Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:math-ph/0408003 · Journal of Physics A Mathematical and General

Decay of a bound state under a time-periodic perturbation: a toy case

Michele Correggi, Gianfausto Dell’Antonio

Published May 18, 2005• 11 citations
arXivHTMLPDF

Abstract

We study the time evolution of a three-dimensional quantum particle, initially in a bound state, under the action of a time-periodic zero range interaction with ‘strength’ α( t ). Under very weak generic conditions on the Fourier coefficients of α( t ), we prove complete ionization as t → ∞. We prove also that, under the same conditions, all the states of the system are scattering states.

DOI
10.1088/0305-4470/38/22/002

Related papers

Ionization for Three Dimensional Timedependent Point Interactions

Michele Correggi, Centro Linceo Interdisciplinare Roma, Rodolfo Figari +1 more

We study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the “strength ” of the interaction α(t) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of α(t), we prove that there is complete ionization as t → ∞, starting from a bound state at time t = 0. Moreover we prove also that, under the same conditions, all the states of the system are scattering states. 1

•Jan 1, 2004•23 citations

Ionization of a model atom by perturbations of the potential

A. Rokhlenko, Joel L. Lebowitz

We study the time evolution of the wave function of a particle bound by an attractive δ-function potential when it is subjected to time-dependent variations of the binding strength (parametric excitation). The simplicity of this model permits certain nonperturbative calculations to be carried out analytically both in one and three dimensions. Thus the survival probability of bound state |θ(t)|2, following a pulse of strength r and duration t, behaves as |θ(t)|2−|θ(∞)|2∼t−α, with both θ(∞) and α depending on r. On the other hand, a sequence of short pulses produces an exponential decay over an intermediate time scale.

arXiv:math-ph/9905015•Journal of Mathematical Physics•Jun 1, 2000•10 citations

Exact results for ionization of a model atom

A. Rokhlenko, Ovidiu Costin, Joel L. Lebowitz

We investigate the ionization of a 1-D quantum system with a zero-range attractive potential at x = 0 when it is subjected to a time dependent harmonic zero range potential forcing of frequency /spl omega/, strength r, at position x = a. We show that the ionization probability goes to 1 as time t /spl rarr/ /spl infin/ for almost all values of r, /spl omega/, and a. The initially exponential decay of the probability to stay in the bound state is followed at later time by a t/sup -3/ law if r is small and /spl omega/ is not close to resonances. When this is not true the separation between the exponential and power law decay regimes disappears and Fermi's golden rule fails. There exist exceptional sets of parameters where the survival probability never decays to zero and complete ionization is impossible.

•Nov 14, 2002•0 citations

Two-dimensional time-dependent point interactions

Raffaele Carlone, Michele Correggi, Rodolfo Figari

We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schrödinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations – the charge equations – involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions.

arXiv:1601.02390•EMS series of congress reports•Apr 19, 2017•12 citations

Quantum scattering with time-decaying harmonic oscillators

Masaki Kawamoto

By controlling coefficients and decaying order of time-decaying harmonic potentials, the velocity of a quantum particle is decelerated by the effect of harmonic potentials but the particle is non-trapping. In this paper, we consider the quantum system with controlled harmonic potentials. By defining the wave operators for this system and suitable range of these, we can prove the existence and completeness of wave operators with respect to the short-range potentials $V(t,x)$ satisfying $|V(t,x)| = o ((1+|x|)^{-1/(1- λ)})$, for some $λ$, $0< λ< 1/2$.

arXiv:1704.03714•Apr 12, 2017•5 citations

Quantum scattering with time-decaying harmonic potentials

Masaki Kawamoto

By controlling coefficients and decaying order of time-decaying harmonic potentials, the velocity of a quantum particle is decelerated by the effect of harmonic potentials but the particle is non-trapping. Deceleration makes the threshold between the short-range class of potentials and long range class of potentials $1/(1- \lambda)$ for some $0 < \lambda < 1/2$, where $\lambda$ is determined by the mass of particle and coefficients of harmonic potentials. In this paper, we consider the quantum system with controlled harmonic potentials. By defining the wave operators for this system and suitable range of these, we prove the existence and completeness of wave operators with respect to such short-range potentials.

•Apr 12, 2017•0 citations
We study the time evolution of a three-dimensional quantum particle, initially in a bound state, under the action of a time-periodic zero range interaction with ‘strength’ α( t ). Under very weak generic conditions on the Fourier coefficients of α( t ), we prove complete ionization as t → ∞. We prove also that, under the same conditions, all the states of the system are scattering states.
We study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the “strength ” of the interaction α(t) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of α(t), we prove that there is complete ionization as t → ∞, starting from a bound state at time t = 0. Moreover we prove also that, under the same conditions, all the states of the system are scattering states. 1
We study the time evolution of the wave function of a particle bound by an attractive δ-function potential when it is subjected to time-dependent variations of the binding strength (parametric excitation). The simplicity of this model permits certain nonperturbative calculations to be carried out analytically both in one and three dimensions. Thus the survival probability of bound state |θ(t)|2, following a pulse of strength r and duration t, behaves as |θ(t)|2−|θ(∞)|2∼t−α, with both θ(∞) and α depending on r. On the other hand, a sequence of short pulses produces an exponential decay over an intermediate time scale.
We investigate the ionization of a 1-D quantum system with a zero-range attractive potential at x = 0 when it is subjected to a time dependent harmonic zero range potential forcing of frequency /spl omega/, strength r, at position x = a. We show that the ionization probability goes to 1 as time t /spl rarr/ /spl infin/ for almost all values of r, /spl omega/, and a. The initially exponential decay of the probability to stay in the bound state is followed at later time by a t/sup -3/ law if r is small and /spl omega/ is not close to resonances. When this is not true the separation between the exponential and power law decay regimes disappears and Fermi's golden rule fails. There exist exceptional sets of parameters where the survival probability never decays to zero and complete ionization is impossible.
We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schrödinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations – the charge equations – involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions.
By controlling coefficients and decaying order of time-decaying harmonic potentials, the velocity of a quantum particle is decelerated by the effect of harmonic potentials but the particle is non-trapping. In this paper, we consider the quantum system with controlled harmonic potentials. By defining the wave operators for this system and suitable range of these, we can prove the existence and completeness of wave operators with respect to the short-range potentials V(t,x)V(t,x)V(t,x) satisfying ∣V(t,x)∣=o((1+∣x∣)−1/(1−λ))|V(t,x)| = o ((1+|x|)^{-1/(1- λ)})∣V(t,x)∣=o((1+∣x∣)−1/(1−λ)), for some λλλ, 0<λ<1/20< λ< 1/20<λ<1/2.
By controlling coefficients and decaying order of time-decaying harmonic potentials, the velocity of a quantum particle is decelerated by the effect of harmonic potentials but the particle is non-trapping. Deceleration makes the threshold between the short-range class of potentials and long range class of potentials 1/(1−λ)1/(1- \lambda)1/(1−λ) for some 0<λ<1/20 < \lambda < 1/20<λ<1/2, where λ\lambdaλ is determined by the mass of particle and coefficients of harmonic potentials. In this paper, we consider the quantum system with controlled harmonic potentials. By defining the wave operators for this system and suitable range of these, we prove the existence and completeness of wave operators with respect to such short-range potentials.