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Journal of Mathematical Physics

Nonexponential decay laws in perturbation theory of near threshold eigenvalues

Victor Dinu, Arne Jensen, G. Nenciu

Published Jan 1, 2009• 11 citations
Publisher

Abstract

We consider a two channel model of the form Hε=[Hop00E0]+ε[0W12W210] on H=Hop⊕C. The operator Hop is assumed to have the properties of a Schrödinger operator in odd dimensions, with a threshold at zero. As the energy parameter E0 is tuned past the threshold, we consider the survival probability |⟨Ψ0,e−itHεΨ0⟩|2, where Ψ0 is the eigenfunction corresponding to eigenvalue E0 for ε=0. We find nonexponential decay laws for ε small and E0 close to zero provided that the resolvent of Hop is not at least Lipschitz continuous at the threshold zero.

DOI
10.1063/1.3046562

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We consider a two channel model of the form Hε=[Hop00E0]+ε[0W12W210] on H=Hop⊕C. The operator Hop is assumed to have the properties of a Schrödinger operator in odd dimensions, with a threshold at zero. As the energy parameter E0 is tuned past the threshold, we consider the survival probability |⟨Ψ0,e−itHεΨ0⟩|2, where Ψ0 is the eigenfunction corresponding to eigenvalue E0 for ε=0. We find nonexponential decay laws for ε small and E0 close to zero provided that the resolvent of Hop is not at least Lipschitz continuous at the threshold zero.
The most unstable quantum states and elementary particles possess more than a single decay channel. At the same time, it is well known that typically the decay law is not simply exponential. Therefore, it is natural to ask how to spot the nonexponential decay when (at least) two decay channels are opened. In this work, we study the tunneling phenomenon of an initially localized particle in two spatially opposite directions through two different barriers, mimicking two decay channels. In this framework, through specific quantum mechanical examples which can be accurately solved, we study the general properties of a two-channel decay that apply for various unstable quantum states (including unstable particles). Apart from small deviations at early times, the survival probability and the partial tunneling probability along the chosen direction are very well described by the exponential-decay model. In contrast, the ratios of the decay probabilities and probability currents are evidently not a simple constant (as they would be in the exponential limit), but display time-persisting oscillations. Hence, these ratios are optimal witnesses of deviations from the exponential-decay law.
We study resonances and time decay in a quantum waveguide with two semitransparent barriers. The model is first analyzed as a one-dimensional Schrödinger operator with two delta interactions and then extended to a two-dimensional strip, where transverse Dirichlet modes reduce the problem, channel by channel, to the one-dimensional setting. We derive explicit scattering amplitudes and characterize resonance poles by analytic continuation. In the strong-barrier regime ρ=α−1→0+, the real parts of the resonances approach the eigenvalues of the decoupled cavity, while the imaginary parts are negative and of order α−2, so the lifetimes grow like α2. We also study the time evolution on short, intermediate, and long time scales: the survival probability is quadratic at short times, in accordance with the quantum Zeno effect; the intermediate regime is resonance-dominated and approximately exponential; and the long-time behavior is governed by the low-energy threshold and becomes algebraic. Numerical simulations complement the analysis by confirming these regimes and by showing the breakdown of exponential decay. They also provide an empirical fit for the breakdown time tbr in terms of the barrier strength α and the distance d1 between the barriers.
A general formula is presented that expresses the temporal evolution of compound systems. It connects the time dependence of the density matrix with the energy dependence of the scattering matrix. Using results of random matrix theory, we then study the decay behavior of stochastic compound systems with arbitrary coupling between bound states and decay channels. As an example we consider the case of one open channel and prove a nonexponential decay law for all times that asymptotically is of the form ${\mathit{t}}^{\mathrm{\ensuremath{-}}3/2}$.
We examine an analytical expression for the survival probability for the time evolution of quantum decay to discuss a regime where quantum decay is nonexponential at all times. We find that the interference between the exponential and nonexponential terms of the survival amplitude modifies the usual exponential decay regime in systems where the ratio of the resonance energy to the decay width is less than 0.3. We suggest that such a regime could be observed in semiconductor double-barrier resonant quantum structures with appropriate parameters.
In quantum field theory it is believed that the spontaneous decay of excited atomic or molecular level is due to the interaction with continuum of field modes. Besides, the atom makes a transition from upper level to lower one so that the probability to find the atom in the excited state tends to zero. In this paper it will be shown that the mathematical model in single-photon approximation may predict another behavior of this probability generally. Namely, the probability to find the atom in the excited state may tend to a nonzero constant so that the atom is not in the pure state finally. This effect is due to that the spectrum of the complete Hamiltonian is not purely absolutely continuous and has a discrete level outside the continuous part. Namely, we state that in the corresponding invariant subspace, determining the time evolution, the spectrum of the complete Hamiltonian when the field is considered in three dimensions may be not purely absolutely continuous and may have an eigenvalue. The appearance of eigenvalue has a threshold character. If the field is considered in two dimensions the spectrum always has an eigenvalue and the decay is absent.