The linear, differential operators that preserve the N-dimensional vector space whose entries are polynomials of fixed, but arbitrary, degrees in one variable are considered herein. They are relevant for the classification of quasi-exactly solvable systems of equations. Generating elements are explicitly constructed and rules for ordering their products are derived. Some examples of equations that can be reduced to such systems are discussed.
We present a general scheme for constructing the Poisson structure of superintegrable dynamical systems of which the rational Calogero-Moser system is the most interesting one. This dynamical system is 2N dimensional with 2N − 1 first integrals and our construction yields 2N − 1 degenerate Poisson tensors that each admit 2(N − 1) Casimirs. Our results are quite generally applicable to all super-integrable systems and form an alternative to the traditional bi-Hamiltonian approach.
The differential realization of the recently proposed deformed Poincaré algebra is considered. The notion of covariant wave functions is introduced and their explicit form in the "minimal" (in Weinberg's sense) case is given. The deformed Dirac equation is constructed.
Abstract. The differential calculus on n-dimensional quantum Minkowski space covariant with respect to left action of κ-Poincaré group is constructed and its uniqueness is shown.
The superintegrability of so called Tremblay-Turbiner-Winternitz (TTW) model has been conjectured on the basis of the fact that all its trajectories are closed. This conjecture has been proven using the method based on solving the partial differential equations for two functions having the same Poisson bracket with the Hamiltonian. In the present short paper we show that superintegrability of TTW model can be established by using well-known elegant techniques of analytical mechanics. Moreover, the resulting expression ( after an appropriate ordering ) can be generalized to the quantum-mechanical case.
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