Perturbation of an eigenvalue from a dense point spectrum: a general Floquet Hamiltonian

TL;DR

This paper investigates how a dense point spectrum in a Floquet Hamiltonian is affected by perturbations, showing that under certain conditions, perturbation theory remains valid with modifications despite the spectrum's density.

Contribution

It demonstrates the applicability of perturbation theory to dense spectra in Floquet Hamiltonians with smooth perturbations, introducing conditions on the coupling constant and series asymptotics.

Findings

Perturbation theory applies for almost all frequencies with smooth potentials.

The coupling constant set may be non-interval but still dense at zero.

Rayleigh-Schrodinger series are asymptotic to eigenvalues and eigenvectors.

Abstract

We consider a perturbed Floquet Hamiltonian $-i\partial_t + H + \beta V(\omega t)$ in the Hilbert space $L^2([0,T],E,dt)$. Here $H$ is a self-adjoint operator in $E$ with a discrete spectrum obeying a growing gap condition, $V(t)$ is a symmetric bounded operator in $E$ depending on $t$ $2\pi$-periodically, $\omega = 2\pi/T$ is a frequency and $\beta$ is a coupling constant. The spectrum $Spec(-i\partial_t + H)$ of the unperturbed part is pure point and dense in $R$ for almost every $\omega$. This fact excludes application of the regular perturbation theory. Nevertheless we show, for almost all $\omega$ and provided $V(t)$ is sufficiently smooth, that the perturbation theory still makes sense, however, with two modifications. First, the coupling constant is restricted to a set $I$ which need not be an interval but 0 is still a point of density of $I$. Second, the Rayleigh-Schrodinger series are asymptotic to the perturbed eigen-value and the perturbed eigen-vector.