Resonances, scattering theory and rigged Hilbert spaces. [Survival amplitude, Cauchy theorem]

Technical Report · Mon Jan 01 04:00:00 EST 1979

DOI:https://doi.org/10.2172/5771839· OSTI ID:5771839

Parravicini, G.; Gorini, V.; Sudarshan, E. C.G.

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: Texas Univ., Austin (USA). Center for Particle Theory
• OSTI ID: 5771839
• Report Number(s): ORO-3992-358
• Country of Publication: United States
• Language: English

Related Subjects

645500* -- High Energy Physics-- Scattering Theory-- (-1987)
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
ANALYTIC FUNCTIONS
BANACH SPACE
CAUCHY PROBLEM
DECAY
EIGENVALUES
EIGENVECTORS
ELEMENTARY PARTICLES
ENERGY-LEVEL TRANSITIONS
FUNCTIONS
HADRONS
HAMILTONIANS
HILBERT SPACE
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MATRICES
PARTICLE DECAY
QUANTUM OPERATORS
RESONANCE
RESONANCE PARTICLES
S MATRIX
SCATTERING
SERIES EXPANSION
SINGULARITY
SPACE
TIME DEPENDENCE