Obstruction To The Global Control Of The Gross-Pitaevskii Equations With Singular Potentials

In this master thesis,we investigate the obstruction to the global control problem of the following Gross-Pitaevskii equations with singular potentials:i(?)tψ+Hψ=u(t)K(x)ψ-σ|ψ|2ψwhere H=—△+|x|2 is the Hamiltonian of the quantum harmonic oscillator on R3,K(x)is a given singular potential,u(t)is the control function and σ>0.We prove that(ⅰ)When K(x)∈W1,3+W1,∞,the attainable set (?) is a countable union of compact subsets of H1(R3);(ⅱ)When K(x)=1/|x|,the attainable set (?) is a countable union of compact subsets of H1(R3).Therefore,for the above two kinds of potentials,It is impossible to realize the global control in Σ through the potential management.