Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay

• Published in: Nonlinearity Vol. 30; no. 4; pp. 1405 - 1448
• Main Authors: Wang, Zhiguo, Liang, Zhenguo
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.04.2017
• Subjects: KAM; logarithmic decay; pure-point spectrum; quantum harmonic oscillator; reducibility
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/1361-6544/aa5d6c

In this paper we prove an infinite dimensional KAM theorem, in which the assumptions on the derivatives of the perturbation in [24] are weakened from polynomial decay to logarithmic decay. As a consequence, we can apply it to 1D quantum harmonic oscillators and prove the reducibility of the linear harmonic oscillator, T=−d2dx2+x2, on L2(R) perturbed by the quasi-periodic in the time potential V(x,ωt;ω) with logarithmic decay. This proves the pure-point nature of the spectrum of the Floquet operator K, where K:=−i∑k=1nωk∂∂θk−d2dx2+x2+εV(x,θ;ω) is defined on L2(R)⊗L2(Tn), and the potential V(x,θ;ω) has logarithmic decay as well as its gradient in ω.