Symplectic Normal Form and Growth of Sobolev Norm

TL;DR

This paper classifies long-term behaviors of solutions to reducible Hamiltonian PDEs across dimensions using symplectic normal forms, revealing new growth rates of Sobolev norms and the uniqueness of stability in one dimension.

Contribution

It introduces a comprehensive classification of Sobolev norm growth for Hamiltonian PDEs via symplectic normal forms, identifying novel growth rates and the special case of stability in one dimension.

Findings

New growth rates for Sobolev norms in quantum harmonic oscillators.

Stability in Sobolev space is unique to one-dimensional cases.

Sobolev norm growth can be linked to classical Hamiltonian solutions.

Abstract

For a class of reducible Hamiltonian partial differential equations (PDEs) with arbitrary spatial dimensions, quantified by a quadratic polynomial with time-dependent coefficients, we present a comprehensive classification of long-term solution behaviors within Sobolev space. This classification is achieved through the utilization of Metaplectic and Schr\"odinger representations. Each pattern of Sobolev norm behavior corresponds to a specific $n-$dimensional symplectic normal form, as detailed in Theorems 1.1 and 1.2. When applied to periodically or quasi-periodically forced $n-$dimensional quantum harmonic oscillators, we identify novel growth rates for the $\mathcal{H}^s-$norm as $t$ tends to infinity, such as $t^{(n-1)s}e^{\lambda st}$ (with $\lambda>0$) and $t^{(2n-1)s}+ \iota t^{2ns}$ (with $\iota\geq 0$). Notably, we demonstrate that stability in Sobolev space, defined as the boundedness of the Sobolev norm, is essentially a unique characteristic of one-dimensional scenarios, as outlined in Theorem 1.3. As a byproduct, we discover that the growth rate of the Sobolev norm for the quantum Hamiltonian can be directly described by that of the solution to the classical Hamiltonian which exhibits the ``fastest" growth, as articulated in Theorem 1.4.