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Journal of the Physical Society of Japan

The Friedrichs Model with Singular Continuous Spectrum

M. Fujiyoshi, Shuichi Tasaki

Published Jan 1, 2003• 0 citations
Publisher

Abstract

The decay of an unstable state is investigated for the Friedrichs model with a singular continuous spectrum. On the basis of a scaling approach, it is shown that the survival amplitude may exhibit the Mittag–Leffler relaxation. The validity of the scaling approach is investigated numerically as well.
DOI
10.1143/jpsjs.72sc.73

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The decay of an unstable quantum state interacting with an environment with a purely singular continuous energy spectrum is studied. Within the framework of the weak coupling Friedrichs-Fano (Newns-Anderson) model, a scaling approach is developed by assuming the local Hölder continuity of the environmental integrated density of states (IDOS). It reduces to the conventional van Hove scaling when IDOS is differentiable at the resonant energy. When the Hölder index at the resonant energy is less than unity, the survival amplitude is shown to be a Mittag-Leffler function of the scaled time, which describes monotonic decay, decay with one local maximum, oscillatory convergence towards a finite value. Then, these predictions are compared with the numerical solution of the integral equation for the survival amplitude and are shown to well reproduce the numerical observations provided that the differential/Hölder coefficients are estimated in an appropriate coarse-grained way.
The long-time behavior of the survival probability for unstable multilevel systems that follows the power-decay law is studied based on the NNN-level Friedrichs model, and is shown to depend on the initial population in unstable states. A special initial state maximizing the asymptote of the survival probability at long times is found and examined by considering the spontaneous-emission process for the hydrogen atom interacting with the electromagnetic field.
Long time behaviour of the survival probability for unstable multilevel systems is studied based on the N-level Friedrichs model in the presence of zero energy resonance, that means the divergence of the resolvent at zero energy. With resort to the technique developed by Jensen and Kato [Duke Math. J. 46, 583 (1979)], the zero energy resonance of this model is characterized by the zero energy eigenstate that does not belong to the Hilbert space. It is then shown that for some kinds of the form factors, logarithmically slower decay of the survival probability can be realized.
The decaying behavior of both the survival S(t) and total P(t) probabilities for unstable multilevel systems at long times is investigated by using the N-level Friedrichs model. The long-time asymptotic-forms of both S(t) and P(t) are obtained for an arbitrary initial-state extending over the unstable levels. It is then clarified how the asymptotic forms depend on the initial population in unstable levels. In particular, a special initial state that maximizes the asymptotic form of both S(t) and P(t) is found. On the other hand, the initial states eliminating the first term of their asymptotic expnasions also exist, which implies that a faster decay rather than expected can be realized. This faster decay for S(t) is numerically confirmed by considering the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field. It is demonstrated that the t-4-decay and a faster decay are realized depending on the initial states, where the latter is estimated as t-8.
Abstract We present here a relation of different types of Friedrichs models and their use in the description and comprehension of resonance phenomena. We first discuss the basic Friedrichs model and obtain its resonance in the case that this is simple or doubly degenerated. Next, we discuss the model with N levels and show how the probability amplitude has an oscillatory behavior. Two generalizations of the Friedrichs model are suitable to introduce resonance behavior in quantum field theory. We also discuss a discrete version of the Friedrichs model and also a resonant interaction between two systems both with continuous spectrum. In an appendix, we review the mathematics of rigged Hilbert spaces.
Abstract We present here a relation of different types of Friedrichs models and their use in the description and comprehension of resonance phenomena. We first discuss the basic Friedrichs model and obtain its resonance in the case that this is simple or doubly degenerated. Next, we discuss the model with N levels and show how the probability amplitude has an oscillatory behavior. Two generalizations of the Friedrichs model are suitable to introduce resonance behavior in quantum field theory. We also discuss a discrete version of the Friedrichs model and also a resonant interaction between two systems both with continuous spectrum. In an appendix, we review the mathematics of rigged Hilbert spaces.