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 explicitly 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.

Quantum models related to fouled Hamiltonians of the harmonic oscillator

ALFINITO, ELEONORA;
2002

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 explicitly 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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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11587/103661
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