Heisenberg-type higher order symmetries of superintegrable systems separable in cartesian coordinates

Heisenberg-type higher order symmetries are studied for both classical and quantum mechanical systems separable in cartesian coordinates. A few particular cases of this type of superintegrable systems were already considered in the literature, but here they are characterized in full generality together with their integrability properties. Some of these systems are defined only in a region of $\mathbb R^n$, and in general they do not include bounded solutions. The quantum symmetries and potentials are shown to reduce to their superintegrable classical analogs in the $\hbar \to0$ limit.Comment: 23 Pages, 3 figures, To appear in Nonlinearit

Superintegrability of the Fock-Darwin system

The Fock-Darwin system is analysed from the point of view of its symmetry properties in the quantum and classical frameworks. The quantum Fock-Darwin system is known to have two sets of ladder operators, a fact which guarantees its solvability. We show that for rational values of the quotient of two relevant frequencies, this system is superintegrable, the quantum symmetries being responsible for the degeneracy of the energy levels. These symmetries are of higher order and close a polynomial algebra. In the classical case, the ladder operators are replaced by ladder functions and the symmetries by constants of motion. We also prove that the rational classical system is superintegrable and its trajectories are closed. The constants of motion are also generators of symmetry transformations in the phase space that have been integrated for some special cases. These transformations connect different trajectories with the same energy. The coherent states of the quantum superintegrable system are found and they reproduce the closed trajectories of the classical one.Comment: 21 pages,16 figure

Two families of superintegrable and isospectral potentials in two dimensions

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.Comment: 22 pages, only four typos are correcte

Anatomical and histological analysis of the salpinx and ovary in Anatolian wild goat (Capra aegagrus aegagrus)

Background: This study determined the anatomical and histological structure of the salpinx and ovary of the Anatolian wild goat (C. aegagrus aegagrus). Materials and methods: The salpinges and ovaries from four wild goats of similar ages were used. These goats were brought to the Kafkas University Wildlife Rescue and Rehabilitation Centre (Kars, Turkey) for various reasons such as traffic accidents or firearm injury, but either could not be saved or were euthanased. Results: The mean length of the ovary was 13.22 +/- 1.27 mm, width was 8.46 +/- +/- 0.88 mm, thickness was 5.67 +/- 0.79 mm, and weight was 0.59 +/- 0.17 g. The mean length of the salpinx was 58.11 +/- 14.02 mm, width was 0.80 +/- 0.22 mm, thickness was 0.41 +/- 0.01 mm, and weight was 0.30 +/- 0.08 g. The salpinx consisted of three different regions (infundibulum, ampulla, and isthmus). These regions consisted of tunica mucosa, tunica muscularis, and tunica serosa. Prismatic epithelial cells were observed in the lamina epithelialis of the tunica mucosa in every region. Conclusions: The data resulting from this study regarding anatomical and histological structures of the salpinx and ovary in the genital system organs of Anatolian wild goat will support other studies seeking to improve reproductive efficiency and eliminate the danger of extinction of this species

Anatomical and histological analysis of the salpinx and ovary in Anatolian wild goat (Capra aegagrus aegagrus)

Background: This study determined the anatomical and histological structure of the salpinx and ovary of the Anatolian wild goat (C. aegagrus aegagrus). Materials and methods: The salpinges and ovaries from four wild goats of similar ages were used. These goats were brought to the Kafkas University Wildlife Rescue and Rehabilitation Centre (Kars, Turkey) for various reasons such as traffic accidents or firearm injury, but either could not be saved or were euthanased. Results: The mean length of the ovary was 13.22 ± 1.27 mm, width was 8.46 ± 0.88 mm, thickness was 5.67 ± 0.79 mm, and weight was 0.59 ± 0.17 g. The mean length of the salpinx was 58.11 ± 14.02 mm, width was 0.80 ± 0.22 mm, thickness was 0.41 ± 0.01 mm, and weight was 0.30 ± 0.08 g. The salpinx consisted of three different regions (infundibulum, ampulla, and isthmus). These regions consisted of tunica mucosa, tunica muscularis, and tunica serosa. Prismatic epithelial cells were observed in the lamina epithelialis of the tunica mucosa in every region. Conclusions: The data resulting from this study regarding anatomical and histological structures of the salpinx and ovary in the genital system organs of Anatolian wild goat will support other studies seeking to improve reproductive efficiency and eliminate the danger of extinction of this species