{
"id": "1711.03950",
"version": "v1",
"published": "2017-11-10T18:25:29.000Z",
"updated": "2017-11-10T18:25:29.000Z",
"title": "Perturbation theory for almost-periodic potentials I. One-dimensional case",
"authors": [
"Leonid Parnovski",
"Roman Shterenberg"
],
"comment": "27 pages, 1 figure",
"categories": [
"math-ph",
"math.MP",
"math.SP"
],
"abstract": "We consider the family of operators $H^{(\\epsilon)}:=-\\frac{d^2}{dx^2}+\\epsilon V$ in ${\\mathbb R}$ with almost-periodic potential $V$. We prove that the length of any spectral gap has a complete asymptotic expansion in natural powers of $\\epsilon$ when $\\epsilon\\to 0$. We also study the behaviour of the integrated density of states (IDS) $N(H^{(\\epsilon)};\\lambda)$ when $\\epsilon\\to 0$ and $\\lambda$ is a fixed energy. When $V$ is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each $\\lambda$ the IDS has a complete asymptotic expansion in powers of $\\epsilon$; these powers are either integer, or in some special cases half-integer. We also prove that when the potential is not quasi-periodic, there is an exceptional set $\\mathcal S$ (which we call $\\hbox{the super-resonance set}$) such that for any $\\sqrt\\lambda\\not\\in\\mathcal S$ there is a complete power asymptotic expansion of IDS, and when $\\sqrt\\lambda\\in\\mathcal S$, then even two-terms power asymptotic expansion does not exist. We also show that the super-resonant set $\\mathcal S$ is uncountable, but has measure zero. These results are new even for periodic $V$.",
"revisions": [
{
"version": "v1",
"updated": "2017-11-10T18:25:29.000Z"
}
],
"analyses": {
"subjects": [
"35P20",
"35J10",
"47A55",
"81Q10"
],
"keywords": [
"almost-periodic potential",
"perturbation theory",
"one-dimensional case",
"complete asymptotic expansion",
"two-terms power asymptotic expansion"
],
"note": {
"typesetting": "TeX",
"pages": 27,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}