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arXiv:2409.14080

A remark on the binding condition by the decay of particle's potential

Toshimitsu Takaesu

Published Sep 21, 2024• 0 citations
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Abstract

The system of a particle interacting with a Bose field is investigated. It is proven that the binding condition holds by the decay of particle's potential. As an application, the exponential decay of the ground state follows.
DOI
10.48550/arxiv.2409.14080

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We investigate the survival probability of a localized 1D quantum particle subjected to a time-dependent potential of the form rU ( x ) sin ω t with U ( x ) = 2δ( x − a ) or U ( x ) = 2δ( x − a ) − 2δ( x + a ). The particle is initially in a bound state produced by the binding potential −2δ( x ). We prove that this probability goes to zero as t → ∞ for almost all values of r , ω and a . The decay is initially exponential followed by a t −3 law if ω is not close to resonances and r is small; otherwise the exponential disappears and Fermi's golden rule fails. For exceptional sets of parameters r , ω and a the survival probability never decays to zero, corresponding to the Floquet operator having a bound state. We show similar behaviour even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential.