Jacobi fields and the stability of minimal foliations of arbitrary codimension

In this article, we investigate the stability of leaves of minimal foliations of arbitrary codimension. We also study relations between Jacobi fields and vector fields which preserves a foliation and we use these results to Killing fields

Generalized Niederer’s transformation for quantum Pais–Uhlenbeck oscillator

We extend, to the quantum domain, the results obtained in [Nucl. Phys. B 885 (2014) 150] and [Phys. Lett. B 738 (2014) 405] concerning Niederer’s transformation for the Pais–Uhlenbeck oscillator. Namely, the quantum counterpart (an unitary operator) of the transformation which maps the free higher derivatives theory into the Pais–Uhlenbeck oscillator is constructed. Some consequences of this transformation are discussed.The author is grateful to Joanna and Cezary Gonera, Piotr Kosiński and Paweł Maślanka for useful comments and remarks. The research was supported by the grant of National Science Center number DEC-2013/09/B/ST2/02205. Funded by SCOAP3

On dynamical realizations of l-conformal Galilei and Newton-Hooke algebras

In two recent papers [N. Aizawa, Y. Kimura, J. Segar, J. Phys. A 46 (2013) 405204] and [N. Aizawa, Z. Kuznetsova, F. Toppan, J. Math. Phys. 56 (2015) 031701], representation theory of the centrally extended l-conformal Galilei algebra with half-integer l has been applied so as to construct second order differential equations exhibiting the corresponding group as kinematical symmetry. It was suggested to treat them as the Schrodinger equations which involve Hamiltonians describing dynamical systems without higher derivatives. The Hamiltonians possess two unusual features, however. First, they involve the standard kinetic term only for one degree of freedom, while the remaining variables provide contributions linear in momenta. This is typical for Ostrogradsky's canonical approach to the description of higher derivative systems. Second, the Hamiltonian in the second paper is not Hermitian in the conventional sense. In this work, we study the classical limit of the quantum Hamiltonians and demonstrate that the first of them is equivalent to the Hamiltonian describing free higher derivative nonrelativistic particles, while the second can be linked to the Pais-Uhlenbeck oscillator whose frequencies form the arithmetic sequence omega_k=(2k-1), k=1,...,n. We also confront the higher derivative models with a genuine second order system constructed in our recent work [A. Galajinsky, I. Masterov, Nucl. Phys. B 866 (2013) 212] which is discussed in detail for l=3/2.Comment: V2:12 pages,clarifying remarks included into the Introduction and Conclusion, the version to appear in NP