Symplectic normal form and growth of Sobolev norm

Abstract

For a class of reducible Hamiltonian partial differential equations (PDEs) with arbitrary spatial dimension, 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 Schrödinger 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 $H^s-$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 $\iota \ge 0$). 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 optimal growth, as articulated in Theorem 1.4.