Quantum models related to fouled Hamiltonians of the harmonic oscillator

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.

24 pages, no figures, accepted for publication on JMP

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Cited by in corpus (3)

• Canonical quantization of so-called non-Lagrangian systems
• On the action principle for a system of differential equations
• Canonoid and Poissonoid Transformations, Symmetries and BiHamiltonian Structures