2006/11/27 by Ovidiu Costin, O. Costin, Joel L. Lebowitz +5
Mathematics · Physics and Astronomy · #35B40 #35Q40 #47A13 #47A45 #81Q05 #81Q10 #81V45 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.SP #msc:35B40 #msc:35Q40 #msc:47A13 #msc:47A45 #msc:81Q05 #msc:81Q10 #msc:81V45
paper · pdf · doi:10.48550/arxiv.math/0611818
openalex publication_date 2006/11/27 · arxiv created 2010/01/03 · arxiv updated 2010/01/07 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
We analyze the long time behavior of solutions of the Schrödinger equation iψt=(-Δ-b/r+V(t,x))ψ, x∈\RR3, r=|x|, describing a Coulomb system subjected to a spatially compactly supported time periodic potential V(t,x)=V(t+2π/ω,x) with zero time average. We show that, for any V(t,x) of the form 2Ω(r)sin (ωt-θ), with Ω(r) nonzero on its support, Floquet bound states do not exist. This implies that the system ionizes, \em i.e. P(t,K)=∫K|ψ(t,x)|2dx→ 0 as t→∞ for any compact set K⊂\RR3. Furthermore, if the initial state is compactly supported and has only finitely many spherical harmonic modes, then P(t,K) decays like t-5/3 as t → ∞ . To prove these statements, we develop a rigorous WKB theory for infinite systems of ordinary differential equations.