Description of Resonances within the Rigged Hilbert Space

Abstract

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 these lectures, I will explain how this is done.

Shell model in the complex energy plane

“…The two complex-energy representations of Green's function given by Eqs. (22,23) and the real-energy expression (21) are mathematically equivalent; hence, they should give the same results in practical applications. From a practical point of view, it is interesting to know if one can choose the complex contours in such a way that the contributions from the integrals in Eqs.…”

A Primer on Resonances in Quantum Mechanics

“…This means no work is done to sustain the oscillations so that there is no supplied energy to preserve the motion any longer. This system can be studied by solving (5) with F = 0. After introducing the ansatz x = Re(ze iwt ) we get a quadratic equation for w, the solution of which reads…”

“…In other words, decaying states are an approximation within the conventional quantum mechanics framework. This fact is usually taken to motivate the study of the rigged (equipped) Hilbert space H [3][4][5] (For a recent review see [6]). The mathematical structure of H lies on the nuclear spectral theorem introduced by Dirac in a heuristic form [18] and studied in formal rigor by Maurin [19] and Gelfand and Vilenkin [20].…”

Localization of Non-Relativistic Particles

“…In the position representation, these states blow up exponentially at infinity. In order to control such exponential blow-up, we need a space Φ exp of test functions that fall off faster than real exponentials [15,13]. The space Φ exp then yields two rigged Hilbert spaces in a natural way:…”

“…The space Φ pol is the largest subspace of the Hilbert space that remains invariant under the action of the observables of the algebra. The spaces Φ ′ pol and Φ × pol respectively contain the bras and the kets of the observables [11,12,13].…”