Resonances, scattering theory and rigged Hilbert spaces. [Survival amplitude, Cauchy theorem]
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 same 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. 48 references.
- Research Organization:
- Univ. of Texas, Austin, TX (United States). Center for Particle Theory
- DOE Contract Number:
- EY-76-S-05-3992
- OSTI ID:
- 5771839
- Report Number(s):
- ORO-3992-358; TRN: 79-022787
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
PARTICLE DECAY
HILBERT SPACE
RESONANCE PARTICLES
DECAY
SCATTERING
RESONANCE
ANALYTIC FUNCTIONS
CAUCHY PROBLEM
EIGENVALUES
EIGENVECTORS
ENERGY-LEVEL TRANSITIONS
HAMILTONIANS
S MATRIX
SERIES EXPANSION
SINGULARITY
TIME DEPENDENCE
BANACH SPACE
ELEMENTARY PARTICLES
FUNCTIONS
HADRONS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MATRICES
QUANTUM OPERATORS
SPACE
645500* - High Energy Physics- Scattering Theory- (-1987)