Reducibility of 1-d Schroedinger equation with time quasiperiodic unbounded perturbations, I

Abstract

We study the Schrödinger equation on $\mathbb{R}$ with a polynomial potential behaving as $x^{2l}$ at infinity, $1\le l\in \mathbb{N}$ and with a small time quasiperiodic perturbation. We prove that if the symbol of the perturbation grows at most like $({\xi }^2+x^{2l}{)}^{\beta /(2l)}$, with $\beta <l+1$, then the system is reducible. Some extensions including cases with $\beta =2l$ are also proved. The result implies boundedness of Sobolev norms. The proof is based on pseudodifferential calculus and KAM theory.

References in corpus (2)

• On time dependent Schr{ö}dinger equations: global well-posedness and growth of Sobolev norms
• Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations

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