Resonances, scattering theory, and rigged Hilbert spaces

Journal Article · Fri Aug 01 00:00:00 EDT 1980 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.524655· OSTI ID:5173335

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 and to the scattering of a spinless particle by a local central potential.

• Research Organization: Department of Physics, Center for Particle Theory, The University of Texas at Austin, Austin, Texas 78712
• DOE Contract Number: EY-76-S-05-3992
• OSTI ID: 5173335
• Journal Information: J. Math. Phys. (N.Y.); (United States), Vol. 21:8
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
SCATTERING
EIGENVECTORS
RESONANCE
EIGENVALUES
HAMILTONIANS
HILBERT SPACE
QUANTUM MECHANICS
S MATRIX
BANACH SPACE
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MATRICES
MECHANICS
QUANTUM OPERATORS
SPACE
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