Reducibility of 1-D Quantum Harmonic Oscillator with Decaying Conditions on the Derivative of Perturbation Potentials

Abstract

We prove the reducibility of 1-D quantum harmonic oscillators in $\mathbb{R}$ perturbed by a quasi-periodic in time potential $V(x,\omega t)$ under the following conditions, namely there is a $C>0$ such that \begin{equation*} |V(x,θ)|\le C,\quad|x\partial_xV(x,θ)|\le C,\quad\forall~(x,θ)\in\mathbb R\times\mathbb T_σ^n. \end{equation*} The corresponding perturbation matrix $(P_i^j(\theta ))$ is proved to satisfy $(1+|i-j|)|P_i^j(\theta )|\le C$ and $\sqrt{ij}|P_{i+1}^{j+1}(\theta )-P_i^j(\theta )|\le C$ for any $\theta \in {\mathbb{T}}_{\sigma }^n$ and $i,j\ge 1$. A new reducibility theorem is set up under this kind of decay in the perturbation matrix element $P_i^j(\theta )$ as well as the discrete difference matrix element $P_{i+1}^{j+1}(\theta )-P_i^j(\theta )$. For the proof the novelty is that we use the decay in the discrete difference matrix element to control the measure estimates for the thrown parameter sets.

References in corpus (2)

• Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations
• Growth of Sobolev Norms in 1-d Quantum Harmonic Oscillator with Polynomial Time Quasi-periodic Perturbation

Cited by in corpus (2)

• Almost reducibility and oscillatory growth of Sobolev norms
• Reducibility of 1-D quantum harmonic oscillator with new unbounded oscillatory perturbations