We consider the semiclassical Schrödinger equation on Rd given by iħ∂tψ=(− [Formula presented] Δ+Wl(x))ψ+V(t,x)ψ, where Wl is an anharmonic trapping of the form Wl(x)= [Formula presented] ∑j=1dxj2l, l≥2 is an integer and ħ is a semiclassical small parameter. We construct a smooth potential V(t,x), bounded in time with its derivatives, and an initial datum such that the Sobolev norms of the solution grow at a logarithmic speed for all times of order log [Formula presented] (ħ−1). The proof relies on two ingredients: first we construct an unbounded solution to a forced mechanical anharmonic oscillator, then we exploit semiclassical approximation with coherent states to obtain growth of Sobolev norms for the quantum system which are valid for semiclassical time scales.
Haus, E., Maspero, A. (2020). Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators. JOURNAL OF FUNCTIONAL ANALYSIS, 278(2), 108316 [10.1016/j.jfa.2019.108316].
Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators
Haus E.;Maspero A.
2020-01-01
Abstract
We consider the semiclassical Schrödinger equation on Rd given by iħ∂tψ=(− [Formula presented] Δ+Wl(x))ψ+V(t,x)ψ, where Wl is an anharmonic trapping of the form Wl(x)= [Formula presented] ∑j=1dxj2l, l≥2 is an integer and ħ is a semiclassical small parameter. We construct a smooth potential V(t,x), bounded in time with its derivatives, and an initial datum such that the Sobolev norms of the solution grow at a logarithmic speed for all times of order log [Formula presented] (ħ−1). The proof relies on two ingredients: first we construct an unbounded solution to a forced mechanical anharmonic oscillator, then we exploit semiclassical approximation with coherent states to obtain growth of Sobolev norms for the quantum system which are valid for semiclassical time scales.File | Dimensione | Formato | |
---|---|---|---|
Haus_Maspero_REVISED.pdf
accesso aperto
Descrizione: Articolo principale
Tipologia:
Documento in Post-print
Dimensione
473.32 kB
Formato
Adobe PDF
|
473.32 kB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.