skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Description of Resonances within the Rigged Hilbert Space

Abstract

The spectrum of a quantum system has in general bound, scattering and resonant parts. The Hilbert space includes only the bound and scattering spectra, and discards the resonances. One must therefore enlarge the Hilbert space to a rigged Hilbert space, within which the physical bound, scattering and resonance spectra are included on the same footing. In this work, I will explain how this is done.

Authors:
 [1]
  1. Department of Physics, University of California at San Diego, La Jolla, CA 92093 (United States)
Publication Date:
OSTI Identifier:
21054789
Resource Type:
Journal Article
Resource Relation:
Journal Name: AIP Conference Proceedings; Journal Volume: 885; Journal Issue: 1; Conference: EAV06: Advanced summer school in physics 2006: Frontiers in contemporary physics, Mexico City (Mexico), 10-14 Jul 2006; Other Information: DOI: 10.1063/1.2563170; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ENERGY SPECTRA; HILBERT SPACE; QUANTUM MECHANICS; RESONANCE; SCATTERING

Citation Formats

Madrid, Rafael de la. Description of Resonances within the Rigged Hilbert Space. United States: N. p., 2007. Web. doi:10.1063/1.2563170.
Madrid, Rafael de la. Description of Resonances within the Rigged Hilbert Space. United States. doi:10.1063/1.2563170.
Madrid, Rafael de la. Fri . "Description of Resonances within the Rigged Hilbert Space". United States. doi:10.1063/1.2563170.
@article{osti_21054789,
title = {Description of Resonances within the Rigged Hilbert Space},
author = {Madrid, Rafael de la},
abstractNote = {The spectrum of a quantum system has in general bound, scattering and resonant parts. The Hilbert space includes only the bound and scattering spectra, and discards the resonances. One must therefore enlarge the Hilbert space to a rigged Hilbert space, within which the physical bound, scattering and resonance spectra are included on the same footing. In this work, I will explain how this is done.},
doi = {10.1063/1.2563170},
journal = {AIP Conference Proceedings},
number = 1,
volume = 885,
place = {United States},
year = {Fri Feb 09 00:00:00 EST 2007},
month = {Fri Feb 09 00:00:00 EST 2007}
}
  • The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,''in,'' and ''out'' eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ''out'' eigenvector family are the samemore » as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ''complete'' sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ''out'' eigenvectors. The free, ''in'' and ''out'' eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee--Friedrichs model and to the scattering of a spinless particle by a local central potential.« less
  • Within the rigged Hilbert space formulation of quantum mechanics idealized resonances (without background) are described by generalized eigenvectors of an essentially self-adjoint Hamiltonian with complex eigenvalue and a Breit--Wigner energy distribution. This establishes the link between the S matrix description of resonances by a pole and the usual description of states by vectors, overcomes theoretical problems connected with the deviation from exponential law and simplifies the calculation of the decay rate formula.
  • The evolution of the universe and that of most of its subsystems are time asymmetric which contrasts the time symmetric laws of physics. The aim of this paper is to prove that, from causality and the global nature of time asymmetry this asymmetry can be derived at the fundamental quantum level, defining a growing entropy and the outcome of equilibrium, explaining the decaying processes, etc.