Reducibility of relativistic Schrödinger equation with unbounded perturbations

The stability of Sobolev norms for the linear wave equation with unbounded perturbations

“…From Lemma 8.6 in [34], we known that the operator Z is a real Fourier multiplies of order −1. Also, we divided the perturbation K(ωt) into two parts, that is (4.20)…”

“…For the bounded perturbations, we mention the results of Eliasson-Kuksin [19] which proved the reducibility of the Schrödinger equation on T d and Grébert et al [24,25] which proved the reducibility of the quantum harmonic oscillator on R d .The reducibility results imply that the Sobolev norms of solutions for the equation considered is uniformly bounded. For the unbounded perturbations, there are several papers devoting to the reducibility of some Schrödinger equations, such as the quantum harmonic oscillator [4][5][6][7], duffing oscillator [27], relativistic Schrödinger equation on torus [34] and the free schrödinger equation on Zoll manifolds [22,23].…”

Reducibility for a linear wave equation with Sobolev smooth fast driven potential

“…In [22] the authors proved the reducibility of the Schrödinger equation with pseudodifferential perturbations of order less or equal than 1{2 on Zoll manifolds. The reducibility was proved also for the relativistic Schrödinger equation on T with time quasi-periodic unbounded perturbations of order 1{2 [46], for a wave equation with time quasi-periodic perturbations of order 1 [45] and for a linear Schrödinger equation with an almost periodic unbounded pertubation [39]. In the context of nonlinear PDEs, recently the existence of KAM reducible tori was proved for quasilinear perturbations the Degasperis-Procesi equation [24], for water waves equations [5,13,14,17], for semilinear perturbation of the defocusing NLS equation [15] and quasilinear perturbations of the KdV equation [16].…”