dorsal/arxiv
View SchemaResummation of anisotropic quartic oscillator. Crossover from anisotropic to isotropic large-order behavior
| Authors | H. Kleinert, S. Thoms |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9605033 |
| URL | https://arxiv.org/abs/quant-ph/9605033 |
| DOI | 10.1103/PhysRevA.55.915 |
Abstract
We present an approximative calculation of the ground-state energy for the anisotropic anharmonic oscillator Using an instanton solution of the isotropic action $\delta = 0$, we obtain the imaginary part of the ground-state energy for small negative $g$ as a series expansion in the anisotropy parameter $\delta$. From this, the large-order behavior of the $g$-expansions accompanying each power of $\delta$ are obtained by means of a dispersion relation in $g$. These $g$-expansions are summed by a Borel transformation, yielding an approximation to the ground-state energy for the region near the isotropic limit. This approximation is found to be excellent in a rather wide region of $\delta$ around $\delta = 0$. Special attention is devoted to the immediate vicinity of the isotropic point. Using a simple model integral we show that the large-order behavior of an $\delta$-dependent series expansion in $g$ undergoes a crossover from an isotropic to an anisotropic regime as the order $k$ of the expansion coefficients passes the value $k_{{\rm cross} \sim 1/ |{\delta}|$.
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"abstract": "We present an approximative calculation of the ground-state energy for the\nanisotropic anharmonic oscillator Using an instanton solution of the isotropic\naction $\\delta = 0$, we obtain the imaginary part of the ground-state energy\nfor small negative $g$ as a series expansion in the anisotropy parameter\n$\\delta$. From this, the large-order behavior of the $g$-expansions\naccompanying each power of $\\delta$ are obtained by means of a dispersion\nrelation in $g$. These $g$-expansions are summed by a Borel transformation,\nyielding an approximation to the ground-state energy for the region near the\nisotropic limit. This approximation is found to be excellent in a rather wide\nregion of $\\delta$ around $\\delta = 0$. Special attention is devoted to the\nimmediate vicinity of the isotropic point. Using a simple model integral we\nshow that the large-order behavior of an $\\delta$-dependent series expansion in\n$g$ undergoes a crossover from an isotropic to an anisotropic regime as the\norder $k$ of the expansion coefficients passes the value $k_{{\\rm cross} \\sim\n1/ |{\\delta}|$.",
"arxiv_id": "quant-ph/9605033",
"authors": [
"H. Kleinert",
"S. Thoms"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.55.915",
"title": "Resummation of anisotropic quartic oscillator. Crossover from anisotropic to isotropic large-order behavior",
"url": "https://arxiv.org/abs/quant-ph/9605033"
},
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