# Reducibility of Schr\"odinger equation on the sphere

## TL;DR

This paper proves a reducibility result for the linear Schr"odinger equation on the sphere with quasi-periodic in time perturbations, including unbounded cases, without using pseudo-differential calculus.

## Contribution

It provides one of the first reducibility results for multi-dimensional Schr"odinger equations with unbounded perturbations, under specific conditions.

## Key findings

- Reduces the Schr"odinger equation to a simpler form
- Handles unbounded perturbations of order less than 1/2
- Does not require pseudo-differential calculus

## Abstract

In this article we prove a reducibility result for the linear Schr\"odinger equation on the sphere $\mathbb{S}^{n}$ with quasi-periodic in time perturbation. Our result includes the case of unbounded perturbation that we assume to be of order strictly less than 1/2 and satisfying some parity condition. As far as we know, this is one of the few reducibility results for an equation in more than one dimension with unbounded perturbations. We notice that our result does not requires the use of the pseudo-differential calculus.