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.

Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators / Haus, E.; Maspero, A.. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - 278:2(2020). [10.1016/j.jfa.2019.108316]

Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

Maspero, A.
2020

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.
278
2
108316
https://www.sciencedirect.com/science/article/pii/S0022123619303106?via=ihub
http://ricerca.mat.uniroma3.it/users/ehaus/Anharmonic.pdf
Haus, E.; Maspero, A.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/103466
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