As an extension of the intertwining operator idea, an algebraic method which provides a link between supersymmetric quantum mechanics and quantum (super)integrability is introduced. By realization of the method in two dimensions, two infinite families of superintegrable and isospectral stationary potentials are generated. The method makes it possible to perform Darboux transformations in such a way that, in addition to the isospectral property, they acquire the superintegrability preserving property. Symmetry generators are second and fourth order in derivatives and all potentials are isospectral with one of the Smorodinsky-Winternitz potentials. Explicit expressions of the potentials, their dynamical symmetry generators and the algebra they obey as well as their degenerate spectra and corresponding normalizable states are presented.

Generation of Wigner functions of Landau levels and determination of their symmetries and generic properties are achieved in the autonomous framework of deformation quantization. Transformation properties of diagonal Wigner functions under space inversion, time reversal and parity transformations are specified and their invariance under a four-parameter subgroup of symplectic transformations are established. A generating function for all Wigner functions is developed and this has been identified as the phase-space coherent state for Landau levels. Integrated forms of generating function are used in generating explicit expressions of marginal probability densities on all possible two dimensional phase-space planes. Phase-space realization of unitary similarity and gauge transformations as well as some general implications for the Wigner function theory are presented.

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We obtain the solutions of two-dimensional singular oscillator which is known as the quantum Calogero–Sutherland model both in cartesian and parabolic coordinates within the framework of quantum Hamilton Jacobi formalism. Solvability conditions and eigenfunctions are obtained by using the singularity structures of quantum momentum functions under some conditions. New potentials are generated by using the first two states of singular oscillator for parabolic coordinates.

A theory of non-unitary-invertible and also unitary canonical transformations is formulated in the context of Weyl's phase space representations. It is shown in the phase space that all quantum canonical transformations without an explicit ℏ dependence are also classical mechanical and vice versa. Contrary to some earlier results, it is also shown that the quantum generators and their classical counterparts are identical in this ℏ-independent universal class. © 2006 Elsevier B.V. All rights reserved