dorsal/arxiv
View SchemaPartial sums and optimal shifts in shifted large-L perturbation expansions for quasi-exact potentials
| Authors | Miloslav Znojil |
|---|---|
| Categories | |
| ArXiv ID | physics/0404123 |
| URL | https://arxiv.org/abs/physics/0404123 |
| Journal | Int. J. Pure Appl. Math. 12 (2004) 79-104 |
Abstract
Exact solvability (typically, of harmonic oscillators) in quantum mechanics usually implies an elementary form of the spectrum while in all the "next-to-solvable" models, the energies E are only available in an implicit form (typically, as eigenvalues of an N-dimensional matrix). We demonstrate here that certain echoes of the unattainable harmonic-oscillator ideal may still survive in the latter (often called quasi-exact) cases exemplified here by the popular sextic anharmonic oscillator. In particular we show that whenever the spatial dimension D (or, equivalently, angular momentum L) happens to be "sufficiently" large, the surprisingly compact semi-explicit energies E remain available. In detail, using the Rayleigh-Schr\"{o}dinger perturbation theory in its appropriate "shifted-L" version we observe that: (1) all the k-th order approximants $E_k$ remain defined in integer arithmetics (i.e., without any errors); (2) an optimal auxiliary N-dependent shift $\beta(N)$ of L exists and is unique; (3) the resulting perturbative E degenerates to the series in powers of $1/[L+\beta(N)]^2$; (4) a certain optimal Pade re-summation formulae exist and possess a generic branched-continued-fraction structure.
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"abstract": "Exact solvability (typically, of harmonic oscillators) in quantum mechanics\nusually implies an elementary form of the spectrum while in all the\n\"next-to-solvable\" models, the energies E are only available in an implicit\nform (typically, as eigenvalues of an N-dimensional matrix). We demonstrate\nhere that certain echoes of the unattainable harmonic-oscillator ideal may\nstill survive in the latter (often called quasi-exact) cases exemplified here\nby the popular sextic anharmonic oscillator. In particular we show that\nwhenever the spatial dimension D (or, equivalently, angular momentum L) happens\nto be \"sufficiently\" large, the surprisingly compact semi-explicit energies E\nremain available. In detail, using the Rayleigh-Schr\\\"{o}dinger perturbation\ntheory in its appropriate \"shifted-L\" version we observe that: (1) all the k-th\norder approximants $E_k$ remain defined in integer arithmetics (i.e., without\nany errors); (2) an optimal auxiliary N-dependent shift $\\beta(N)$ of L exists\nand is unique; (3) the resulting perturbative E degenerates to the series in\npowers of $1/[L+\\beta(N)]^2$; (4) a certain optimal Pade re-summation formulae\nexist and possess a generic branched-continued-fraction structure.",
"arxiv_id": "physics/0404123",
"authors": [
"Miloslav Znojil"
],
"categories": [
"physics.comp-ph"
],
"journal_ref": "Int. J. Pure Appl. Math. 12 (2004) 79-104",
"title": "Partial sums and optimal shifts in shifted large-L perturbation expansions for quasi-exact potentials",
"url": "https://arxiv.org/abs/physics/0404123"
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