dorsal/arxiv
View SchemaQuantum models related to fouled Hamiltonians of the harmonic oscillator
| Authors | P. Tempesta, E. Alfinito, R. A. Leo, G. Soliani |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0203121 |
| URL | https://arxiv.org/abs/quant-ph/0203121 |
| DOI | 10.1063/1.1479300 |
| Journal | J.Math.Phys. 43 (2002) 3538-3553 |
Abstract
We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator which provide, at the classical level, the same equation of motion as the conventional Hamiltonian. These Hamiltonians, say $K_{1}$ and $K_{2}$, result to be explicitly time-dependent and can be expressed as a formal rotation of two cubic polynomial functions, $H_{1}$ and $H_{2}$, of the canonical variables (q,p). We investigate the role of these fouled Hamiltonians at the quantum level. Adopting a canonical quantization procedure, we construct some quantum models and analyze the related eigenvalue equations. One of these models is described by a Hamiltonian admitting infinite self-adjoint extensions, each of them has a discrete spectrum on the real line. A self-adjoint extension is fixed by choosing the spectral parameter $\epsilon$ of the associated eigenvalue equation equal to zero. The spectral problem is discussed in the context of three different representations. For $\epsilon =0$, the eigenvalue equation is exactly solved in all these representations, in which square-integrable solutions are explicity found. A set of constants of motion corresponding to these quantum models is also obtained. Furthermore, the algebraic structure underlying the quantum models is explored. This turns out to be a nonlinear (quadratic) algebra, which could be applied for the determination of approximate solutions to the eigenvalue equations.
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"abstract": "We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator\nwhich provide, at the classical level, the same equation of motion as the\nconventional Hamiltonian. These Hamiltonians, say $K_{1}$ and $K_{2}$, result\nto be explicitly time-dependent and can be expressed as a formal rotation of\ntwo cubic polynomial functions, $H_{1}$ and $H_{2}$, of the canonical variables\n(q,p).\n We investigate the role of these fouled Hamiltonians at the quantum level.\nAdopting a canonical quantization procedure, we construct some quantum models\nand analyze the related eigenvalue equations. One of these models is described\nby a Hamiltonian admitting infinite self-adjoint extensions, each of them has a\ndiscrete spectrum on the real line. A self-adjoint extension is fixed by\nchoosing the spectral parameter $\\epsilon$ of the associated eigenvalue\nequation equal to zero. The spectral problem is discussed in the context of\nthree different representations. For $\\epsilon =0$, the eigenvalue equation is\nexactly solved in all these representations, in which square-integrable\nsolutions are explicity found. A set of constants of motion corresponding to\nthese quantum models is also obtained. Furthermore, the algebraic structure\nunderlying the quantum models is explored. This turns out to be a nonlinear\n(quadratic) algebra, which could be applied for the determination of\napproximate solutions to the eigenvalue equations.",
"arxiv_id": "quant-ph/0203121",
"authors": [
"P. Tempesta",
"E. Alfinito",
"R. A. Leo",
"G. Soliani"
],
"categories": [
"quant-ph",
"hep-th"
],
"doi": "10.1063/1.1479300",
"journal_ref": "J.Math.Phys. 43 (2002) 3538-3553",
"title": "Quantum models related to fouled Hamiltonians of the harmonic oscillator",
"url": "https://arxiv.org/abs/quant-ph/0203121"
},
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