dorsal/arxiv
View SchemaPerturbation of an eigenvalue from a dense point spectrum: a general Floquet Hamiltonian
| Authors | P. Duclos, P. Stovicek, M. Vittot |
|---|---|
| Categories | |
| ArXiv ID | physics/9712006 |
| URL | https://arxiv.org/abs/physics/9712006 |
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.
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"abstract": "We consider a perturbed Floquet Hamiltonian $-i\\partial_t + H + \\beta\nV(\\omega t)$ in the Hilbert space $L^2([0,T],E,dt)$. Here $H$ is a self-adjoint\noperator in $E$ with a discrete spectrum obeying a growing gap condition,\n$V(t)$ is a symmetric bounded operator in $E$ depending on $t$\n$2\\pi$-periodically, $\\omega = 2\\pi/T$ is a frequency and $\\beta$ is a coupling\nconstant. The spectrum $Spec(-i\\partial_t + H)$ of the unperturbed part is pure\npoint and dense in $R$ for almost every $\\omega$. This fact excludes\napplication of the regular perturbation theory. Nevertheless we show, for\nalmost all $\\omega$ and provided $V(t)$ is sufficiently smooth, that the\nperturbation theory still makes sense, however, with two modifications. First,\nthe coupling constant is restricted to a set $I$ which need not be an interval\nbut 0 is still a point of density of $I$. Second, the Rayleigh-Schrodinger\nseries are asymptotic to the perturbed eigen-value and the perturbed\neigen-vector.",
"arxiv_id": "physics/9712006",
"authors": [
"P. Duclos",
"P. Stovicek",
"M. Vittot"
],
"categories": [
"math-ph",
"math.FA",
"math.MP",
"nlin.CD",
"quant-ph"
],
"title": "Perturbation of an eigenvalue from a dense point spectrum: a general Floquet Hamiltonian",
"url": "https://arxiv.org/abs/physics/9712006"
},
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