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The concept of extended Hamiltonian systems allows a geometric interpretation of several integrable and superintegrable systems with polynomial first integrals of a degree depending on a rational parameter. Until now, the extension procedure has been applied only in the case of natural Hamiltonians. We give several examples of application to nonnatural Hamiltonians, such as the Hamiltonian of a system of two point-vortices, the Hamiltonian of the Lotka–Volterra model, and some Hamiltonians quartic in the momenta. We effectively obtain extended Hamiltonians in some cases, fail in other cases, and briefly discuss the reasons for these results. [ABSTRACT FROM AUTHOR]

Based on fundamental principles of quantum mechanics, we present a method for a rigorous analytic construction of the spectra of the one- and two-photon Jaynes-Cummings models in a Kerr medium. To obtain an idea of the method, we consider a first generalized Jaynes-Cummings model with a real, linear superpotential using techniques of supersymmetric quantum mechanics. The Hamiltonian of this model is written as a combination of operators generating the underlying superalgebra whose elements are defined as differential matrix operators. Based on the formalism of supersymmetric quantum mechanics and the properties of sets of common observables, we derive solutions of the one- and two-photon models expressed in terms of confluent hypergeometric functions. Finally, using numerical analysis, we study the influence of the Kerr effect on the energy spectra of the physical system. [ABSTRACT FROM AUTHOR]

We develop an analytic formalism using basic quantum mechanics techniques to successfully solve the multiphoton Jaynes–Cummings and the generalized Dicke Hamiltonians. For this, we split the Hamiltonians of these models into two operators that have the properties of constants of motion for these systems. We then use some well-known operator properties to obtain complete analytic spectra for the considered models. [ABSTRACT FROM AUTHOR]