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Minimal time for the approximate bilinear control of Schrödinger equations
Author(s) -
Beauchard Karine,
Coron JeanMichel,
Teismann Holger
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4710
Subject(s) - controllability , mathematics , unit sphere , bilinear interpolation , quantum , schrödinger equation , mathematical analysis , class (philosophy) , pure mathematics , quantum mechanics , physics , statistics , artificial intelligence , computer science
We consider a quantum particle in a potential V ( x ) subject to a time‐dependent (and spatially homogeneous) electric field E ( t ) (the control). Boscain, Caponigro, Chambrion, and Sigalotti proved that, under generic assumptions on V , this system is approximately controllable on the L 2 ( R N , C ) unit sphere, in sufficiently large time T . In the present article, we show that, for a large class of initial states (dense in L 2 ( R N , C ) unit sphere), approximate controllability does not hold in arbitrarily small time. This generalizes our previous result for Gaussian initial conditions. Furthermore, we prove that the minimal time can in fact be arbitrarily large.