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Reports on Mathematical Physics

Scattering of a Light Particle by a System of Harmonic Oscillators

Domenico Finco, Alessandro Teta

Published Jun 1, 2007• 0 citations
Publisher
DOI
10.1016/s0034-4877(07)80073-5

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The irreversible behavior of simple systems is studied. An example is a system composed of coupled many harmonic oscillators. It is shown that the amplitude of a specified oscillator under appropriate initial condition, and the response function vanish after a long time, as the number of oscillators tends to infinity. Another example is a system of gaseous particle which is subject to collisions due to fixed scattering centers. It is shown that if there is only one scatterer the occupation number for the initial state changes periodically, and if there are many scatterers the occupation number exhibits exponential decay.
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.