arXiv quant-ph/02031212002-03-25quant-phhep-th

与谐振子纠缠哈密顿量有关的量子模型

Quantum models related to fouled Hamiltonians of the harmonic oscillator

P. Tempesta, E. Alfinito, R. A. Leo, G. Soliani

备注：: 24 pages, no figures, accepted for publication on JMP
发表信息：: J.Math.Phys. 43 (2002) 3538-3553
DOI：: 10.1063/1.1479300

摘要：

本文研究了谐振子的一对广义哈密顿量，它们在经典水平上提供了与常规哈密顿量相同的运动方程。这些汉密尔顿算子，例如$K_{1}$和$K_{2}$，导致显式地与时间相关，并且可以表示为典范变量（q，p）的两个三次多项式函数$H_{1}$和$H_{2}$的形式旋转。我们调查的作用，这些犯规哈密顿在量子水平。采用正则量子化方法，我们构造了一些量子模型，并分析了相关的本征方程。其中一个模型是用一个容许无限自伴扩张的哈密顿量来描述的，每个自伴扩张在实直线上都有一个离散谱。通过选择相关联的本征值方程的谱参数$ε$等于零来固定自伴扩张。在三种不同的表示的背景下讨论的频谱问题。对于$ε=0$，在所有这些表象下精确求解了本征方程，得到了其平方可积解，并得到了对应于这些量子模型的运动常数.此外，量子模型的代数结构进行了探讨。这原来是一个非线性（二次）代数，它可以应用于确定近似解的特征值方程。

英文摘要：

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 $ε$ of the associated eigenvalue equation equal to zero. The spectral problem is discussed in the context of three different representations. For $ε=0$, 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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摘要：

英文摘要：

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