into the N Q-eigenchannels corresponding to these eigenvalues, and not to any other Q-eigenchannels. Thus, the set of Q-eigenchannels can be divided into 'resonance channel space', i.e. the subset of N Q-eigenchannels associated with the resonances and its complement completely irrelevant to the resonances asymptotically.