Generalized Reducibility and Growth of Sobolev Norms

TL;DR

This paper introduces generalized reducibility to analyze long-term behavior of quadratic quantum Hamiltonians and constructs perturbations causing Sobolev norms to grow at prescribed sub-exponential rates.

Contribution

It develops a new framework for analyzing quantum Hamiltonians and explicitly constructs perturbations inducing specific Sobolev norm growth rates.

Findings

Established the concept of generalized reducibility.

Constructed perturbations for prescribed Sobolev norm growth.

Demonstrated growth at various sub-exponential rates.

Abstract

We introduce the concept of {\it generalized reducibility}, which provides a flexible framework for analyzing the long-time behavior of solutions to quadratic quantum Hamiltonians. As an application of this notion, for many prescribed sub-exponential growth rates $f(t)$, either monotone or oscillatory, we explicitly construct time-decaying perturbations of the one-dimensional quantum harmonic oscillator such that the Sobolev norms of solutions grow at the rate $f(t)$.